Results of Orthogonal Polynomials
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| DLMF | Formula | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|
| 18.1.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{0}{n}@{x} = \frac{2}{n}\ChebyshevpolyT{n}@{x}} | GegenbauerC(n, 0, x)=(2)/(n)*ChebyshevT(n, x) |
GegenbauerC[n, 0, x]=Divide[2,n]*ChebyshevT[n, x] |
Failure | Failure | Successful | Fail
-0.6666666666666666 <- {Rule[n, 3], Rule[x, 1]}-17.333333333333332 <- {Rule[n, 3], Rule[x, 2]}-66.0 <- {Rule[n, 3], Rule[x, 3]} |
| 18.1.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2}{n}\ChebyshevpolyT{n}@{x} = \frac{2(n-1)!}{\Pochhammersym{\tfrac{1}{2}}{n}}\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{x}} | (2)/(n)*ChebyshevT(n, x)=(2*factorial(n - 1))/(pochhammer((1)/(2), n))*JacobiP(n, -(1)/(2), -(1)/(2), x) |
Divide[2,n]*ChebyshevT[n, x]=Divide[2*(n - 1)!,Pochhammer[Divide[1,2], n]]*JacobiP[n, -Divide[1,2], -Divide[1,2], x] |
Successful | Successful | - | - |
| 18.1.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \shiftJacobipolyG{n}@{p}{q}{x} = \frac{n!}{\Pochhammersym{n+p}{n}}\JacobipolyP{p-q}{q-1}{n}@{2x-1}} | JacobiP(n, p-q, q-1, 2*(x)-1)*((n)!)/pochhammer(n+p, n)=(factorial(n))/(pochhammer(n + p, n))*JacobiP(n, p - q, q - 1, 2*x - 1) |
Error |
Successful | Error | - | - |
| 18.2.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\diff{x} = 0} | int(p[n]*(x)* p[m]*(x)* w*(x), x = a..b)= 0 |
Integrate[Subscript[p, n]*(x)* Subscript[p, m]*(x)* w*(x), {x, a, b}]= 0 |
Failure | Failure | Skip | Successful |
| 18.3.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x_{N+1,n} = \cos@{(n-\tfrac{1}{2})\pi/(N+1)}} | x[N + 1 , n]= cos((n -(1)/(2))* Pi/(N + 1)) |
Subscript[x, N + 1 , n]= Cos[(n -Divide[1,2])* Pi/(N + 1)] |
Failure | Failure | Fail .4933988023+1.280284738*I <- {N = 2^(1/2)+I*2^(1/2), x[N+1,n] = 2^(1/2)+I*2^(1/2), n = 1}1.251822237+.4629104109*I <- {N = 2^(1/2)+I*2^(1/2), x[N+1,n] = 2^(1/2)+I*2^(1/2), n = 2}3.059241197+.132349918*I <- {N = 2^(1/2)+I*2^(1/2), x[N+1,n] = 2^(1/2)+I*2^(1/2), n = 3}.4933988023-1.548142386*I <- {N = 2^(1/2)+I*2^(1/2), x[N+1,n] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Successful |
| 18.5.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{n}@{x} = \cos@{n\theta}} | ChebyshevT(n, x)= cos(n*theta) |
ChebyshevT[n, x]= Cos[n*\[Theta]] |
Failure | Failure | Fail .6603260076+1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 1}1.660326008+1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 2}2.660326008+1.911393109*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 3}9.076090394+2.597002114*I <- {theta = 2^(1/2)+I*2^(1/2), n = 2, x = 1}... skip entries to safe data |
Fail
Complex[0.6603260083052754, 1.9113931101642103] <- {Rule[n, 1], Rule[x, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.6603260083052755, 1.9113931101642103] <- {Rule[n, 1], Rule[x, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[2.6603260083052755, 1.9113931101642103] <- {Rule[n, 1], Rule[x, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[9.076090401898771, 2.5970021097090865] <- {Rule[n, 2], Rule[x, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 18.5.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyU{n}@{x} = \ifrac{(\sin@@{(n+1)\theta})}{\sin@@{\theta}}} | ChebyshevU(n, x)=(sin((n + 1)* theta))/(sin(theta)) |
ChebyshevU[n, x]=Divide[Sin[(n + 1)* \[Theta]],Sin[\[Theta]]] |
Failure | Failure | Fail 1.320652015+3.822786219*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 1}3.320652015+3.822786219*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 2}5.320652015+3.822786219*I <- {theta = 2^(1/2)+I*2^(1/2), n = 1, x = 3}18.15218079+5.194004229*I <- {theta = 2^(1/2)+I*2^(1/2), n = 2, x = 1}... skip entries to safe data |
Fail
Complex[1.3206520166105502, 3.82278622032842] <- {Rule[n, 1], Rule[x, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[3.32065201661055, 3.82278622032842] <- {Rule[n, 1], Rule[x, 2], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[5.32065201661055, 3.82278622032842] <- {Rule[n, 1], Rule[x, 3], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[18.152180803797542, 5.194004219418172] <- {Rule[n, 2], Rule[x, 1], Rule[θ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 18.5.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle p_{n}(x) = \frac{1}{\kappa_{n}w(x)}\deriv[n]{}{x}\left(w(x)(F(x))^{n}\right)} | p[n]*(x)=(1)/(kappa[n]*w*(x))*diff(w*(x)*(F*(x))^(n), [x$(n)]) |
Subscript[p, n]*(x)=Divide[1,Subscript[\[Kappa], n]*w*(x)]*D[w*(x)*(F*(x))^(n), {x, n}] |
Failure | Failure | Skip | Skip |
| 18.5.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \JacobipolyP{\alpha}{\beta}{n}@{x} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{\alpha+\ell+1}{n-\ell}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}} | JacobiP(n, alpha, beta, x)= sum((pochhammer(n + alpha + beta + 1, ell)*pochhammer(alpha + ell + 1, n - ell))/(factorial(ell)*factorial(n - ell))*((x - 1)/(2))^(ell), ell = 0..n) |
JacobiP[n, \[Alpha], \[Beta], x]= Sum[Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(Divide[x - 1,2])^(\[ScriptL]), {\[ScriptL], 0, n}] |
Successful | Successful | - | - |
| 18.5.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{n}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{\alpha+\ell+1}{n-\ell}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\genhyperF{2}{1}@@{-n,n+\alpha+\beta+1}{\alpha+1}{\frac{1-x}{2}}} | sum((pochhammer(n + alpha + beta + 1, ell)*pochhammer(alpha + ell + 1, n - ell))/(factorial(ell)*factorial(n - ell))*((x - 1)/(2))^(ell), ell = 0..n)=(pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n , n + alpha + beta + 1], [alpha + 1], (1 - x)/(2)) |
Sum[Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(Divide[x - 1,2])^(\[ScriptL]), {\[ScriptL], 0, n}]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n , n + \[Alpha]+ \[Beta]+ 1}, {\[Alpha]+ 1}, Divide[1 - x,2]] |
Successful | Successful | - | - |
| 18.5.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}} | JacobiP(n, alpha, beta, x)= (2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n) |
JacobiP[n, \[Alpha], \[Beta], x]= (2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^(\[ScriptL]), {\[ScriptL], 0, n}] |
Failure | Failure | Skip | Skip |
| 18.5.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}} | (2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n)=(pochhammer(alpha + 1, n))/(factorial(n))*((x + 1)/(2))^(n)* hypergeom([- n , - n - beta], [alpha + 1], (x - 1)/(x + 1)) |
(2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^(\[ScriptL]), {\[ScriptL], 0, n}]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(Divide[x + 1,2])^(n)* HypergeometricPFQ[{- n , - n - \[Beta]}, {\[Alpha]+ 1}, Divide[x - 1,x + 1]] |
Failure | Failure | Skip | Skip |
| 18.5.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{\lambda}{n}@{x} = \frac{\Pochhammersym{2\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,n+2\lambda}{\lambda+\tfrac{1}{2}}{\frac{1-x}{2}}} | GegenbauerC(n, lambda, x)=(pochhammer(2*lambda, n))/(factorial(n))*hypergeom([- n , n + 2*lambda], [lambda +(1)/(2)], (1 - x)/(2)) |
GegenbauerC[n, \[Lambda], x]=Divide[Pochhammer[2*\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , n + 2*\[Lambda]}, {\[Lambda]+Divide[1,2]}, Divide[1 - x,2]] |
Successful | Successful | - | - |
| 18.5.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{\lambda}{n}@{x} = \sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}} | GegenbauerC(n, lambda, x)= sum(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/ 2)) |
GegenbauerC[n, \[Lambda], x]= Sum[Divide[(- 1)^(\[ScriptL])* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/ 2]}] |
Failure | Successful | Skip | - |
| 18.5.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell} = (2x)^{n}\frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}}{1-\lambda-n}{\frac{1}{x^{2}}}} | sum(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/ 2))=(2*x)^(n)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [1 - lambda - n], (1)/((x)^(2))) |
Sum[Divide[(- 1)^(\[ScriptL])* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/ 2]}]=(2*x)^(n)*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{-Divide[1,2]*n , -Divide[1,2]*n +Divide[1,2]}, {1 - \[Lambda]- n}, Divide[1,(x)^(2)]] |
Failure | Failure | Skip | Successful |
| 18.5.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{\lambda}{n}@{\cos@@{\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-\ell)!}\cos@{(n-2\ell)\theta}} | GegenbauerC(n, lambda, cos(theta))= sum((pochhammer(lambda, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*cos((n - 2*ell)* theta), ell = 0..n) |
GegenbauerC[n, \[Lambda], Cos[\[Theta]]]= Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])* \[Theta]], {\[ScriptL], 0, n}] |
Failure | Failure | Skip | Successful |
| 18.5.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-\ell)!}\cos@{(n-2\ell)\theta} = e^{\iunit n\theta}\frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,\lambda}{1-\lambda-n}{e^{-2\iunit\theta}}} | sum((pochhammer(lambda, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*cos((n - 2*ell)* theta), ell = 0..n)= exp(I*n*theta)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([- n , lambda], [1 - lambda - n], exp(- 2*I*theta)) |
Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])* \[Theta]], {\[ScriptL], 0, n}]= Exp[I*n*\[Theta]]*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , \[Lambda]}, {1 - \[Lambda]- n}, Exp[- 2*I*\[Theta]]] |
Failure | Failure | Skip | Skip |
| 18.5.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{n}\frac{\Pochhammersym{\alpha+\ell+1}{n-\ell}}{(n-\ell)!\;\ell!}(-x)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\genhyperF{1}{1}@@{-n}{\alpha+1}{x}} | sum((pochhammer(alpha + ell + 1, n - ell))/(factorial(n - ell)*factorial(ell))*(- x)^(ell), ell = 0..n)=(pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n], [alpha + 1], x) |
Sum[Divide[Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(n - \[ScriptL])!*(\[ScriptL])!]*(- x)^(\[ScriptL]), {\[ScriptL], 0, n}]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n}, {\[Alpha]+ 1}, x] |
Successful | Successful | - | - |
| 18.5.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{n}@{x} = n!\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}} | HermiteH(n, x)= factorial(n)*sum(((- 1)^(ell)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/ 2)) |
HermiteH[n, x]= (n)!*Sum[Divide[(- 1)^(\[ScriptL])*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/ 2]}] |
Failure | Failure | Skip | Successful |
| 18.5.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n!\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!} = (2x)^{n}\genhyperF{2}{0}@@{-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}}{-}{-\frac{1}{x^{2}}}} | factorial(n)*sum(((- 1)^(ell)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/ 2))=(2*x)^(n)* hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [-], -(1)/((x)^(2))) |
(n)!*Sum[Divide[(- 1)^(\[ScriptL])*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/ 2]}]=(2*x)^(n)* HypergeometricPFQ[{-Divide[1,2]*n , -Divide[1,2]*n +Divide[1,2]}, {-}, -Divide[1,(x)^(2)]] |
Error | Failure | - | Error |
| 18.5#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{0}@{x} = 1} | ChebyshevT(0, x)= 1 |
ChebyshevT[0, x]= 1 |
Successful | Successful | - | - |
| 18.5#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{1}@{x} = x} | ChebyshevT(1, x)= x |
ChebyshevT[1, x]= x |
Successful | Successful | - | - |
| 18.5#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{2}@{x} = 2x^{2}-1} | ChebyshevT(2, x)= 2*(x)^(2)- 1 |
ChebyshevT[2, x]= 2*(x)^(2)- 1 |
Successful | Successful | - | - |
| 18.5#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{3}@{x} = 4x^{3}-3x} | ChebyshevT(3, x)= 4*(x)^(3)- 3*x |
ChebyshevT[3, x]= 4*(x)^(3)- 3*x |
Successful | Successful | - | - |
| 18.5#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{4}@{x} = 8x^{4}-8x^{2}+1} | ChebyshevT(4, x)= 8*(x)^(4)- 8*(x)^(2)+ 1 |
ChebyshevT[4, x]= 8*(x)^(4)- 8*(x)^(2)+ 1 |
Successful | Successful | - | - |
| 18.5#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{5}@{x} = 16x^{5}-20x^{3}+5x} | ChebyshevT(5, x)= 16*(x)^(5)- 20*(x)^(3)+ 5*x |
ChebyshevT[5, x]= 16*(x)^(5)- 20*(x)^(3)+ 5*x |
Successful | Successful | - | - |
| 18.5#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{6}@{x} = 32x^{6}-48x^{4}+18x^{2}-1} | ChebyshevT(6, x)= 32*(x)^(6)- 48*(x)^(4)+ 18*(x)^(2)- 1 |
ChebyshevT[6, x]= 32*(x)^(6)- 48*(x)^(4)+ 18*(x)^(2)- 1 |
Successful | Successful | - | - |
| 18.5#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyU{0}@{x} = 1} | ChebyshevU(0, x)= 1 |
ChebyshevU[0, x]= 1 |
Successful | Successful | - | - |
| 18.5#Ex9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyU{1}@{x} = 2x} | ChebyshevU(1, x)= 2*x |
ChebyshevU[1, x]= 2*x |
Successful | Successful | - | - |
| 18.5#Ex10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyU{2}@{x} = 4x^{2}-1} | ChebyshevU(2, x)= 4*(x)^(2)- 1 |
ChebyshevU[2, x]= 4*(x)^(2)- 1 |
Successful | Successful | - | - |
| 18.5#Ex11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyU{3}@{x} = 8x^{3}-4x} | ChebyshevU(3, x)= 8*(x)^(3)- 4*x |
ChebyshevU[3, x]= 8*(x)^(3)- 4*x |
Successful | Successful | - | - |
| 18.5#Ex12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyU{4}@{x} = 16x^{4}-12x^{2}+1} | ChebyshevU(4, x)= 16*(x)^(4)- 12*(x)^(2)+ 1 |
ChebyshevU[4, x]= 16*(x)^(4)- 12*(x)^(2)+ 1 |
Successful | Successful | - | - |
| 18.5#Ex13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyU{5}@{x} = 32x^{5}-32x^{3}+6x} | ChebyshevU(5, x)= 32*(x)^(5)- 32*(x)^(3)+ 6*x |
ChebyshevU[5, x]= 32*(x)^(5)- 32*(x)^(3)+ 6*x |
Successful | Successful | - | - |
| 18.5#Ex14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyU{6}@{x} = 64x^{6}-80x^{4}+24x^{2}-1} | ChebyshevU(6, x)= 64*(x)^(6)- 80*(x)^(4)+ 24*(x)^(2)- 1 |
ChebyshevU[6, x]= 64*(x)^(6)- 80*(x)^(4)+ 24*(x)^(2)- 1 |
Successful | Successful | - | - |
| 18.5#Ex15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{0}@{x} = 1} | LegendreP(0, x)= 1 |
LegendreP[0, x]= 1 |
Successful | Successful | - | - |
| 18.5#Ex16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{1}@{x} = x} | LegendreP(1, x)= x |
LegendreP[1, x]= x |
Successful | Successful | - | - |
| 18.5#Ex17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{2}@{x} = \tfrac{3}{2}x^{2}-\tfrac{1}{2}} | LegendreP(2, x)=(3)/(2)*(x)^(2)-(1)/(2) |
LegendreP[2, x]=Divide[3,2]*(x)^(2)-Divide[1,2] |
Successful | Successful | - | - |
| 18.5#Ex18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{3}@{x} = \tfrac{5}{2}x^{3}-\tfrac{3}{2}x} | LegendreP(3, x)=(5)/(2)*(x)^(3)-(3)/(2)*x |
LegendreP[3, x]=Divide[5,2]*(x)^(3)-Divide[3,2]*x |
Successful | Successful | - | - |
| 18.5#Ex19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{4}@{x} = \tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8}} | LegendreP(4, x)=(35)/(8)*(x)^(4)-(15)/(4)*(x)^(2)+(3)/(8) |
LegendreP[4, x]=Divide[35,8]*(x)^(4)-Divide[15,4]*(x)^(2)+Divide[3,8] |
Successful | Successful | - | - |
| 18.5#Ex20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{5}@{x} = \tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x} | LegendreP(5, x)=(63)/(8)*(x)^(5)-(35)/(4)*(x)^(3)+(15)/(8)*x |
LegendreP[5, x]=Divide[63,8]*(x)^(5)-Divide[35,4]*(x)^(3)+Divide[15,8]*x |
Successful | Successful | - | - |
| 18.5#Ex21 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{6}@{x} = \tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-\tfrac{5}{16}} | LegendreP(6, x)=(231)/(16)*(x)^(6)-(315)/(16)*(x)^(4)+(105)/(16)*(x)^(2)-(5)/(16) |
LegendreP[6, x]=Divide[231,16]*(x)^(6)-Divide[315,16]*(x)^(4)+Divide[105,16]*(x)^(2)-Divide[5,16] |
Successful | Successful | - | - |
| 18.5#Ex22 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LaguerrepolyL[]{0}@{x} = 1} | LaguerreL(0, x)= 1 |
Error |
Successful | Error | - | - |
| 18.5#Ex23 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LaguerrepolyL[]{1}@{x} = -x+1} | LaguerreL(1, x)= - x + 1 |
Error |
Successful | Error | - | - |
| 18.5#Ex24 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LaguerrepolyL[]{2}@{x} = \tfrac{1}{2}x^{2}-2x+1} | LaguerreL(2, x)=(1)/(2)*(x)^(2)- 2*x + 1 |
Error |
Successful | Error | - | - |
| 18.5#Ex25 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LaguerrepolyL[]{3}@{x} = -\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1} | LaguerreL(3, x)= -(1)/(6)*(x)^(3)+(3)/(2)*(x)^(2)- 3*x + 1 |
Error |
Successful | Error | - | - |
| 18.5#Ex26 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LaguerrepolyL[]{4}@{x} = \tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1} | LaguerreL(4, x)=(1)/(24)*(x)^(4)-(2)/(3)*(x)^(3)+ 3*(x)^(2)- 4*x + 1 |
Error |
Successful | Error | - | - |
| 18.5#Ex27 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LaguerrepolyL[]{5}@{x} = -\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}-5x+1} | LaguerreL(5, x)= -(1)/(120)*(x)^(5)+(5)/(24)*(x)^(4)-(5)/(3)*(x)^(3)+ 5*(x)^(2)- 5*x + 1 |
Error |
Successful | Error | - | - |
| 18.5#Ex28 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LaguerrepolyL[]{6}@{x} = \tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1} | LaguerreL(6, x)=(1)/(720)*(x)^(6)-(1)/(20)*(x)^(5)+(5)/(8)*(x)^(4)-(10)/(3)*(x)^(3)+(15)/(2)*(x)^(2)- 6*x + 1 |
Error |
Successful | Error | - | - |
| 18.5#Ex29 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{0}@{x} = 1} | HermiteH(0, x)= 1 |
HermiteH[0, x]= 1 |
Successful | Successful | - | - |
| 18.5#Ex30 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{1}@{x} = 2x} | HermiteH(1, x)= 2*x |
HermiteH[1, x]= 2*x |
Successful | Successful | - | - |
| 18.5#Ex31 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{2}@{x} = 4x^{2}-2} | HermiteH(2, x)= 4*(x)^(2)- 2 |
HermiteH[2, x]= 4*(x)^(2)- 2 |
Successful | Successful | - | - |
| 18.5#Ex32 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{3}@{x} = 8x^{3}-12x} | HermiteH(3, x)= 8*(x)^(3)- 12*x |
HermiteH[3, x]= 8*(x)^(3)- 12*x |
Successful | Successful | - | - |
| 18.5#Ex33 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{4}@{x} = 16x^{4}-48x^{2}+12} | HermiteH(4, x)= 16*(x)^(4)- 48*(x)^(2)+ 12 |
HermiteH[4, x]= 16*(x)^(4)- 48*(x)^(2)+ 12 |
Successful | Successful | - | - |
| 18.5#Ex34 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{5}@{x} = 32x^{5}-160x^{3}+120x} | HermiteH(5, x)= 32*(x)^(5)- 160*(x)^(3)+ 120*x |
HermiteH[5, x]= 32*(x)^(5)- 160*(x)^(3)+ 120*x |
Successful | Successful | - | - |
| 18.5#Ex35 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{6}@{x} = 64x^{6}-480x^{4}+720x^{2}-120} | HermiteH(6, x)= 64*(x)^(6)- 480*(x)^(4)+ 720*(x)^(2)- 120 |
HermiteH[6, x]= 64*(x)^(6)- 480*(x)^(4)+ 720*(x)^(2)- 120 |
Successful | Successful | - | - |
| 18.6.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{\alpha\to\infty}\frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{1}} = \left(\frac{1+x}{2}\right)^{n}} | limit((JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, 1)), alpha = infinity)=((1 + x)/(2))^(n) |
Limit[Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], 1]], \[Alpha] -> Infinity]=(Divide[1 + x,2])^(n) |
Failure | Failure | Skip | Error |
| 18.6.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{\beta\to\infty}\frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{-1}} = \left(\frac{1-x}{2}\right)^{n}} | limit((JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, - 1)), beta = infinity)=((1 - x)/(2))^(n) |
Limit[Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], - 1]], \[Beta] -> Infinity]=(Divide[1 - x,2])^(n) |
Failure | Failure | Skip | Skip |
| 18.6.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{\lambda\to\infty}\frac{\ultrasphpoly{\lambda}{n}@{x}}{\ultrasphpoly{\lambda}{n}@{1}} = x^{n}} | limit((GegenbauerC(n, lambda, x))/(GegenbauerC(n, lambda, 1)), lambda = infinity)= (x)^(n) |
Limit[Divide[GegenbauerC[n, \[Lambda], x],GegenbauerC[n, \[Lambda], 1]], \[Lambda] -> Infinity]= (x)^(n) |
Failure | Failure | Skip | Error |
| 18.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{\lambda}{n}@{x} = \frac{\Pochhammersym{2\lambda}{n}}{\Pochhammersym{\lambda+\frac{1}{2}}{n}}\JacobipolyP{\lambda-\frac{1}{2}}{\lambda-\frac{1}{2}}{n}@{x}} | GegenbauerC(n, lambda, x)=(pochhammer(2*lambda, n))/(pochhammer(lambda +(1)/(2), n))*JacobiP(n, lambda -(1)/(2), lambda -(1)/(2), x) |
GegenbauerC[n, \[Lambda], x]=Divide[Pochhammer[2*\[Lambda], n],Pochhammer[\[Lambda]+Divide[1,2], n]]*JacobiP[n, \[Lambda]-Divide[1,2], \[Lambda]-Divide[1,2], x] |
Successful | Successful | - | - |
| 18.7.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \JacobipolyP{\alpha}{\alpha}{n}@{x} = \frac{\Pochhammersym{\alpha+1}{n}}{\Pochhammersym{2\alpha+1}{n}}\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{x}} | JacobiP(n, alpha, alpha, x)=(pochhammer(alpha + 1, n))/(pochhammer(2*alpha + 1, n))*GegenbauerC(n, alpha +(1)/(2), x) |
JacobiP[n, \[Alpha], \[Alpha], x]=Divide[Pochhammer[\[Alpha]+ 1, n],Pochhammer[2*\[Alpha]+ 1, n]]*GegenbauerC[n, \[Alpha]+Divide[1,2], x] |
Successful | Successful | - | - |
| 18.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{n}@{x} = \ifrac{\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{x}}{\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{1}}} | ChebyshevT(n, x)=(JacobiP(n, -(1)/(2), -(1)/(2), x))/(JacobiP(n, -(1)/(2), -(1)/(2), 1)) |
ChebyshevT[n, x]=Divide[JacobiP[n, -Divide[1,2], -Divide[1,2], x],JacobiP[n, -Divide[1,2], -Divide[1,2], 1]] |
Successful | Successful | - | - |
| 18.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyU{n}@{x} = \ultrasphpoly{1}{n}@{x}} | ChebyshevU(n, x)= GegenbauerC(n, 1, x) |
ChebyshevU[n, x]= GegenbauerC[n, 1, x] |
Successful | Successful | - | - |
| 18.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{1}{n}@{x} = \ifrac{(n+1)\JacobipolyP{\frac{1}{2}}{\frac{1}{2}}{n}@{x}}{\JacobipolyP{\frac{1}{2}}{\frac{1}{2}}{n}@{1}}} | GegenbauerC(n, 1, x)=((n + 1)* JacobiP(n, (1)/(2), (1)/(2), x))/(JacobiP(n, (1)/(2), (1)/(2), 1)) |
GegenbauerC[n, 1, x]=Divide[(n + 1)* JacobiP[n, Divide[1,2], Divide[1,2], x],JacobiP[n, Divide[1,2], Divide[1,2], 1]] |
Successful | Successful | - | - |
| 18.7.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{n}@{x} = \ultrasphpoly{\frac{1}{2}}{n}@{x}} | LegendreP(n, x)= GegenbauerC(n, (1)/(2), x) |
LegendreP[n, x]= GegenbauerC[n, Divide[1,2], x] |
Successful | Successful | - | - |
| 18.7.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{\frac{1}{2}}{n}@{x} = \JacobipolyP{0}{0}{n}@{x}} | GegenbauerC(n, (1)/(2), x)= JacobiP(n, 0, 0, x) |
GegenbauerC[n, Divide[1,2], x]= JacobiP[n, 0, 0, x] |
Successful | Successful | - | - |
| 18.7.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \shiftLegendrepolyP{n}@{x} = \LegendrepolyP{n}@{2x-1}} | LegendreP(n, 2*(x) - 1)= LegendreP(n, 2*x - 1) |
Error |
Successful | Error | - | - |
| 18.7.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\JacobipolyP{\alpha}{\alpha}{2n}@{x}}{\JacobipolyP{\alpha}{\alpha}{2n}@{1}} = \frac{\JacobipolyP{\alpha}{-\frac{1}{2}}{n}@{2x^{2}-1}}{\JacobipolyP{\alpha}{-\frac{1}{2}}{n}@{1}}} | (JacobiP(2*n, alpha, alpha, x))/(JacobiP(2*n, alpha, alpha, 1))=(JacobiP(n, alpha, -(1)/(2), 2*(x)^(2)- 1))/(JacobiP(n, alpha, -(1)/(2), 1)) |
Divide[JacobiP[2*n, \[Alpha], \[Alpha], x],JacobiP[2*n, \[Alpha], \[Alpha], 1]]=Divide[JacobiP[n, \[Alpha], -Divide[1,2], 2*(x)^(2)- 1],JacobiP[n, \[Alpha], -Divide[1,2], 1]] |
Failure | Failure | Successful | Successful |
| 18.7.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\JacobipolyP{\alpha}{\alpha}{2n+1}@{x}}{\JacobipolyP{\alpha}{\alpha}{2n+1}@{1}} = \frac{x\JacobipolyP{\alpha}{\frac{1}{2}}{n}@{2x^{2}-1}}{\JacobipolyP{\alpha}{\frac{1}{2}}{n}@{1}}} | (JacobiP(2*n + 1, alpha, alpha, x))/(JacobiP(2*n + 1, alpha, alpha, 1))=(x*JacobiP(n, alpha, (1)/(2), 2*(x)^(2)- 1))/(JacobiP(n, alpha, (1)/(2), 1)) |
Divide[JacobiP[2*n + 1, \[Alpha], \[Alpha], x],JacobiP[2*n + 1, \[Alpha], \[Alpha], 1]]=Divide[x*JacobiP[n, \[Alpha], Divide[1,2], 2*(x)^(2)- 1],JacobiP[n, \[Alpha], Divide[1,2], 1]] |
Failure | Failure | Successful | Successful |
| 18.7.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{\lambda}{2n}@{x} = \frac{\Pochhammersym{\lambda}{n}}{\Pochhammersym{\tfrac{1}{2}}{n}}\JacobipolyP{\lambda-\frac{1}{2}}{-\frac{1}{2}}{n}@{2x^{2}-1}} | GegenbauerC(2*n, lambda, x)=(pochhammer(lambda, n))/(pochhammer((1)/(2), n))*JacobiP(n, lambda -(1)/(2), -(1)/(2), 2*(x)^(2)- 1) |
GegenbauerC[2*n, \[Lambda], x]=Divide[Pochhammer[\[Lambda], n],Pochhammer[Divide[1,2], n]]*JacobiP[n, \[Lambda]-Divide[1,2], -Divide[1,2], 2*(x)^(2)- 1] |
Failure | Failure | Successful | Successful |
| 18.7.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{\lambda}{2n+1}@{x} = \frac{\Pochhammersym{\lambda}{n+1}}{\Pochhammersym{\frac{1}{2}}{n+1}}x\JacobipolyP{\lambda-\frac{1}{2}}{\frac{1}{2}}{n}@{2x^{2}-1}} | GegenbauerC(2*n + 1, lambda, x)=(pochhammer(lambda, n + 1))/(pochhammer((1)/(2), n + 1))*x*JacobiP(n, lambda -(1)/(2), (1)/(2), 2*(x)^(2)- 1) |
GegenbauerC[2*n + 1, \[Lambda], x]=Divide[Pochhammer[\[Lambda], n + 1],Pochhammer[Divide[1,2], n + 1]]*x*JacobiP[n, \[Lambda]-Divide[1,2], Divide[1,2], 2*(x)^(2)- 1] |
Failure | Failure | Successful | Successful |
| 18.7.E23 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}\JacobipolyP{\alpha}{\alpha}{n}@{\alpha^{-\frac{1}{2}}x} = \frac{\HermitepolyH{n}@{x}}{2^{n}n!}} | limit((alpha)^(-(1)/(2)*n)* JacobiP(n, alpha, alpha, (alpha)^(-(1)/(2))* x), alpha = infinity)=(HermiteH(n, x))/((2)^(n)* factorial(n)) |
Limit[(\[Alpha])^(-Divide[1,2]*n)* JacobiP[n, \[Alpha], \[Alpha], (\[Alpha])^(-Divide[1,2])* x], \[Alpha] -> Infinity]=Divide[HermiteH[n, x],(2)^(n)* (n)!] |
Failure | Failure | Skip | Error |
| 18.7.E24 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}\ultrasphpoly{\lambda}{n}@{\lambda^{-\frac{1}{2}}x} = \frac{\HermitepolyH{n}@{x}}{n!}} | limit((lambda)^(-(1)/(2)*n)* GegenbauerC(n, lambda, (lambda)^(-(1)/(2))* x), lambda = infinity)=(HermiteH(n, x))/(factorial(n)) |
Limit[(\[Lambda])^(-Divide[1,2]*n)* GegenbauerC[n, \[Lambda], (\[Lambda])^(-Divide[1,2])* x], \[Lambda] -> Infinity]=Divide[HermiteH[n, x],(n)!] |
Failure | Failure | Skip | Skip |
| 18.7.E25 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{\lambda\to 0}\frac{1}{\lambda}\ultrasphpoly{\lambda}{n}@{x} = \frac{2}{n}\ChebyshevpolyT{n}@{x}} | limit((1)/(lambda)*GegenbauerC(n, lambda, x), lambda = 0)=(2)/(n)*ChebyshevT(n, x) |
Limit[Divide[1,\[Lambda]]*GegenbauerC[n, \[Lambda], x], \[Lambda] -> 0]=Divide[2,n]*ChebyshevT[n, x] |
Failure | Failure | Skip | Successful |
| 18.9.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \JacobipolyP{\alpha}{\beta-1}{n}@{x}-\JacobipolyP{\alpha-1}{\beta}{n}@{x} = \JacobipolyP{\alpha}{\beta}{n-1}@{x}} | JacobiP(n, alpha, beta - 1, x)- JacobiP(n, alpha - 1, beta, x)= JacobiP(n - 1, alpha, beta, x) |
JacobiP[n, \[Alpha], \[Beta]- 1, x]- JacobiP[n, \[Alpha]- 1, \[Beta], x]= JacobiP[n - 1, \[Alpha], \[Beta], x] |
Failure | Successful | Successful | - |
| 18.9.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (1-x)\JacobipolyP{\alpha+1}{\beta}{n}@{x}+(1+x)\JacobipolyP{\alpha}{\beta+1}{n}@{x} = 2\!\JacobipolyP{\alpha}{\beta}{n}@{x}} | (1 - x)* JacobiP(n, alpha + 1, beta, x)+(1 + x)* JacobiP(n, alpha, beta + 1, x)= 2*JacobiP(n, alpha, beta, x) |
(1 - x)* JacobiP[n, \[Alpha]+ 1, \[Beta], x]+(1 + x)* JacobiP[n, \[Alpha], \[Beta]+ 1, x]= 2*JacobiP[n, \[Alpha], \[Beta], x] |
Failure | Successful | Successful | - |
| 18.9.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (2n+\alpha+\beta+1)\JacobipolyP{\alpha}{\beta}{n}@{x} = (n+\alpha+\beta+1)\JacobipolyP{\alpha}{\beta+1}{n}@{x}+(n+\alpha)\JacobipolyP{\alpha}{\beta+1}{n-1}@{x}} | (2*n + alpha + beta + 1)* JacobiP(n, alpha, beta, x)=(n + alpha + beta + 1)* JacobiP(n, alpha, beta + 1, x)+(n + alpha)* JacobiP(n - 1, alpha, beta + 1, x) |
(2*n + \[Alpha]+ \[Beta]+ 1)* JacobiP[n, \[Alpha], \[Beta], x]=(n + \[Alpha]+ \[Beta]+ 1)* JacobiP[n, \[Alpha], \[Beta]+ 1, x]+(n + \[Alpha])* JacobiP[n - 1, \[Alpha], \[Beta]+ 1, x] |
Failure | Successful | Successful | - |
| 18.9.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)\JacobipolyP{\alpha}{\beta+1}{n}@{x} = (n+1)\JacobipolyP{\alpha}{\beta}{n+1}@{x}+(n+\beta+1)\JacobipolyP{\alpha}{\beta}{n}@{x}} | (n +(1)/(2)*alpha +(1)/(2)*beta + 1)*(1 + x)* JacobiP(n, alpha, beta + 1, x)=(n + 1)* JacobiP(n + 1, alpha, beta, x)+(n + beta + 1)* JacobiP(n, alpha, beta, x) |
(n +Divide[1,2]*\[Alpha]+Divide[1,2]*\[Beta]+ 1)*(1 + x)* JacobiP[n, \[Alpha], \[Beta]+ 1, x]=(n + 1)* JacobiP[n + 1, \[Alpha], \[Beta], x]+(n + \[Beta]+ 1)* JacobiP[n, \[Alpha], \[Beta], x] |
Failure | Successful | Successful | - |
| 18.9.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (n+\lambda)\ultrasphpoly{\lambda}{n}@{x} = \lambda\left(\ultrasphpoly{\lambda+1}{n}@{x}-\ultrasphpoly{\lambda+1}{n-2}@{x}\right)} | (n + lambda)* GegenbauerC(n, lambda, x)= lambda*(GegenbauerC(n, lambda + 1, x)- GegenbauerC(n - 2, lambda + 1, x)) |
(n + \[Lambda])* GegenbauerC[n, \[Lambda], x]= \[Lambda]*(GegenbauerC[n, \[Lambda]+ 1, x]- GegenbauerC[n - 2, \[Lambda]+ 1, x]) |
Successful | Successful | - | - |
| 18.9.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 4\lambda(n+\lambda+1)(1-x^{2})\ultrasphpoly{\lambda+1}{n}@{x} = -(n+1)(n+2)\ultrasphpoly{\lambda}{n+2}@{x}+(n+2\lambda)(n+2\lambda+1)\ultrasphpoly{\lambda}{n}@{x}} | 4*lambda*(n + lambda + 1)*(1 - (x)^(2))* GegenbauerC(n, lambda + 1, x)= -(n + 1)*(n + 2)* GegenbauerC(n + 2, lambda, x)+(n + 2*lambda)*(n + 2*lambda + 1)* GegenbauerC(n, lambda, x) |
4*\[Lambda]*(n + \[Lambda]+ 1)*(1 - (x)^(2))* GegenbauerC[n, \[Lambda]+ 1, x]= -(n + 1)*(n + 2)* GegenbauerC[n + 2, \[Lambda], x]+(n + 2*\[Lambda])*(n + 2*\[Lambda]+ 1)* GegenbauerC[n, \[Lambda], x] |
Successful | Successful | - | - |
| 18.9.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{n}@{x} = \tfrac{1}{2}\left(\ChebyshevpolyU{n}@{x}-\ChebyshevpolyU{n-2}@{x}\right)} | ChebyshevT(n, x)=(1)/(2)*(ChebyshevU(n, x)- ChebyshevU(n - 2, x)) |
ChebyshevT[n, x]=Divide[1,2]*(ChebyshevU[n, x]- ChebyshevU[n - 2, x]) |
Successful | Failure | - | Successful |
| 18.9.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (1-x^{2})\ChebyshevpolyU{n}@{x} = -\tfrac{1}{2}\left(\ChebyshevpolyT{n+2}@{x}-\ChebyshevpolyT{n}@{x}\right)} | (1 - (x)^(2))* ChebyshevU(n, x)= -(1)/(2)*(ChebyshevT(n + 2, x)- ChebyshevT(n, x)) |
(1 - (x)^(2))* ChebyshevU[n, x]= -Divide[1,2]*(ChebyshevT[n + 2, x]- ChebyshevT[n, x]) |
Successful | Failure | - | Successful |
| 18.9.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\JacobipolyP{\alpha}{\beta}{n}@{x} = \tfrac{1}{2}(n+\alpha+\beta+1)\JacobipolyP{\alpha+1}{\beta+1}{n-1}@{x}} | diff(JacobiP(n, alpha, beta, x), x)=(1)/(2)*(n + alpha + beta + 1)* JacobiP(n - 1, alpha + 1, beta + 1, x) |
D[JacobiP[n, \[Alpha], \[Beta], x], x]=Divide[1,2]*(n + \[Alpha]+ \[Beta]+ 1)* JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, x] |
Failure | Successful | Fail Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 1, x = 1}Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 2, x = 1}Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 3, x = 1}Float(infinity)+Float(infinity)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)-I*2^(1/2), n = 1, x = 1}... skip entries to safe data |
- |
| 18.9.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\left((1-x)^{\alpha}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}\right) = -2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}\JacobipolyP{\alpha-1}{\beta-1}{n+1}@{x}} | diff((1 - x)^(alpha)*(1 + x)^(beta)* JacobiP(n, alpha, beta, x), x)= - 2*(n + 1)*(1 - x)^(alpha - 1)*(1 + x)^(beta - 1)* JacobiP(n + 1, alpha - 1, beta - 1, x) |
D[(1 - x)^(\[Alpha])*(1 + x)^(\[Beta])* JacobiP[n, \[Alpha], \[Beta], x], x]= - 2*(n + 1)*(1 - x)^(\[Alpha]- 1)*(1 + x)^(\[Beta]- 1)* JacobiP[n + 1, \[Alpha]- 1, \[Beta]- 1, x] |
Failure | Successful | Fail Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 1, x = 1}Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 2, x = 1}Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), n = 3, x = 1}Float(undefined)+Float(undefined)*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)-I*2^(1/2), n = 1, x = 1}... skip entries to safe data |
- |
| 18.9.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (2n+\alpha+\beta)(1-x^{2})\deriv{}{x}\JacobipolyP{\alpha}{\beta}{n}@{x} = n\left(\alpha-\beta-(2n+\alpha+\beta)x\right)\JacobipolyP{\alpha}{\beta}{n}@{x}+2(n+\alpha)(n+\beta)\JacobipolyP{\alpha}{\beta}{n-1}@{x}} | (2*n + alpha + beta)*(1 - (x)^(2))* diff(JacobiP(n, alpha, beta, x), x)= n*(alpha - beta -(2*n + alpha + beta)*x)* JacobiP(n, alpha, beta, x)+ 2*(n + alpha)*(n + beta)* JacobiP(n - 1, alpha, beta, x) |
(2*n + \[Alpha]+ \[Beta])*(1 - (x)^(2))* D[JacobiP[n, \[Alpha], \[Beta], x], x]= n*(\[Alpha]- \[Beta]-(2*n + \[Alpha]+ \[Beta])*x)* JacobiP[n, \[Alpha], \[Beta], x]+ 2*(n + \[Alpha])*(n + \[Beta])* JacobiP[n - 1, \[Alpha], \[Beta], x] |
Failure | Successful | Successful | - |
| 18.9.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (2n+\alpha+\beta+2)(1-x^{2})\deriv{}{x}\JacobipolyP{\alpha}{\beta}{n}@{x} = (n+\alpha+\beta+1)\left(\alpha-\beta+(2n+\alpha+\beta+2)x\right)\JacobipolyP{\alpha}{\beta}{n}@{x}-2(n+1)(n+\alpha+\beta+1)\JacobipolyP{\alpha}{\beta}{n+1}@{x}} | (2*n + alpha + beta + 2)*(1 - (x)^(2))* diff(JacobiP(n, alpha, beta, x), x)=(n + alpha + beta + 1)*(alpha - beta +(2*n + alpha + beta + 2)*x)* JacobiP(n, alpha, beta, x)- 2*(n + 1)*(n + alpha + beta + 1)* JacobiP(n + 1, alpha, beta, x) |
(2*n + \[Alpha]+ \[Beta]+ 2)*(1 - (x)^(2))* D[JacobiP[n, \[Alpha], \[Beta], x], x]=(n + \[Alpha]+ \[Beta]+ 1)*(\[Alpha]- \[Beta]+(2*n + \[Alpha]+ \[Beta]+ 2)*x)* JacobiP[n, \[Alpha], \[Beta], x]- 2*(n + 1)*(n + \[Alpha]+ \[Beta]+ 1)* JacobiP[n + 1, \[Alpha], \[Beta], x] |
Failure | Successful | Successful | - |
| 18.9.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\ultrasphpoly{\lambda}{n}@{x} = 2\lambda\ultrasphpoly{\lambda+1}{n-1}@{x}} | diff(GegenbauerC(n, lambda, x), x)= 2*lambda*GegenbauerC(n - 1, lambda + 1, x) |
D[GegenbauerC[n, \[Lambda], x], x]= 2*\[Lambda]*GegenbauerC[n - 1, \[Lambda]+ 1, x] |
Successful | Successful | - | - |
| 18.9.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\left((1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{n}@{x}\right) = -\frac{(n+1)(n+2\lambda-1)}{2(\lambda-1)}{(1-x^{2})^{\lambda-\frac{3}{2}}}\ultrasphpoly{\lambda-1}{n+1}@{x}} | diff((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(n, lambda, x), x)= -((n + 1)*(n + 2*lambda - 1))/(2*(lambda - 1))*(1 - (x)^(2))^(lambda -(3)/(2))*GegenbauerC(n + 1, lambda - 1, x) |
D[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[n, \[Lambda], x], x]= -Divide[(n + 1)*(n + 2*\[Lambda]- 1),2*(\[Lambda]- 1)]*(1 - (x)^(2))^(\[Lambda]-Divide[3,2])*GegenbauerC[n + 1, \[Lambda]- 1, x] |
Failure | Successful | Fail Float(infinity)+Float(infinity)*I <- {lambda = 2^(1/2)+I*2^(1/2), n = 1, x = 1}Float(infinity)+Float(infinity)*I <- {lambda = 2^(1/2)+I*2^(1/2), n = 2, x = 1}Float(infinity)+Float(infinity)*I <- {lambda = 2^(1/2)+I*2^(1/2), n = 3, x = 1}Float(infinity)+Float(infinity)*I <- {lambda = 2^(1/2)-I*2^(1/2), n = 1, x = 1}... skip entries to safe data |
- |
| 18.9.E21 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\ChebyshevpolyT{n}@{x} = n\ChebyshevpolyU{n-1}@{x}} | diff(ChebyshevT(n, x), x)= n*ChebyshevU(n - 1, x) |
D[ChebyshevT[n, x], x]= n*ChebyshevU[n - 1, x] |
Successful | Successful | - | - |
| 18.9.E22 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\left((1-x^{2})^{\frac{1}{2}}\ChebyshevpolyU{n}@{x}\right) = -(n+1){(1-x^{2})^{-\frac{1}{2}}}\ChebyshevpolyT{n+1}@{x}} | diff((1 - (x)^(2))^((1)/(2))* ChebyshevU(n, x), x)= -(n + 1)*(1 - (x)^(2))^(-(1)/(2))*ChebyshevT(n + 1, x) |
D[(1 - (x)^(2))^(Divide[1,2])* ChebyshevU[n, x], x]= -(n + 1)*(1 - (x)^(2))^(-Divide[1,2])*ChebyshevT[n + 1, x] |
Successful | Successful | - | - |
| 18.9.E25 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\HermitepolyH{n}@{x} = 2n\HermitepolyH{n-1}@{x}} | diff(HermiteH(n, x), x)= 2*n*HermiteH(n - 1, x) |
D[HermiteH[n, x], x]= 2*n*HermiteH[n - 1, x] |
Successful | Successful | - | - |
| 18.9.E26 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\left(e^{-x^{2}}\HermitepolyH{n}@{x}\right) = -e^{-x^{2}}\HermitepolyH{n+1}@{x}} | diff(exp(- (x)^(2))*HermiteH(n, x), x)= - exp(- (x)^(2))*HermiteH(n + 1, x) |
D[Exp[- (x)^(2)]*HermiteH[n, x], x]= - Exp[- (x)^(2)]*HermiteH[n + 1, x] |
Successful | Successful | - | - |
| 18.10.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\JacobipolyP{\alpha}{\alpha}{n}@{\cos@@{\theta}}}{\JacobipolyP{\alpha}{\alpha}{n}@{1}} = \frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}}} | (JacobiP(n, alpha, alpha, cos(theta)))/(JacobiP(n, alpha, alpha, 1))=(GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1)) |
Divide[JacobiP[n, \[Alpha], \[Alpha], Cos[\[Theta]]],JacobiP[n, \[Alpha], \[Alpha], 1]]=Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]] |
Successful | Successful | - | - |
| 18.10.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}} = \frac{2^{\alpha+\frac{1}{2}}\EulerGamma@{\alpha+1}}{\pi^{\frac{1}{2}}\EulerGamma@{\alpha+\frac{1}{2}}}(\sin@@{\theta})^{-2\alpha}\int_{0}^{\theta}\frac{\cos@{(n+\alpha+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{-\alpha+\frac{1}{2}}}\diff{\phi}} | (GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1))=((2)^(alpha +(1)/(2))* GAMMA(alpha + 1))/((Pi)^((1)/(2))* GAMMA(alpha +(1)/(2)))*(sin(theta))^(- 2*alpha)* int((cos((n + alpha +(1)/(2))* phi))/((cos(phi)- cos(theta))^(- alpha +(1)/(2))), phi = 0..theta) |
Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]]=Divide[(2)^(\[Alpha]+Divide[1,2])* Gamma[\[Alpha]+ 1],(Pi)^(Divide[1,2])* Gamma[\[Alpha]+Divide[1,2]]]*(Sin[\[Theta]])^(- 2*\[Alpha])* Integrate[Divide[Cos[(n + \[Alpha]+Divide[1,2])* \[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(- \[Alpha]+Divide[1,2])], {\[Phi], 0, \[Theta]}] |
Failure | Failure | Skip | Successful |
| 18.10.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{n}@{\cos@@{\theta}} = \frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac{\cos@{(n+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{\frac{1}{2}}}\diff{\phi}} | LegendreP(n, cos(theta))=((2)^((1)/(2)))/(Pi)*int((cos((n +(1)/(2))* phi))/((cos(phi)- cos(theta))^((1)/(2))), phi = 0..theta) |
LegendreP[n, Cos[\[Theta]]]=Divide[(2)^(Divide[1,2]),Pi]*Integrate[Divide[Cos[(n +Divide[1,2])* \[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(Divide[1,2])], {\[Phi], 0, \[Theta]}] |
Failure | Failure | Skip | Successful |
| 18.10.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{n}@{\cos@@{\theta}} = \frac{1}{\pi}\int_{0}^{\pi}(\cos@@{\theta}+i\sin@@{\theta}\cos@@{\phi})^{n}\diff{\phi}} | LegendreP(n, cos(theta))=(1)/(Pi)*int((cos(theta)+ I*sin(theta)*cos(phi))^(n), phi = 0..Pi) |
LegendreP[n, Cos[\[Theta]]]=Divide[1,Pi]*Integrate[(Cos[\[Theta]]+ I*Sin[\[Theta]]*Cos[\[Phi]])^(n), {\[Phi], 0, Pi}] |
Failure | Failure | Skip | Error |
| 18.10.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{n}@{x} = \frac{2^{n}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}(x+it)^{n}e^{-t^{2}}\diff{t}} | HermiteH(n, x)=((2)^(n))/((Pi)^((1)/(2)))*int((x + I*t)^(n)* exp(- (t)^(2)), t = - infinity..infinity) |
HermiteH[n, x]=Divide[(2)^(n),(Pi)^(Divide[1,2])]*Integrate[(x + I*t)^(n)* Exp[- (t)^(2)], {t, - Infinity, Infinity}] |
Failure | Failure | Skip | Skip |
| 18.10.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{n}@{x} = \frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t}} | HermiteH(n, x)=((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity) |
HermiteH[n, x]=Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}] |
Failure | Failure | Skip | Successful |
| 18.10.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t} = \frac{2^{n+1}}{\pi^{\frac{1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos@{2xt-\tfrac{1}{2}n\pi}\diff{t}} | ((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity)=((2)^(n + 1))/((Pi)^((1)/(2)))*exp((x)^(2))*int(exp(- (t)^(2))*(t)^(n)* cos(2*x*t -(1)/(2)*n*Pi), t = 0..infinity) |
Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}]=Divide[(2)^(n + 1),(Pi)^(Divide[1,2])]*Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Cos[2*x*t -Divide[1,2]*n*Pi], {t, 0, Infinity}] |
Successful | Failure | - | Error |
| 18.11.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \FerrersP[m]{n}@{x} = \Pochhammersym{\tfrac{1}{2}}{m}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}\ultrasphpoly{m+\frac{1}{2}}{n-m}@{x}} | LegendreP(n, m, x)= pochhammer((1)/(2), m)*(- 2)^(m)*(1 - (x)^(2))^((1)/(2)*m)* GegenbauerC(n - m, m +(1)/(2), x) |
LegendreP[n, m, x]= Pochhammer[Divide[1,2], m]*(- 2)^(m)*(1 - (x)^(2))^(Divide[1,2]*m)* GegenbauerC[n - m, m +Divide[1,2], x] |
Failure | Failure | Skip | Successful |
| 18.11.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Pochhammersym{\tfrac{1}{2}}{m}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}\ultrasphpoly{m+\frac{1}{2}}{n-m}@{x} = \Pochhammersym{n+1}{m}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}\JacobipolyP{m}{m}{n-m}@{x}} | pochhammer((1)/(2), m)*(- 2)^(m)*(1 - (x)^(2))^((1)/(2)*m)* GegenbauerC(n - m, m +(1)/(2), x)= pochhammer(n + 1, m)*(- 2)^(- m)*(1 - (x)^(2))^((1)/(2)*m)* JacobiP(n - m, m, m, x) |
Pochhammer[Divide[1,2], m]*(- 2)^(m)*(1 - (x)^(2))^(Divide[1,2]*m)* GegenbauerC[n - m, m +Divide[1,2], x]= Pochhammer[n + 1, m]*(- 2)^(- m)*(1 - (x)^(2))^(Divide[1,2]*m)* JacobiP[n - m, m, m, x] |
Failure | Failure | Skip | Successful |
| 18.11.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\Pochhammersym{\alpha+1}{n}}{n!}\KummerconfhyperM@{-n}{\alpha+1}{x} = \frac{(-1)^{n}}{n!}\KummerconfhyperU@{-n}{\alpha+1}{x}} | (pochhammer(alpha + 1, n))/(factorial(n))*KummerM(- n, alpha + 1, x)=((- 1)^(n))/(factorial(n))*KummerU(- n, alpha + 1, x) |
Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*Hypergeometric1F1[- n, \[Alpha]+ 1, x]=Divide[(- 1)^(n),(n)!]*HypergeometricU[- n, \[Alpha]+ 1, x] |
Failure | Failure | Successful | Successful |
| 18.11.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{(-1)^{n}}{n!}\KummerconfhyperU@{-n}{\alpha+1}{x} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\WhittakerconfhyperM{n+\frac{1}{2}(\alpha+1)}{\frac{1}{2}\alpha}@{x}} | ((- 1)^(n))/(factorial(n))*KummerU(- n, alpha + 1, x)=(pochhammer(alpha + 1, n))/(factorial(n))*(x)^(-(1)/(2)*(alpha + 1))* exp((1)/(2)*x)*WhittakerM(n +(1)/(2)*(alpha + 1), (1)/(2)*alpha, x) |
Divide[(- 1)^(n),(n)!]*HypergeometricU[- n, \[Alpha]+ 1, x]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(x)^(-Divide[1,2]*(\[Alpha]+ 1))* Exp[Divide[1,2]*x]*WhittakerM[n +Divide[1,2]*(\[Alpha]+ 1), Divide[1,2]*\[Alpha], x] |
Failure | Failure | Skip | Successful |
| 18.11.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\Pochhammersym{\alpha+1}{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\WhittakerconfhyperM{n+\frac{1}{2}(\alpha+1)}{\frac{1}{2}\alpha}@{x} = \frac{(-1)^{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\WhittakerconfhyperW{n+\frac{1}{2}(\alpha+1)}{\frac{1}{2}\alpha}@{x}} | (pochhammer(alpha + 1, n))/(factorial(n))*(x)^(-(1)/(2)*(alpha + 1))* exp((1)/(2)*x)*WhittakerM(n +(1)/(2)*(alpha + 1), (1)/(2)*alpha, x)=((- 1)^(n))/(factorial(n))*(x)^(-(1)/(2)*(alpha + 1))* exp((1)/(2)*x)*WhittakerW(n +(1)/(2)*(alpha + 1), (1)/(2)*alpha, x) |
Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(x)^(-Divide[1,2]*(\[Alpha]+ 1))* Exp[Divide[1,2]*x]*WhittakerM[n +Divide[1,2]*(\[Alpha]+ 1), Divide[1,2]*\[Alpha], x]=Divide[(- 1)^(n),(n)!]*(x)^(-Divide[1,2]*(\[Alpha]+ 1))* Exp[Divide[1,2]*x]*WhittakerW[n +Divide[1,2]*(\[Alpha]+ 1), Divide[1,2]*\[Alpha], x] |
Failure | Failure | Skip | Successful |
| 18.11.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{n}@{x} = 2^{n}\KummerconfhyperU@{-\tfrac{1}{2}n}{\tfrac{1}{2}}{x^{2}}} | HermiteH(n, x)= (2)^(n)* KummerU(-(1)/(2)*n, (1)/(2), (x)^(2)) |
HermiteH[n, x]= (2)^(n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], (x)^(2)] |
Failure | Failure | Successful | Successful |
| 18.11.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2^{n}\KummerconfhyperU@{-\tfrac{1}{2}n}{\tfrac{1}{2}}{x^{2}} = 2^{n}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{x^{2}}} | (2)^(n)* KummerU(-(1)/(2)*n, (1)/(2), (x)^(2))= (2)^(n)* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (x)^(2)) |
(2)^(n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], (x)^(2)]= (2)^(n)* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], (x)^(2)] |
Failure | Failure | Successful | Successful |
| 18.11.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2^{n}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{x^{2}} = 2^{\frac{1}{2}n}e^{\frac{1}{2}x^{2}}\paraU@{-n-\tfrac{1}{2}}{2^{\frac{1}{2}}x}} | (2)^(n)* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (x)^(2))= (2)^((1)/(2)*n)* exp((1)/(2)*(x)^(2))*CylinderU(- n -(1)/(2), (2)^((1)/(2))* x) |
(2)^(n)* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], (x)^(2)]= (2)^(Divide[1,2]*n)* Exp[Divide[1,2]*(x)^(2)]*ParabolicCylinderD[-- n -Divide[1,2] - 1/2, (2)^(Divide[1,2])* x] |
Failure | Failure | Skip | Successful |
| 18.11.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2^{\frac{1}{2}n}\KummerconfhyperU@{-\tfrac{1}{2}n}{\tfrac{1}{2}}{\tfrac{1}{2}x^{2}} = 2^{\frac{1}{2}(n-1)}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}x^{2}}} | (2)^((1)/(2)*n)* KummerU(-(1)/(2)*n, (1)/(2), (1)/(2)*(x)^(2))= (2)^((1)/(2)*(n - 1))* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (1)/(2)*(x)^(2)) |
(2)^(Divide[1,2]*n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], Divide[1,2]*(x)^(2)]= (2)^(Divide[1,2]*(n - 1))* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], Divide[1,2]*(x)^(2)] |
Failure | Failure | Successful | Successful |
| 18.11.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2^{\frac{1}{2}(n-1)}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}x^{2}} = e^{\tfrac{1}{4}x^{2}}\paraU@{-n-\tfrac{1}{2}}{x}} | (2)^((1)/(2)*(n - 1))* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (1)/(2)*(x)^(2))= exp((1)/(4)*(x)^(2))*CylinderU(- n -(1)/(2), x) |
(2)^(Divide[1,2]*(n - 1))* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], Divide[1,2]*(x)^(2)]= Exp[Divide[1,4]*(x)^(2)]*ParabolicCylinderD[-- n -Divide[1,2] - 1/2, x] |
Failure | Failure | Skip | Successful |
| 18.11.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\frac{1}{n^{\alpha}}\JacobipolyP{\alpha}{\beta}{n}@{1-\frac{z^{2}}{2n^{2}}} = \lim_{n\to\infty}\frac{1}{n^{\alpha}}\JacobipolyP{\alpha}{\beta}{n}@{\cos@@{\frac{z}{n}}}} | limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, 1 -((z)^(2))/(2*(n)^(2))), n = infinity)= limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, cos((z)/(n))), n = infinity) |
Limit[Divide[1,(n)^(\[Alpha])]*JacobiP[n, \[Alpha], \[Beta], 1 -Divide[(z)^(2),2*(n)^(2)]], n -> Infinity]= Limit[Divide[1,(n)^(\[Alpha])]*JacobiP[n, \[Alpha], \[Beta], Cos[Divide[z,n]]], n -> Infinity] |
Failure | Failure | Skip | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Plus[1, Times[Rational[-1, 2], Power[n, -2], Power[z, 2]]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Cos[Times[Power[n, -1], z]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Plus[1, Times[Rational[-1, 2], Power[n, -2], Power[z, 2]]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Cos[Times[Power[n, -1], z]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Plus[1, Times[Rational[-1, 2], Power[n, -2], Power[z, 2]]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Cos[Times[Power[n, -1], z]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Plus[1, Times[Rational[-1, 2], Power[n, -2], Power[z, 2]]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Limit[Times[Power[n, Times[-1, α]], JacobiP[n, α, β, Cos[Times[Power[n, -1], z]]]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 18.11.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\frac{1}{n^{\alpha}}\JacobipolyP{\alpha}{\beta}{n}@{\cos@@{\frac{z}{n}}} = \frac{2^{\alpha}}{z^{\alpha}}\BesselJ{\alpha}@{z}} | limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, cos((z)/(n))), n = infinity)=((2)^(alpha))/((z)^(alpha))*BesselJ(alpha, z) |
Limit[Divide[1,(n)^(\[Alpha])]*JacobiP[n, \[Alpha], \[Beta], Cos[Divide[z,n]]], n -> Infinity]=Divide[(2)^(\[Alpha]),(z)^(\[Alpha])]*BesselJ[\[Alpha], z] |
Failure | Failure | Skip | Skip |
| 18.11.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\frac{(-1)^{n}n^{\frac{1}{2}}}{2^{2n}n!}\HermitepolyH{2n}@{\frac{z}{2n^{\frac{1}{2}}}} = \frac{1}{\pi^{\frac{1}{2}}}\cos@@{z}} | limit(((- 1)^(n)* (n)^((1)/(2)))/((2)^(2*n)* factorial(n))*HermiteH(2*n, (z)/(2*(n)^((1)/(2)))), n = infinity)=(1)/((Pi)^((1)/(2)))*cos(z) |
Limit[Divide[(- 1)^(n)* (n)^(Divide[1,2]),(2)^(2*n)* (n)!]*HermiteH[2*n, Divide[z,2*(n)^(Divide[1,2])]], n -> Infinity]=Divide[1,(Pi)^(Divide[1,2])]*Cos[z] |
Failure | Failure | Skip | - |
| 18.11.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\frac{(-1)^{n}}{2^{2n}n!}\HermitepolyH{2n+1}@{\frac{z}{2n^{\frac{1}{2}}}} = \frac{2}{\pi^{\frac{1}{2}}}\sin@@{z}} | limit(((- 1)^(n))/((2)^(2*n)* factorial(n))*HermiteH(2*n + 1, (z)/(2*(n)^((1)/(2)))), n = infinity)=(2)/((Pi)^((1)/(2)))*sin(z) |
Limit[Divide[(- 1)^(n),(2)^(2*n)* (n)!]*HermiteH[2*n + 1, Divide[z,2*(n)^(Divide[1,2])]], n -> Infinity]=Divide[2,(Pi)^(Divide[1,2])]*Sin[z] |
Failure | Failure | Skip | Skip |
| 18.12.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2^{\alpha+\beta}}{R(1+R-z)^{\alpha}(1+R+z)^{\beta}} = \sum_{n=0}^{\infty}\JacobipolyP{\alpha}{\beta}{n}@{x}z^{n}} | ((2)^(alpha + beta))/(R*(1 + R - z)^(alpha)*(1 + R + z)^(beta))= sum(JacobiP(n, alpha, beta, x)*(z)^(n), n = 0..infinity) |
Divide[(2)^(\[Alpha]+ \[Beta]),R*(1 + R - z)^(\[Alpha])*(1 + R + z)^(\[Beta])]= Sum[JacobiP[n, \[Alpha], \[Beta], x]*(z)^(n), {n, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 18.12.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (1+z)^{-\alpha-\beta-1}\*\genhyperF{2}{1}@@{\tfrac{1}{2}(\alpha+\beta+1),\tfrac{1}{2}(\alpha+\beta+2)}{\beta+1}{\frac{2(x+1)z}{(1+z)^{2}}} = \sum_{n=0}^{\infty}\frac{\Pochhammersym{\alpha+\beta+1}{n}}{\Pochhammersym{\beta+1}{n}}\JacobipolyP{\alpha}{\beta}{n}@{x}z^{n}} | (1 + z)^(- alpha - beta - 1)* hypergeom([(1)/(2)*(alpha + beta + 1),(1)/(2)*(alpha + beta + 2)], [beta + 1], (2*(x + 1)* z)/((1 + z)^(2)))= sum((pochhammer(alpha + beta + 1, n))/(pochhammer(beta + 1, n))*JacobiP(n, alpha, beta, x)*(z)^(n), n = 0..infinity) |
(1 + z)^(- \[Alpha]- \[Beta]- 1)* HypergeometricPFQ[{Divide[1,2]*(\[Alpha]+ \[Beta]+ 1),Divide[1,2]*(\[Alpha]+ \[Beta]+ 2)}, {\[Beta]+ 1}, Divide[2*(x + 1)* z,(1 + z)^(2)]]= Sum[Divide[Pochhammer[\[Alpha]+ \[Beta]+ 1, n],Pochhammer[\[Beta]+ 1, n]]*JacobiP[n, \[Alpha], \[Beta], x]*(z)^(n), {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 18.12.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (1-2xz+z^{2})^{-\lambda} = \sum_{n=0}^{\infty}\ultrasphpoly{\lambda}{n}@{x}z^{n}} | (1 - 2*x*z + (z)^(2))^(- lambda)= sum(GegenbauerC(n, lambda, x)*(z)^(n), n = 0..infinity) |
(1 - 2*x*z + (z)^(2))^(- \[Lambda])= Sum[GegenbauerC[n, \[Lambda], x]*(z)^(n), {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 18.12.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}\ultrasphpoly{\lambda}{n}@{x}z^{n} = \sum_{n=0}^{\infty}\frac{\Pochhammersym{2\lambda}{n}}{\Pochhammersym{\lambda+\tfrac{1}{2}}{n}}\JacobipolyP{\lambda-\frac{1}{2}}{\lambda-\frac{1}{2}}{n}@{x}z^{n}} | sum(GegenbauerC(n, lambda, x)*(z)^(n), n = 0..infinity)= sum((pochhammer(2*lambda, n))/(pochhammer(lambda +(1)/(2), n))*JacobiP(n, lambda -(1)/(2), lambda -(1)/(2), x)*(z)^(n), n = 0..infinity) |
Sum[GegenbauerC[n, \[Lambda], x]*(z)^(n), {n, 0, Infinity}]= Sum[Divide[Pochhammer[2*\[Lambda], n],Pochhammer[\[Lambda]+Divide[1,2], n]]*JacobiP[n, \[Lambda]-Divide[1,2], \[Lambda]-Divide[1,2], x]*(z)^(n), {n, 0, Infinity}] |
Successful | Failure | - | Skip |
| 18.12.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1-xz}{(1-2xz+z^{2})^{\lambda+1}} = \sum_{n=0}^{\infty}\frac{n+2\lambda}{2\lambda}\ultrasphpoly{\lambda}{n}@{x}z^{n}} | (1 - x*z)/((1 - 2*x*z + (z)^(2))^(lambda + 1))= sum((n + 2*lambda)/(2*lambda)*GegenbauerC(n, lambda, x)*(z)^(n), n = 0..infinity) |
Divide[1 - x*z,(1 - 2*x*z + (z)^(2))^(\[Lambda]+ 1)]= Sum[Divide[n + 2*\[Lambda],2*\[Lambda]]*GegenbauerC[n, \[Lambda], x]*(z)^(n), {n, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 18.12.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{\lambda+\tfrac{1}{2}}e^{z\cos@@{\theta}}(\tfrac{1}{2}z\sin@@{\theta})^{\frac{1}{2}-\lambda}\BesselJ{\lambda-\frac{1}{2}}@{z\sin@@{\theta}} = \sum_{n=0}^{\infty}\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta}}}{\Pochhammersym{2\lambda}{n}}z^{n}} | GAMMA(lambda +(1)/(2))*exp(z*cos(theta))*((1)/(2)*z*sin(theta))^((1)/(2)- lambda)* BesselJ(lambda -(1)/(2), z*sin(theta))= sum((GegenbauerC(n, lambda, cos(theta)))/(pochhammer(2*lambda, n))*(z)^(n), n = 0..infinity) |
Gamma[\[Lambda]+Divide[1,2]]*Exp[z*Cos[\[Theta]]]*(Divide[1,2]*z*Sin[\[Theta]])^(Divide[1,2]- \[Lambda])* BesselJ[\[Lambda]-Divide[1,2], z*Sin[\[Theta]]]= Sum[Divide[GegenbauerC[n, \[Lambda], Cos[\[Theta]]],Pochhammer[2*\[Lambda], n]]*(z)^(n), {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 18.12.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1-z^{2}}{1-2xz+z^{2}} = 1+2\sum_{n=1}^{\infty}\ChebyshevpolyT{n}@{x}z^{n}} | (1 - (z)^(2))/(1 - 2*x*z + (z)^(2))= 1 + 2*sum(ChebyshevT(n, x)*(z)^(n), n = 1..infinity) |
Divide[1 - (z)^(2),1 - 2*x*z + (z)^(2)]= 1 + 2*Sum[ChebyshevT[n, x]*(z)^(n), {n, 1, Infinity}] |
Failure | Successful | Skip | - |
| 18.12.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1-xz}{1-2xz+z^{2}} = \sum_{n=0}^{\infty}\ChebyshevpolyT{n}@{x}z^{n}} | (1 - x*z)/(1 - 2*x*z + (z)^(2))= sum(ChebyshevT(n, x)*(z)^(n), n = 0..infinity) |
Divide[1 - x*z,1 - 2*x*z + (z)^(2)]= Sum[ChebyshevT[n, x]*(z)^(n), {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 18.12.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\ln@{1-2xz+z^{2}} = 2\sum_{n=1}^{\infty}\frac{\ChebyshevpolyT{n}@{x}}{n}z^{n}} | - ln(1 - 2*x*z + (z)^(2))= 2*sum((ChebyshevT(n, x))/(n)*(z)^(n), n = 1..infinity) |
- Log[1 - 2*x*z + (z)^(2)]= 2*Sum[Divide[ChebyshevT[n, x],n]*(z)^(n), {n, 1, Infinity}] |
Failure | Failure | Skip | Successful |
| 18.12.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{1-2xz+z^{2}} = \sum_{n=0}^{\infty}\ChebyshevpolyU{n}@{x}z^{n}} | (1)/(1 - 2*x*z + (z)^(2))= sum(ChebyshevU(n, x)*(z)^(n), n = 0..infinity) |
Divide[1,1 - 2*x*z + (z)^(2)]= Sum[ChebyshevU[n, x]*(z)^(n), {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 18.12.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\sqrt{1-2xz+z^{2}}} = \sum_{n=0}^{\infty}\LegendrepolyP{n}@{x}z^{n}} | (1)/(sqrt(1 - 2*x*z + (z)^(2)))= sum(LegendreP(n, x)*(z)^(n), n = 0..infinity) |
Divide[1,Sqrt[1 - 2*x*z + (z)^(2)]]= Sum[LegendreP[n, x]*(z)^(n), {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 18.12.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{xz}\BesselJ{0}@{z\sqrt{1-x^{2}}} = \sum_{n=0}^{\infty}\frac{\LegendrepolyP{n}@{x}}{n!}z^{n}} | exp(x*z)*BesselJ(0, z*sqrt(1 - (x)^(2)))= sum((LegendreP(n, x))/(factorial(n))*(z)^(n), n = 0..infinity) |
Exp[x*z]*BesselJ[0, z*Sqrt[1 - (x)^(2)]]= Sum[Divide[LegendreP[n, x],(n)!]*(z)^(n), {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 18.12.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{2xz-z^{2}} = \sum_{n=0}^{\infty}\frac{\HermitepolyH{n}@{x}}{n!}z^{n}} | exp(2*x*z - (z)^(2))= sum((HermiteH(n, x))/(factorial(n))*(z)^(n), n = 0..infinity) |
Exp[2*x*z - (z)^(2)]= Sum[Divide[HermiteH[n, x],(n)!]*(z)^(n), {n, 0, Infinity}] |
Error | Successful | - | - |
| 18.14.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\JacobipolyP{\alpha}{\beta}{n}@{x}| <= \JacobipolyP{\alpha}{\beta}{n}@{1}} | abs(JacobiP(n, alpha, beta, x))< = JacobiP(n, alpha, beta, 1) |
Abs[JacobiP[n, \[Alpha], \[Beta], x]]< = JacobiP[n, \[Alpha], \[Beta], 1] |
Failure | Failure | Fail 3.528680673 <= 1.500000000+0.*I <- {beta = 2^(1/2)+I*2^(1/2), n = 1, x = 2, alpha = 1/2}5.595865303 <= 1.500000000+0.*I <- {beta = 2^(1/2)+I*2^(1/2), n = 1, x = 3, alpha = 1/2}11.98055245 <= 1.875000000+0.*I <- {beta = 2^(1/2)+I*2^(1/2), n = 2, x = 2, alpha = 1/2}29.86867781 <= 1.875000000+0.*I <- {beta = 2^(1/2)+I*2^(1/2), n = 2, x = 3, alpha = 1/2}... skip entries to safe data |
Successful |
| 18.14.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \JacobipolyP{\alpha}{\beta}{n}@{1} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}} | JacobiP(n, alpha, beta, 1)=(pochhammer(alpha + 1, n))/(factorial(n)) |
JacobiP[n, \[Alpha], \[Beta], 1]=Divide[Pochhammer[\[Alpha]+ 1, n],(n)!] |
Successful | Successful | - | - |
| 18.14.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\JacobipolyP{\alpha}{\beta}{n}@{x}| <= |\JacobipolyP{\alpha}{\beta}{n}@{-1}|=\frac{\Pochhammersym{\beta+1}{n}}{n!}} | abs(JacobiP(n, alpha, beta, x))< =abs(JacobiP(n, alpha, beta, - 1))=(pochhammer(beta + 1, n))/(factorial(n)) |
Abs[JacobiP[n, \[Alpha], \[Beta], x]]< =Abs[JacobiP[n, \[Alpha], \[Beta], - 1]]=Divide[Pochhammer[\[Beta]+ 1, n],(n)!] |
Failure | Failure | Error | Fail
1.5 <- {Rule[n, 1], Rule[β, Rational[1, 2]]}1.875 <- {Rule[n, 2], Rule[β, Rational[1, 2]]}2.1875 <- {Rule[n, 3], Rule[β, Rational[1, 2]]} |
| 18.14.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\ultrasphpoly{\lambda}{n}@{x}| <= \ultrasphpoly{\lambda}{n}@{1}} | abs(GegenbauerC(n, lambda, x))< = GegenbauerC(n, lambda, 1) |
Abs[GegenbauerC[n, \[Lambda], x]]< = GegenbauerC[n, \[Lambda], 1] |
Failure | Failure | Fail 2.000000000 <= 1.000000000 <- {n = 1, x = 2, lambda = 1/2}3.000000000 <= 1.000000000 <- {n = 1, x = 3, lambda = 1/2}5.500000000 <= 1.000000000 <- {n = 2, x = 2, lambda = 1/2}13.00000000 <= 1.000000000 <- {n = 2, x = 3, lambda = 1/2}... skip entries to safe data |
Successful |
| 18.14.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{\lambda}{n}@{1} = \frac{\Pochhammersym{2\lambda}{n}}{n!}} | GegenbauerC(n, lambda, 1)=(pochhammer(2*lambda, n))/(factorial(n)) |
GegenbauerC[n, \[Lambda], 1]=Divide[Pochhammer[2*\[Lambda], n],(n)!] |
Successful | Successful | - | - |
| 18.14.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\ultrasphpoly{\lambda}{2m}@{x}| <= |\ultrasphpoly{\lambda}{2m}@{0}|=\left|\frac{\Pochhammersym{\lambda}{m}}{m!}\right|} | abs(GegenbauerC(2*m, lambda, x))< =abs(GegenbauerC(2*m, lambda, 0))=abs((pochhammer(lambda, m))/(factorial(m))) |
Abs[GegenbauerC[2*m, \[Lambda], x]]< =Abs[GegenbauerC[2*m, \[Lambda], 0]]=Abs[Divide[Pochhammer[\[Lambda], m],(m)!]] |
Failure | Failure | Error | Successful |
| 18.14.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\ultrasphpoly{\lambda}{2m+1}@{x}| < \frac{-2\Pochhammersym{\lambda}{m+1}}{\left((2m+1)(2\lambda+2m+1)\right)^{\frac{1}{2}}m!}} | abs(GegenbauerC(2*m + 1, lambda, x))<(- 2*pochhammer(lambda, m + 1))/(((2*m + 1)*(2*lambda + 2*m + 1))^((1)/(2))* factorial(m)) |
Abs[GegenbauerC[2*m + 1, \[Lambda], x]]<Divide[- 2*Pochhammer[\[Lambda], m + 1],((2*m + 1)*(2*\[Lambda]+ 2*m + 1))^(Divide[1,2])* (m)!] |
Failure | Failure | Skip | Successful |
| 18.14.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{(2^{n}n!)^{\frac{1}{2}}}e^{-\frac{1}{2}x^{2}}|\HermitepolyH{n}@{x}| <= 1} | (1)/(((2)^(n)* factorial(n))^((1)/(2)))*exp(-(1)/(2)*(x)^(2))*abs(HermiteH(n, x))< = 1 |
Divide[1,((2)^(n)* (n)!)^(Divide[1,2])]*Exp[-Divide[1,2]*(x)^(2)]*Abs[HermiteH[n, x]]< = 1 |
Failure | Failure | Skip | Successful |
| 18.14.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (\LegendrepolyP{n}@{x})^{2} >= \LegendrepolyP{n-1}@{x}\LegendrepolyP{n+1}@{x}} | (LegendreP(n, x))^(2)> = LegendreP(n - 1, x)*LegendreP(n + 1, x) |
(LegendreP[n, x])^(2)> = LegendreP[n - 1, x]*LegendreP[n + 1, x] |
Failure | Failure | Skip | Successful |
| 18.14.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (\HermitepolyH{n}@{x})^{2} >= \HermitepolyH{n-1}@{x}\HermitepolyH{n+1}@{x}} | (HermiteH(n, x))^(2)> = HermiteH(n - 1, x)*HermiteH(n + 1, x) |
(HermiteH[n, x])^(2)> = HermiteH[n - 1, x]*HermiteH[n + 1, x] |
Failure | Failure | Skip | Successful |
| 18.14#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| > |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|} | abs(JacobiP(n, alpha, beta, x[n , 0]))>abs(JacobiP(n, alpha, beta, x[n , 1])) |
Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]]>Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]] |
Failure | Failure | Skip | Successful |
| 18.14#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n}}| > |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n-1}}|} | abs(JacobiP(n, alpha, beta, x[n , n]))>abs(JacobiP(n, alpha, beta, x[n , n - 1])) |
Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n]]]>Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - 1]]] |
Failure | Failure | Skip | Successful |
| 18.14#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| < |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|} | abs(JacobiP(n, alpha, beta, x[n , 0]))<abs(JacobiP(n, alpha, beta, x[n , 1])) |
Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]]<Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]] |
Failure | Failure | Skip | Successful |
| 18.14#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n}}| < |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n-1}}|} | abs(JacobiP(n, alpha, beta, x[n , n]))<abs(JacobiP(n, alpha, beta, x[n , n - 1])) |
Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n]]]<Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - 1]]] |
Failure | Failure | Skip | Successful |
| 18.14.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| < |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|} | abs(JacobiP(n, alpha, beta, x[n , 0]))<abs(JacobiP(n, alpha, beta, x[n , 1])) |
Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]]<Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]] |
Failure | Failure | Skip | Successful |
| 18.14.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| > |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|} | abs(JacobiP(n, alpha, beta, x[n , 0]))>abs(JacobiP(n, alpha, beta, x[n , 1])) |
Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]]>Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]] |
Failure | Failure | Skip | Successful |
| 18.14.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left|\frac{\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n-m}}}{\JacobipolyP{\alpha}{\beta}{n}@{1}}\right| > \left|\frac{\JacobipolyP{\alpha}{\beta}{n+1}@{x_{n+1,n-m+1}}}{\JacobipolyP{\alpha}{\beta}{n+1}@{1}}\right|} | abs((JacobiP(n, alpha, beta, x[n , n - m]))/(JacobiP(n, alpha, beta, 1)))>abs((JacobiP(n + 1, alpha, beta, x[n + 1 , n - m + 1]))/(JacobiP(n + 1, alpha, beta, 1))) |
Abs[Divide[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - m]],JacobiP[n, \[Alpha], \[Beta], 1]]]>Abs[Divide[JacobiP[n + 1, \[Alpha], \[Beta], Subscript[x, n + 1 , n - m + 1]],JacobiP[n + 1, \[Alpha], \[Beta], 1]]] |
Failure | Failure | Skip | Skip |
| 18.15.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (\sin@@{\tfrac{1}{2}\theta})^{\alpha+\frac{1}{2}}(\cos@@{\tfrac{1}{2}\theta})^{\beta+\frac{1}{2}}\JacobipolyP{\alpha}{\beta}{n}@{\cos@@{\theta}} = \frac{\EulerGamma@{n+\alpha+1}}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*\left(\theta^{\frac{1}{2}}\BesselJ{\alpha}@{\rho\theta}\sum_{m=0}^{M}\dfrac{A_{m}(\theta)}{\rho^{2m}}+\theta^{\frac{3}{2}}\BesselJ{\alpha+1}@{\rho\theta}\sum_{m=0}^{M-1}\dfrac{B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M}(\rho,\theta)\right)} | (sin((1)/(2)*theta))^(alpha +(1)/(2))*(cos((1)/(2)*theta))^(beta +(1)/(2))* JacobiP(n, alpha, beta, cos(theta))=(GAMMA(n + alpha + 1))/((2)^((1)/(2))* (rho)^(alpha)* factorial(n))*((theta)^((1)/(2))* BesselJ(alpha, rho*theta)*sum((A[m]*(theta))/((rho)^(2*m)), m = 0..M)+ (theta)^((3)/(2))* BesselJ(alpha + 1, rho*theta)*sum((B[m]*(theta))/((rho)^(2*m + 1)), m = 0..M - 1)+ varepsilon[M]*(rho , theta)) |
(Sin[Divide[1,2]*\[Theta]])^(\[Alpha]+Divide[1,2])*(Cos[Divide[1,2]*\[Theta]])^(\[Beta]+Divide[1,2])* JacobiP[n, \[Alpha], \[Beta], Cos[\[Theta]]]=Divide[Gamma[n + \[Alpha]+ 1],(2)^(Divide[1,2])* (\[Rho])^(\[Alpha])* (n)!]*((\[Theta])^(Divide[1,2])* BesselJ[\[Alpha], \[Rho]*\[Theta]]*Sum[Divide[Subscript[A, m]*(\[Theta]),(\[Rho])^(2*m)], {m, 0, M}]+ (\[Theta])^(Divide[3,2])* BesselJ[\[Alpha]+ 1, \[Rho]*\[Theta]]*Sum[Divide[Subscript[B, m]*(\[Theta]),(\[Rho])^(2*m + 1)], {m, 0, M - 1}]+ Subscript[\[CurlyEpsilon], M]*(\[Rho], \[Theta])) |
Failure | Failure | Skip | Error |
| 18.15.E28 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{n}@{x} = 2^{\frac{1}{4}(\mu^{2}-1)}e^{\frac{1}{2}\mu^{2}t^{2}}\paraU@{-\tfrac{1}{2}\mu^{2}}{\mu t\sqrt{2}}} | HermiteH(n, x)= (2)^((1)/(4)*((mu)^(2)- 1))* exp((1)/(2)*(mu)^(2)* (t)^(2))*CylinderU(-(1)/(2)*(mu)^(2), mu*t*sqrt(2)) |
HermiteH[n, x]= (2)^(Divide[1,4]*((\[Mu])^(2)- 1))* Exp[Divide[1,2]*(\[Mu])^(2)* (t)^(2)]*ParabolicCylinderD[--Divide[1,2]*(\[Mu])^(2) - 1/2, \[Mu]*t*Sqrt[2]] |
Failure | Failure | Fail 2.014197760+.2431743734e-2*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), n = 1, x = 1}4.014197760+.2431743734e-2*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), n = 1, x = 2}6.014197760+.2431743734e-2*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), n = 1, x = 3}2.014197760+.2431743734e-2*I <- {mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), n = 2, x = 1}... skip entries to safe data |
Skip |
| 18.16.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (2n+1)^{\frac{1}{2}} > x_{n,1}} | (2*n + 1)^((1)/(2))> x[n , 1] |
(2*n + 1)^(Divide[1,2])> Subscript[x, n , 1] |
Failure | Failure | Successful | Successful |
| 18.16.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x_{n,1} > x_{n,2}} | x[n , 1]> x[n , 2] |
Subscript[x, n , 1]> Subscript[x, n , 2] |
Failure | Failure | Fail 1.414213562+1.414213562*I < 1.414213562+1.414213562*I <- {x[n,1] = 2^(1/2)+I*2^(1/2), x[n,2] = 2^(1/2)+I*2^(1/2)}1.414213562-1.414213562*I < 1.414213562-1.414213562*I <- {x[n,1] = 2^(1/2)-I*2^(1/2), x[n,2] = 2^(1/2)-I*2^(1/2)}-1.414213562-1.414213562*I < -1.414213562-1.414213562*I <- {x[n,1] = -2^(1/2)-I*2^(1/2), x[n,2] = -2^(1/2)-I*2^(1/2)}-1.414213562+1.414213562*I < -1.414213562+1.414213562*I <- {x[n,1] = -2^(1/2)+I*2^(1/2), x[n,2] = -2^(1/2)+I*2^(1/2)} |
Successful |
| 18.16.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x_{n,2} > \cdots} | x[n , 2]> .. |
Subscript[x, n , 2]> ... |
Failure | Failure | Skip | Successful |
| 18.17.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{y}\diff{y} = \JacobipolyP{\alpha+1}{\beta+1}{n-1}@{0}-(1-x)^{\alpha+1}(1+x)^{\beta+1}\JacobipolyP{\alpha+1}{\beta+1}{n-1}@{x}} | 2*n*int((1 - y)^(alpha)*(1 + y)^(beta)* JacobiP(n, alpha, beta, y), y = 0..x)= JacobiP(n - 1, alpha + 1, beta + 1, 0)-(1 - x)^(alpha + 1)*(1 + x)^(beta + 1)* JacobiP(n - 1, alpha + 1, beta + 1, x) |
2*n*Integrate[(1 - y)^(\[Alpha])*(1 + y)^(\[Beta])* JacobiP[n, \[Alpha], \[Beta], y], {y, 0, x}]= JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, 0]-(1 - x)^(\[Alpha]+ 1)*(1 + x)^(\[Beta]+ 1)* JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, x] |
Failure | Failure | Skip | Skip |
| 18.17.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{x}\LaguerrepolyL[]{m}@{y}\LaguerrepolyL[]{n}@{x-y}\diff{y} = \int_{0}^{x}\LaguerrepolyL[]{m+n}@{y}\diff{y}} | int(LaguerreL(m, y)*LaguerreL(n, x - y), y = 0..x)= int(LaguerreL(m + n, y), y = 0..x) |
Error |
Failure | Error | - | - |
| 18.17.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{x}\LaguerrepolyL[]{m+n}@{y}\diff{y} = \LaguerrepolyL[]{m+n}@{x}-\LaguerrepolyL[]{m+n+1}@{x}} | int(LaguerreL(m + n, y), y = 0..x)= LaguerreL(m + n, x)- LaguerreL(m + n + 1, x) |
Error |
Successful | Error | - | - |
| 18.17.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{x}\HermitepolyH{n}@{y}\diff{y} = \frac{1}{2(n+1)}(\HermitepolyH{n+1}@{x}-\HermitepolyH{n+1}@{0})} | int(HermiteH(n, y), y = 0..x)=(1)/(2*(n + 1))*(HermiteH(n + 1, x)- HermiteH(n + 1, 0)) |
Integrate[HermiteH[n, y], {y, 0, x}]=Divide[1,2*(n + 1)]*(HermiteH[n + 1, x]- HermiteH[n + 1, 0]) |
Failure | Successful | Skip | - |
| 18.17.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{x}e^{-y^{2}}\HermitepolyH{n}@{y}\diff{y} = \HermitepolyH{n-1}@{0}-e^{-x^{2}}\HermitepolyH{n-1}@{x}} | int(exp(- (y)^(2))*HermiteH(n, y), y = 0..x)= HermiteH(n - 1, 0)- exp(- (x)^(2))*HermiteH(n - 1, x) |
Integrate[Exp[- (y)^(2)]*HermiteH[n, y], {y, 0, x}]= HermiteH[n - 1, 0]- Exp[- (x)^(2)]*HermiteH[n - 1, x] |
Failure | Successful | Skip | - |
| 18.17.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{n}@{\cos@@{\theta_{1}}}\LegendrepolyP{n}@{\cos@@{\theta_{2}}} = \frac{1}{\pi}\int_{0}^{\pi}\LegendrepolyP{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}}\diff{\phi}} | LegendreP(n, cos(theta[1]))*LegendreP(n, cos(theta[2]))=(1)/(Pi)*int(LegendreP(n, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi)), phi = 0..Pi) |
LegendreP[n, Cos[Subscript[\[Theta], 1]]]*LegendreP[n, Cos[Subscript[\[Theta], 2]]]=Divide[1,Pi]*Integrate[LegendreP[n, Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]], {\[Phi], 0, Pi}] |
Failure | Failure | Skip | Skip |
| 18.17.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\LegendrepolyP{n}@{x}\right)^{2}+4\pi^{-2}\left(\FerrersQ[]{n}@{x}\right)^{2} = 4\pi^{-2}\*\int_{1}^{\infty}\assLegendreQ[]{n}@{x^{2}+(1-x^{2})t}(t^{2}-1)^{-\frac{1}{2}}\diff{t}} | (LegendreP(n, x))^(2)+ 4*(Pi)^(- 2)*(LegendreQ(n, x))^(2)= 4*(Pi)^(- 2)* int(LegendreQ(n, (x)^(2)+(1 - (x)^(2))* t)*((t)^(2)- 1)^(-(1)/(2)), t = 1..infinity) |
(LegendreP[n, x])^(2)+ 4*(Pi)^(- 2)*(LegendreQ[n, x])^(2)= 4*(Pi)^(- 2)* Integrate[LegendreQ[n, 0, 3, (x)^(2)+(1 - (x)^(2))* t]*((t)^(2)- 1)^(-Divide[1,2]), {t, 1, Infinity}] |
Failure | Failure | - | - |
| 18.17.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\HermitepolyH{n}@{x}\right)^{2}+2^{n}(n!)^{2}e^{x^{2}}\left(\paraV@{-n-\tfrac{1}{2}}{2^{\frac{1}{2}}x}\right)^{2} = \frac{2^{n+\frac{3}{2}}n!\,e^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}\tanh@@{t}}}{(\sinh@@{2t})^{\frac{1}{2}}}\diff{t}} | (HermiteH(n, x))^(2)+ (2)^(n)*(factorial(n))^(2)* exp((x)^(2))*(CylinderV(- n -(1)/(2), (2)^((1)/(2))* x))^(2)=((2)^(n +(3)/(2))* factorial(n)*exp((x)^(2)))/(Pi)*int((exp(-(2*n + 1)* t + (x)^(2)* tanh(t)))/((sinh(2*t))^((1)/(2))), t = 0..infinity) |
Error |
Failure | Error | Skip | - |
| 18.17.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{(1-x)^{\alpha+\mu}\JacobipolyP{\alpha+\mu}{\beta-\mu}{n}@{x}}{\EulerGamma@{\alpha+\mu+n+1}} = \int_{x}^{1}\frac{(1-y)^{\alpha}\JacobipolyP{\alpha}{\beta}{n}@{y}}{\EulerGamma@{\alpha+n+1}}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}} | ((1 - x)^(alpha + mu)* JacobiP(n, alpha + mu, beta - mu, x))/(GAMMA(alpha + mu + n + 1))= int(((1 - y)^(alpha)* JacobiP(n, alpha, beta, y))/(GAMMA(alpha + n + 1))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..1) |
Divide[(1 - x)^(\[Alpha]+ \[Mu])* JacobiP[n, \[Alpha]+ \[Mu], \[Beta]- \[Mu], x],Gamma[\[Alpha]+ \[Mu]+ n + 1]]= Integrate[Divide[(1 - y)^(\[Alpha])* JacobiP[n, \[Alpha], \[Beta], y],Gamma[\[Alpha]+ n + 1]]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, 1}] |
Failure | Failure | Skip | Error |
| 18.17.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{x^{\beta+\mu}(x+1)^{n}}{\EulerGamma@{\beta+\mu+n+1}}\JacobipolyP{\alpha}{\beta+\mu}{n}@{\frac{x-1}{x+1}} = \int_{0}^{x}\frac{y^{\beta}(y+1)^{n}}{\EulerGamma@{\beta+n+1}}\JacobipolyP{\alpha}{\beta}{n}@{\frac{y-1}{y+1}}\*\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}} | ((x)^(beta + mu)*(x + 1)^(n))/(GAMMA(beta + mu + n + 1))*JacobiP(n, alpha, beta + mu, (x - 1)/(x + 1))= int(((y)^(beta)*(y + 1)^(n))/(GAMMA(beta + n + 1))*JacobiP(n, alpha, beta, (y - 1)/(y + 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 0..x) |
Divide[(x)^(\[Beta]+ \[Mu])*(x + 1)^(n),Gamma[\[Beta]+ \[Mu]+ n + 1]]*JacobiP[n, \[Alpha], \[Beta]+ \[Mu], Divide[x - 1,x + 1]]= Integrate[Divide[(y)^(\[Beta])*(y + 1)^(n),Gamma[\[Beta]+ n + 1]]*JacobiP[n, \[Alpha], \[Beta], Divide[y - 1,y + 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 0, x}] |
Failure | Failure | Skip | Error |
| 18.17.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\EulerGamma@{n+\alpha+\beta-\mu+1}}{x^{n+\alpha+\beta-\mu+1}}\JacobipolyP{\alpha}{\beta-\mu}{n}@{1-2x^{-1}} = \int_{x}^{\infty}\frac{\EulerGamma@{n+\alpha+\beta+1}}{y^{n+\alpha+\beta+1}}\JacobipolyP{\alpha}{\beta}{n}@{1-2y^{-1}}\*\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}} | (GAMMA(n + alpha + beta - mu + 1))/((x)^(n + alpha + beta - mu + 1))*JacobiP(n, alpha, beta - mu, 1 - 2*(x)^(- 1))= int((GAMMA(n + alpha + beta + 1))/((y)^(n + alpha + beta + 1))*JacobiP(n, alpha, beta, 1 - 2*(y)^(- 1))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity) |
Divide[Gamma[n + \[Alpha]+ \[Beta]- \[Mu]+ 1],(x)^(n + \[Alpha]+ \[Beta]- \[Mu]+ 1)]*JacobiP[n, \[Alpha], \[Beta]- \[Mu], 1 - 2*(x)^(- 1)]= Integrate[Divide[Gamma[n + \[Alpha]+ \[Beta]+ 1],(y)^(n + \[Alpha]+ \[Beta]+ 1)]*JacobiP[n, \[Alpha], \[Beta], 1 - 2*(y)^(- 1)]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}] |
Failure | Failure | Skip | Error |
| 18.17.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\EulerGamma@{\lambda-\mu}\ultrasphpoly{\lambda-\mu}{n}@{x^{-\frac{1}{2}}}}{x^{\lambda-\mu+\frac{1}{2}n}} = \int_{x}^{\infty}\frac{\EulerGamma@{\lambda}\ultrasphpoly{\lambda}{n}@{y^{-\frac{1}{2}}}}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}} | (GAMMA(lambda - mu)*GegenbauerC(n, lambda - mu, (x)^(-(1)/(2))))/((x)^(lambda - mu +(1)/(2)*n))= int((GAMMA(lambda)*GegenbauerC(n, lambda, (y)^(-(1)/(2))))/((y)^(lambda +(1)/(2)*n))*((y - x)^(mu - 1))/(GAMMA(mu)), y = x..infinity) |
Divide[Gamma[\[Lambda]- \[Mu]]*GegenbauerC[n, \[Lambda]- \[Mu], (x)^(-Divide[1,2])],(x)^(\[Lambda]- \[Mu]+Divide[1,2]*n)]= Integrate[Divide[Gamma[\[Lambda]]*GegenbauerC[n, \[Lambda], (y)^(-Divide[1,2])],(y)^(\[Lambda]+Divide[1,2]*n)]*Divide[(y - x)^(\[Mu]- 1),Gamma[\[Mu]]], {y, x, Infinity}] |
Failure | Failure | Skip | Error |
| 18.17.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{2}}}{\EulerGamma@{\lambda+\mu+\tfrac{1}{2}}}\frac{\ultrasphpoly{\lambda+\mu}{n}@{x^{-\frac{1}{2}}}}{\ultrasphpoly{\lambda+\mu}{n}@{1}} = \int_{1}^{x}\frac{y^{\frac{1}{2}n}(y-1)^{\lambda-\frac{1}{2}}}{\EulerGamma@{\lambda+\tfrac{1}{2}}}\frac{\ultrasphpoly{\lambda}{n}@{y^{-\frac{1}{2}}}}{\ultrasphpoly{\lambda}{n}@{1}}\frac{(x-y)^{\mu-1}}{\EulerGamma@{\mu}}\diff{y}} | ((x)^((1)/(2)*n)*(x - 1)^(lambda + mu -(1)/(2)))/(GAMMA(lambda + mu +(1)/(2)))*(GegenbauerC(n, lambda + mu, (x)^(-(1)/(2))))/(GegenbauerC(n, lambda + mu, 1))= int(((y)^((1)/(2)*n)*(y - 1)^(lambda -(1)/(2)))/(GAMMA(lambda +(1)/(2)))*(GegenbauerC(n, lambda, (y)^(-(1)/(2))))/(GegenbauerC(n, lambda, 1))*((x - y)^(mu - 1))/(GAMMA(mu)), y = 1..x) |
Divide[(x)^(Divide[1,2]*n)*(x - 1)^(\[Lambda]+ \[Mu]-Divide[1,2]),Gamma[\[Lambda]+ \[Mu]+Divide[1,2]]]*Divide[GegenbauerC[n, \[Lambda]+ \[Mu], (x)^(-Divide[1,2])],GegenbauerC[n, \[Lambda]+ \[Mu], 1]]= Integrate[Divide[(y)^(Divide[1,2]*n)*(y - 1)^(\[Lambda]-Divide[1,2]),Gamma[\[Lambda]+Divide[1,2]]]*Divide[GegenbauerC[n, \[Lambda], (y)^(-Divide[1,2])],GegenbauerC[n, \[Lambda], 1]]*Divide[(x - y)^(\[Mu]- 1),Gamma[\[Mu]]], {y, 1, x}] |
Failure | Failure | Skip | Error |
| 18.17.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}e^{ixy}\diff{x} = \frac{(iy)^{n}e^{iy}}{n!}2^{n+\alpha+\beta+1}\EulerBeta@{n+\alpha+1}{n+\beta+1}\genhyperF{1}{1}@{n+\alpha+1}{2n+\alpha+\beta+2}{-2iy}} | int((1 - x)^(alpha)*(1 + x)^(beta)* JacobiP(n, alpha, beta, x)*exp(I*x*y), x = - 1..1)=((I*y)^(n)* exp(I*y))/(factorial(n))*(2)^(n + alpha + beta + 1)* Beta(n + alpha + 1, n + beta + 1)*hypergeom([n + alpha + 1], [2*n + alpha + beta + 2], - 2*I*y) |
Integrate[(1 - x)^(\[Alpha])*(1 + x)^(\[Beta])* JacobiP[n, \[Alpha], \[Beta], x]*Exp[I*x*y], {x, - 1, 1}]=Divide[(I*y)^(n)* Exp[I*y],(n)!]*(2)^(n + \[Alpha]+ \[Beta]+ 1)* Beta[n + \[Alpha]+ 1, n + \[Beta]+ 1]*HypergeometricPFQ[{n + \[Alpha]+ 1}, {2*n + \[Alpha]+ \[Beta]+ 2}, - 2*I*y] |
Failure | Failure | Skip | Error |
| 18.17.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{2n}@{x}\cos@{xy}\diff{x} = \frac{(-1)^{n}\pi\EulerGamma@{2n+2\lambda}\BesselJ{\lambda+2n}@{y}}{(2n)!\EulerGamma@{\lambda}(2y)^{\lambda}}} | int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(2*n, lambda, x)*cos(x*y), x = 0..1)=((- 1)^(n)* Pi*GAMMA(2*n + 2*lambda)*BesselJ(lambda + 2*n, y))/(factorial(2*n)*GAMMA(lambda)*(2*y)^(lambda)) |
Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[2*n, \[Lambda], x]*Cos[x*y], {x, 0, 1}]=Divide[(- 1)^(n)* Pi*Gamma[2*n + 2*\[Lambda]]*BesselJ[\[Lambda]+ 2*n, y],(2*n)!*Gamma[\[Lambda]]*(2*y)^(\[Lambda])] |
Failure | Failure | Skip | Error |
| 18.17.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{2n+1}@{x}\sin@{xy}\diff{x} = \frac{(-1)^{n}\pi\EulerGamma@{2n+2\lambda+1}\BesselJ{2n+\lambda+1}@{y}}{(2n+1)!\EulerGamma@{\lambda}(2y)^{\lambda}}} | int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(2*n + 1, lambda, x)*sin(x*y), x = 0..1)=((- 1)^(n)* Pi*GAMMA(2*n + 2*lambda + 1)*BesselJ(2*n + lambda + 1, y))/(factorial(2*n + 1)*GAMMA(lambda)*(2*y)^(lambda)) |
Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[2*n + 1, \[Lambda], x]*Sin[x*y], {x, 0, 1}]=Divide[(- 1)^(n)* Pi*Gamma[2*n + 2*\[Lambda]+ 1]*BesselJ[2*n + \[Lambda]+ 1, y],(2*n + 1)!*Gamma[\[Lambda]]*(2*y)^(\[Lambda])] |
Failure | Failure | Skip | Skip |
| 18.17.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-1}^{1}\LegendrepolyP{n}@{x}e^{ixy}\diff{x} = i^{n}\sqrt{\frac{2\pi}{y}}\BesselJ{n+\frac{1}{2}}@{y}} | int(LegendreP(n, x)*exp(I*x*y), x = - 1..1)= (I)^(n)*sqrt((2*Pi)/(y))*BesselJ(n +(1)/(2), y) |
Integrate[LegendreP[n, x]*Exp[I*x*y], {x, - 1, 1}]= (I)^(n)*Sqrt[Divide[2*Pi,y]]*BesselJ[n +Divide[1,2], y] |
Failure | Failure | Skip | Error |
| 18.17.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\LegendrepolyP{n}@{1-2x^{2}}\cos@{xy}\diff{x} = (-1)^{n}\tfrac{1}{2}\pi\BesselJ{n+\frac{1}{2}}@{\tfrac{1}{2}y}\BesselJ{-n-\frac{1}{2}}@{\tfrac{1}{2}y}} | int(LegendreP(n, 1 - 2*(x)^(2))*cos(x*y), x = 0..1)=(- 1)^(n)*(1)/(2)*Pi*BesselJ(n +(1)/(2), (1)/(2)*y)*BesselJ(- n -(1)/(2), (1)/(2)*y) |
Integrate[LegendreP[n, 1 - 2*(x)^(2)]*Cos[x*y], {x, 0, 1}]=(- 1)^(n)*Divide[1,2]*Pi*BesselJ[n +Divide[1,2], Divide[1,2]*y]*BesselJ[- n -Divide[1,2], Divide[1,2]*y] |
Failure | Failure | Skip | Successful |
| 18.17.E21 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\LegendrepolyP{n}@{1-2x^{2}}\sin@{xy}\diff{x} = \tfrac{1}{2}\pi\left(\BesselJ{n+\frac{1}{2}}@{\tfrac{1}{2}y}\right)^{2}} | int(LegendreP(n, 1 - 2*(x)^(2))*sin(x*y), x = 0..1)=(1)/(2)*Pi*(BesselJ(n +(1)/(2), (1)/(2)*y))^(2) |
Integrate[LegendreP[n, 1 - 2*(x)^(2)]*Sin[x*y], {x, 0, 1}]=Divide[1,2]*Pi*(BesselJ[n +Divide[1,2], Divide[1,2]*y])^(2) |
Failure | Failure | Skip | Successful |
| 18.17.E33 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-1}^{1}e^{-(x+1)z}\JacobipolyP{\alpha}{\beta}{n}@{x}(1-x)^{\alpha}(1+x)^{\beta}\diff{x} = \frac{(-1)^{n}2^{\alpha+\beta+n+1}\EulerGamma@{\alpha+n+1}\EulerGamma@{\beta+n+1}}{\EulerGamma@{\alpha+\beta+2n+2}n!}z^{n}\genhyperF{1}{1}@@{\beta+n+1}{\alpha+\beta+2n+2}{-2z}} | int(exp(-(x + 1)* z)*JacobiP(n, alpha, beta, x)*(1 - x)^(alpha)*(1 + x)^(beta), x = - 1..1)=((- 1)^(n)* (2)^(alpha + beta + n + 1)* GAMMA(alpha + n + 1)*GAMMA(beta + n + 1))/(GAMMA(alpha + beta + 2*n + 2)*factorial(n))*(z)^(n)* hypergeom([beta + n + 1], [alpha + beta + 2*n + 2], - 2*z) |
Integrate[Exp[-(x + 1)* z]*JacobiP[n, \[Alpha], \[Beta], x]*(1 - x)^(\[Alpha])*(1 + x)^(\[Beta]), {x, - 1, 1}]=Divide[(- 1)^(n)* (2)^(\[Alpha]+ \[Beta]+ n + 1)* Gamma[\[Alpha]+ n + 1]*Gamma[\[Beta]+ n + 1],Gamma[\[Alpha]+ \[Beta]+ 2*n + 2]*(n)!]*(z)^(n)* HypergeometricPFQ[{\[Beta]+ n + 1}, {\[Alpha]+ \[Beta]+ 2*n + 2}, - 2*z] |
Failure | Failure | Skip | Error |
| 18.17.E35 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-\infty}^{\infty}e^{-xz}\HermitepolyH{n}@{x}e^{-x^{2}}\diff{x} = \pi^{\frac{1}{2}}(-z)^{n}e^{\frac{1}{4}z^{2}}} | int(exp(- x*z)*HermiteH(n, x)*exp(- (x)^(2)), x = - infinity..infinity)= (Pi)^((1)/(2))*(- z)^(n)* exp((1)/(4)*(z)^(2)) |
Integrate[Exp[- x*z]*HermiteH[n, x]*Exp[- (x)^(2)], {x, - Infinity, Infinity}]= (Pi)^(Divide[1,2])*(- z)^(n)* Exp[Divide[1,4]*(z)^(2)] |
Failure | Failure | Skip | Error |
| 18.17.E36 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-1}^{1}(1-x)^{z-1}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}\diff{x} = \frac{2^{\beta+z}\EulerGamma@{z}\EulerGamma@{1+\beta+n}\Pochhammersym{1+\alpha-z}{n}}{n!\EulerGamma@{1+\beta+z+n}}} | int((1 - x)^(z - 1)*(1 + x)^(beta)* JacobiP(n, alpha, beta, x), x = - 1..1)=((2)^(beta + z)* GAMMA(z)*GAMMA(1 + beta + n)*pochhammer(1 + alpha - z, n))/(factorial(n)*GAMMA(1 + beta + z + n)) |
Integrate[(1 - x)^(z - 1)*(1 + x)^(\[Beta])* JacobiP[n, \[Alpha], \[Beta], x], {x, - 1, 1}]=Divide[(2)^(\[Beta]+ z)* Gamma[z]*Gamma[1 + \[Beta]+ n]*Pochhammer[1 + \[Alpha]- z, n],(n)!*Gamma[1 + \[Beta]+ z + n]] |
Failure | Failure | Skip | Error |
| 18.17.E37 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{n}@{x}x^{z-1}\diff{x} = \frac{\pi\,2^{1-2\lambda-z}\EulerGamma@{n+2\lambda}\EulerGamma@{z}}{n!\EulerGamma@{\lambda}\EulerGamma@{\frac{1}{2}+\frac{1}{2}n+\lambda+\frac{1}{2}z}\EulerGamma@{\frac{1}{2}+\frac{1}{2}z-\frac{1}{2}n}}} | int((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(n, lambda, x)*(x)^(z - 1), x = 0..1)=(Pi*(2)^(1 - 2*lambda - z)* GAMMA(n + 2*lambda)*GAMMA(z))/(factorial(n)*GAMMA(lambda)*GAMMA((1)/(2)+(1)/(2)*n + lambda +(1)/(2)*z)*GAMMA((1)/(2)+(1)/(2)*z -(1)/(2)*n)) |
Integrate[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[n, \[Lambda], x]*(x)^(z - 1), {x, 0, 1}]=Divide[Pi*(2)^(1 - 2*\[Lambda]- z)* Gamma[n + 2*\[Lambda]]*Gamma[z],(n)!*Gamma[\[Lambda]]*Gamma[Divide[1,2]+Divide[1,2]*n + \[Lambda]+Divide[1,2]*z]*Gamma[Divide[1,2]+Divide[1,2]*z -Divide[1,2]*n]] |
Failure | Failure | Skip | Error |
| 18.17.E38 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\LegendrepolyP{2n}@{x}x^{z-1}\diff{x} = \frac{(-1)^{n}\Pochhammersym{\frac{1}{2}-\frac{1}{2}z}{n}}{2\Pochhammersym{\frac{1}{2}z}{n+1}}} | int(LegendreP(2*n, x)*(x)^(z - 1), x = 0..1)=((- 1)^(n)* pochhammer((1)/(2)-(1)/(2)*z, n))/(2*pochhammer((1)/(2)*z, n + 1)) |
Integrate[LegendreP[2*n, x]*(x)^(z - 1), {x, 0, 1}]=Divide[(- 1)^(n)* Pochhammer[Divide[1,2]-Divide[1,2]*z, n],2*Pochhammer[Divide[1,2]*z, n + 1]] |
Failure | Failure | Skip | Successful |
| 18.17.E39 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\LegendrepolyP{2n+1}@{x}x^{z-1}\diff{x} = \frac{(-1)^{n}\Pochhammersym{1-\frac{1}{2}z}{n}}{2\Pochhammersym{\frac{1}{2}+\frac{1}{2}z}{n+1}}} | int(LegendreP(2*n + 1, x)*(x)^(z - 1), x = 0..1)=((- 1)^(n)* pochhammer(1 -(1)/(2)*z, n))/(2*pochhammer((1)/(2)+(1)/(2)*z, n + 1)) |
Integrate[LegendreP[2*n + 1, x]*(x)^(z - 1), {x, 0, 1}]=Divide[(- 1)^(n)* Pochhammer[1 -Divide[1,2]*z, n],2*Pochhammer[Divide[1,2]+Divide[1,2]*z, n + 1]] |
Failure | Failure | Skip | Successful |
| 18.17.E45 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}\LegendrepolyP{n}@{t}\diff{t} = \ChebyshevpolyT{n}@{x}+\ChebyshevpolyT{n+1}@{x}} | (n +(1)/(2))*(1 + x)^((1)/(2))* int((x - t)^(-(1)/(2))* LegendreP(n, t), t = - 1..x)= ChebyshevT(n, x)+ ChebyshevT(n + 1, x) |
(n +Divide[1,2])*(1 + x)^(Divide[1,2])* Integrate[(x - t)^(-Divide[1,2])* LegendreP[n, t], {t, - 1, x}]= ChebyshevT[n, x]+ ChebyshevT[n + 1, x] |
Failure | Failure | Skip | Skip |
| 18.17.E46 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (n+\tfrac{1}{2})(1-x)^{\frac{1}{2}}\int_{x}^{1}(t-x)^{-\frac{1}{2}}\LegendrepolyP{n}@{t}\diff{t} = \ChebyshevpolyT{n}@{x}-\ChebyshevpolyT{n+1}@{x}} | (n +(1)/(2))*(1 - x)^((1)/(2))* int((t - x)^(-(1)/(2))* LegendreP(n, t), t = x..1)= ChebyshevT(n, x)- ChebyshevT(n + 1, x) |
(n +Divide[1,2])*(1 - x)^(Divide[1,2])* Integrate[(t - x)^(-Divide[1,2])* LegendreP[n, t], {t, x, 1}]= ChebyshevT[n, x]- ChebyshevT[n + 1, x] |
Failure | Failure | Skip | Successful |
| 18.17.E48 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-\infty}^{\infty}\HermitepolyH{m}@{y}e^{-y^{2}}\HermitepolyH{n}@{x-y}e^{-(x-y)^{2}}\diff{y} = \pi^{\frac{1}{2}}2^{-\frac{1}{2}(m+n+1)}\HermitepolyH{m+n}@{2^{-\frac{1}{2}}x}e^{-\frac{1}{2}x^{2}}} | int(HermiteH(m, y)*exp(- (y)^(2))*HermiteH(n, x - y)*exp(-(x - y)^(2)), y = - infinity..infinity)= (Pi)^((1)/(2))* (2)^(-(1)/(2)*(m + n + 1))* HermiteH(m + n, (2)^(-(1)/(2))* x)*exp(-(1)/(2)*(x)^(2)) |
Integrate[HermiteH[m, y]*Exp[- (y)^(2)]*HermiteH[n, x - y]*Exp[-(x - y)^(2)], {y, - Infinity, Infinity}]= (Pi)^(Divide[1,2])* (2)^(-Divide[1,2]*(m + n + 1))* HermiteH[m + n, (2)^(-Divide[1,2])* x]*Exp[-Divide[1,2]*(x)^(2)] |
Failure | Failure | Skip | Error |
| 18.17.E49 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-\infty}^{\infty}\HermitepolyH{\ell}@{x}\HermitepolyH{m}@{x}\HermitepolyH{n}@{x}e^{-x^{2}}\diff{x} = \frac{2^{\frac{1}{2}(\ell+m+n)}\ell\,!\,m\,!\,n\,!\,\sqrt{\pi}}{(\tfrac{1}{2}\ell+\tfrac{1}{2}m-\tfrac{1}{2}n)\,!\,(\tfrac{1}{2}m+\tfrac{1}{2}n-\tfrac{1}{2}\ell\,)\,!\,(\tfrac{1}{2}n+\tfrac{1}{2}\ell-\tfrac{1}{2}m\,)\,!}} | int(HermiteH(ell, x)*HermiteH(m, x)*HermiteH(n, x)*exp(- (x)^(2)), x = - infinity..infinity)=((2)^((1)/(2)*(ell + m + n))* factorial(ell)*factorial(m)*factorial(n)*sqrt(Pi))/(factorial((1)/(2)*ell +(1)/(2)*m -(1)/(2)*n)*factorial((1)/(2)*m +(1)/(2)*n -(1)/(2)*ell)*factorial((1)/(2)*n +(1)/(2)*ell -(1)/(2)*m)) |
Integrate[HermiteH[\[ScriptL], x]*HermiteH[m, x]*HermiteH[n, x]*Exp[- (x)^(2)], {x, - Infinity, Infinity}]=Divide[(2)^(Divide[1,2]*(\[ScriptL]+ m + n))* (\[ScriptL])!*(m)!*(n)!*Sqrt[Pi],(Divide[1,2]*\[ScriptL]+Divide[1,2]*m -Divide[1,2]*n)!*(Divide[1,2]*m +Divide[1,2]*n -Divide[1,2]*\[ScriptL])!*(Divide[1,2]*n +Divide[1,2]*\[ScriptL]-Divide[1,2]*m)!] |
Failure | Failure | Skip | Error |
| 18.18.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ultrasphpoly{\lambda}{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = \sum_{\ell=0}^{n}2^{2\ell}(n-\ell)!\frac{2\lambda+2\ell-1}{2\lambda-1}\frac{(\Pochhammersym{\lambda}{\ell})^{2}}{\Pochhammersym{2\lambda}{n+\ell}}(\sin@@{\theta_{1}})^{\ell}\ultrasphpoly{\lambda+\ell}{n-\ell}@{\cos@@{\theta_{1}}}(\sin@@{\theta_{2}})^{\ell}\ultrasphpoly{\lambda+\ell}{n-\ell}@{\cos@@{\theta_{2}}}\ultrasphpoly{\lambda-\frac{1}{2}}{\ell}@{\cos@@{\phi}}} | GegenbauerC(n, lambda, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi))= sum((2)^(2*ell)*factorial(n - ell)*(2*lambda + 2*ell - 1)/(2*lambda - 1)*((pochhammer(lambda, ell))^(2))/(pochhammer(2*lambda, n + ell))*(sin(theta[1]))^(ell)* GegenbauerC(n - ell, lambda + ell, cos(theta[1]))*(sin(theta[2]))^(ell)* GegenbauerC(n - ell, lambda + ell, cos(theta[2]))*GegenbauerC(ell, lambda -(1)/(2), cos(phi)), ell = 0..n) |
GegenbauerC[n, \[Lambda], Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]]= Sum[(2)^(2*\[ScriptL])*(n - \[ScriptL])!*Divide[2*\[Lambda]+ 2*\[ScriptL]- 1,2*\[Lambda]- 1]*Divide[(Pochhammer[\[Lambda], \[ScriptL]])^(2),Pochhammer[2*\[Lambda], n + \[ScriptL]]]*(Sin[Subscript[\[Theta], 1]])^(\[ScriptL])* GegenbauerC[n - \[ScriptL], \[Lambda]+ \[ScriptL], Cos[Subscript[\[Theta], 1]]]*(Sin[Subscript[\[Theta], 2]])^(\[ScriptL])* GegenbauerC[n - \[ScriptL], \[Lambda]+ \[ScriptL], Cos[Subscript[\[Theta], 2]]]*GegenbauerC[\[ScriptL], \[Lambda]-Divide[1,2], Cos[\[Phi]]], {\[ScriptL], 0, n}] |
Failure | Failure | Skip | Error |
| 18.18.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LegendrepolyP{n}@{\cos@@{\theta_{1}}\cos@@{\theta_{2}}+\sin@@{\theta_{1}}\sin@@{\theta_{2}}\cos@@{\phi}} = {\LegendrepolyP{n}@{\cos@@{\theta_{1}}}\LegendrepolyP{n}@{\cos@@{\theta_{2}}}+2\sum_{\ell=1}^{n}\frac{(n-\ell)!\;(n+\ell)!}{2^{2\ell}(n!)^{2}}(\sin@@{\theta_{1}})^{\ell}\JacobipolyP{\ell}{\ell}{n-\ell}@{\cos@@{\theta_{1}}}(\sin@@{\theta_{2}})^{\ell}\JacobipolyP{\ell}{\ell}{n-\ell}@{\cos@@{\theta_{2}}}\cos@{\ell\phi}}} | LegendreP(n, cos(theta[1])*cos(theta[2])+ sin(theta[1])*sin(theta[2])*cos(phi))=LegendreP(n, cos(theta[1]))*LegendreP(n, cos(theta[2]))+ 2*sum((factorial(n - ell)*factorial(n + ell))/((2)^(2*ell)*(factorial(n))^(2))*(sin(theta[1]))^(ell)* JacobiP(n - ell, ell, ell, cos(theta[1]))*(sin(theta[2]))^(ell)* JacobiP(n - ell, ell, ell, cos(theta[2]))*cos(ell*phi), ell = 1..n) |
LegendreP[n, Cos[Subscript[\[Theta], 1]]*Cos[Subscript[\[Theta], 2]]+ Sin[Subscript[\[Theta], 1]]*Sin[Subscript[\[Theta], 2]]*Cos[\[Phi]]]=LegendreP[n, Cos[Subscript[\[Theta], 1]]]*LegendreP[n, Cos[Subscript[\[Theta], 2]]]+ 2*Sum[Divide[(n - \[ScriptL])!*(n + \[ScriptL])!,(2)^(2*\[ScriptL])*((n)!)^(2)]*(Sin[Subscript[\[Theta], 1]])^(\[ScriptL])* JacobiP[n - \[ScriptL], \[ScriptL], \[ScriptL], Cos[Subscript[\[Theta], 1]]]*(Sin[Subscript[\[Theta], 2]])^(\[ScriptL])* JacobiP[n - \[ScriptL], \[ScriptL], \[ScriptL], Cos[Subscript[\[Theta], 2]]]*Cos[\[ScriptL]*\[Phi]], {\[ScriptL], 1, n}] |
Failure | Failure | Skip | Error |
| 18.18.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \HermitepolyH{n}@{\lambda x} = \lambda^{n}\sum_{\ell=0}^{\floor{n/2}}\frac{\Pochhammersym{-n}{2\ell}}{\ell!}(1-\lambda^{-2})^{\ell}\HermitepolyH{n-2\ell}@{x}} | HermiteH(n, lambda*x)= (lambda)^(n)* sum((pochhammer(- n, 2*ell))/(factorial(ell))*(1 - (lambda)^(- 2))^(ell)* HermiteH(n - 2*ell, x), ell = 0..floor(n/ 2)) |
HermiteH[n, \[Lambda]*x]= (\[Lambda])^(n)* Sum[Divide[Pochhammer[- n, 2*\[ScriptL]],(\[ScriptL])!]*(1 - (\[Lambda])^(- 2))^(\[ScriptL])* HermiteH[n - 2*\[ScriptL], x], {\[ScriptL], 0, Floor[n/ 2]}] |
Failure | Failure | Skip | Fail
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[n, 1], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[5.656854249492381, 5.656854249492381] <- {Rule[n, 1], Rule[x, 2], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[8.485281374238571, 8.485281374238571] <- {Rule[n, 1], Rule[x, 3], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-2.0, 16.0] <- {Rule[n, 2], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 18.18.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (2x)^{n} = \sum_{\ell=0}^{\floor{n/2}}\frac{\Pochhammersym{-n}{2\ell}}{\ell!}\HermitepolyH{n-2\ell}@{x}} | (2*x)^(n)= sum((pochhammer(- n, 2*ell))/(factorial(ell))*HermiteH(n - 2*ell, x), ell = 0..floor(n/ 2)) |
(2*x)^(n)= Sum[Divide[Pochhammer[- n, 2*\[ScriptL]],(\[ScriptL])!]*HermiteH[n - 2*\[ScriptL], x], {\[ScriptL], 0, Floor[n/ 2]}] |
Failure | Failure | Skip | Fail
2.0 <- {Rule[n, 1], Rule[x, 1]}4.0 <- {Rule[n, 1], Rule[x, 2]}6.0 <- {Rule[n, 1], Rule[x, 3]}4.0 <- {Rule[n, 2], Rule[x, 1]}... skip entries to safe data |
| 18.18.E21 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ChebyshevpolyT{m}@{x}\ChebyshevpolyT{n}@{x} = \tfrac{1}{2}(\ChebyshevpolyT{m+n}@{x}+\ChebyshevpolyT{m-n}@{x})} | ChebyshevT(m, x)*ChebyshevT(n, x)=(1)/(2)*(ChebyshevT(m + n, x)+ ChebyshevT(m - n, x)) |
ChebyshevT[m, x]*ChebyshevT[n, x]=Divide[1,2]*(ChebyshevT[m + n, x]+ ChebyshevT[m - n, x]) |
Failure | Failure | Successful | Successful |
| 18.18.E24 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle b_{n,\ell} = \binom{n}{\ell}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{-\beta-n}{n-\ell}}{2^{\ell}\Pochhammersym{\alpha+1}{n}}} | b[n , ell]=binomial(n,ell)*(pochhammer(n + alpha + beta + 1, ell)*pochhammer(- beta - n, n - ell))/((2)^(ell)* pochhammer(alpha + 1, n)) |
Subscript[b, n , \[ScriptL]]=Binomial[n,\[ScriptL]]*Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[- \[Beta]- n, n - \[ScriptL]],(2)^(\[ScriptL])* Pochhammer[\[Alpha]+ 1, n]] |
Failure | Failure | Fail .414213562+1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), b[n,ell] = 2^(1/2)+I*2^(1/2), ell = 1, n = 1}3.722604190+1.233562514*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), b[n,ell] = 2^(1/2)+I*2^(1/2), ell = 1, n = 2}-2.510958325+1.956166705*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), b[n,ell] = 2^(1/2)+I*2^(1/2), ell = 1, n = 3}1.414213562+1.414213562*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), b[n,ell] = 2^(1/2)+I*2^(1/2), ell = 2, n = 1}... skip entries to safe data |
Skip |
| 18.18.E25 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{1}}\frac{\JacobipolyP{\alpha}{\beta}{n}@{y}}{\JacobipolyP{\alpha}{\beta}{n}@{1}} = \sum_{\ell=0}^{n}b_{n,\ell}(x+y)^{\ell}\*\frac{\JacobipolyP{\alpha}{\beta}{\ell}@{\ifrac{(1+xy)}{(x+y)}}}{\JacobipolyP{\alpha}{\beta}{\ell}@{1}}} | (JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, 1))*(JacobiP(n, alpha, beta, y))/(JacobiP(n, alpha, beta, 1))= sum(b[n , ell]*(x + y)^(ell)*(JacobiP(ell, alpha, beta, (1 + x*y)/(x + y)))/(JacobiP(ell, alpha, beta, 1)), ell = 0..n) |
Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], 1]]*Divide[JacobiP[n, \[Alpha], \[Beta], y],JacobiP[n, \[Alpha], \[Beta], 1]]= Sum[Subscript[b, n , \[ScriptL]]*(x + y)^(\[ScriptL])*Divide[JacobiP[\[ScriptL], \[Alpha], \[Beta], Divide[1 + x*y,x + y]],JacobiP[\[ScriptL], \[Alpha], \[Beta], 1]], {\[ScriptL], 0, n}] |
Failure | Failure | Skip | Skip |
| 18.18.E26 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{1}} = \sum_{\ell=0}^{n}b_{n,\ell}(x+1)^{\ell}} | (JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, 1))= sum(b[n , ell]*(x + 1)^(ell), ell = 0..n) |
Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], 1]]= Sum[Subscript[b, n , \[ScriptL]]*(x + 1)^(\[ScriptL]), {\[ScriptL], 0, n}] |
Failure | Failure | Skip | Skip |
| 18.18.E28 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}\frac{\HermitepolyH{n}@{x}\HermitepolyH{n}@{y}}{2^{n}n!}z^{n} = (1-z^{2})^{-\frac{1}{2}}\exp@{\frac{2xyz-(x^{2}+y^{2})z^{2}}{1-z^{2}}}} | sum((HermiteH(n, x)*HermiteH(n, y))/((2)^(n)* factorial(n))*(z)^(n), n = 0..infinity)=(1 - (z)^(2))^(-(1)/(2))* exp((2*x*y*z -((x)^(2)+ (y)^(2))* (z)^(2))/(1 - (z)^(2))) |
Sum[Divide[HermiteH[n, x]*HermiteH[n, y],(2)^(n)* (n)!]*(z)^(n), {n, 0, Infinity}]=(1 - (z)^(2))^(-Divide[1,2])* Exp[Divide[2*x*y*z -((x)^(2)+ (y)^(2))* (z)^(2),1 - (z)^(2)]] |
Error | Failure | - | Skip |
| 18.18.E29 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{n}\ultrasphpoly{\lambda}{\ell}@{x}\ultrasphpoly{\mu}{n-\ell}@{x} = \ultrasphpoly{\lambda+\mu}{n}@{x}} | sum(GegenbauerC(ell, lambda, x)*GegenbauerC(n - ell, mu, x), ell = 0..n)= GegenbauerC(n, lambda + mu, x) |
Sum[GegenbauerC[\[ScriptL], \[Lambda], x]*GegenbauerC[n - \[ScriptL], \[Mu], x], {\[ScriptL], 0, n}]= GegenbauerC[n, \[Lambda]+ \[Mu], x] |
Failure | Successful | Skip | - |
| 18.18.E30 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{n}\frac{\ell+2\lambda}{2\lambda}\ultrasphpoly{\lambda}{\ell}@{x}x^{n-\ell} = \ultrasphpoly{\lambda+1}{n}@{x}} | sum((ell + 2*lambda)/(2*lambda)*GegenbauerC(ell, lambda, x)*(x)^(n - ell), ell = 0..n)= GegenbauerC(n, lambda + 1, x) |
Sum[Divide[\[ScriptL]+ 2*\[Lambda],2*\[Lambda]]*GegenbauerC[\[ScriptL], \[Lambda], x]*(x)^(n - \[ScriptL]), {\[ScriptL], 0, n}]= GegenbauerC[n, \[Lambda]+ 1, x] |
Failure | Failure | Skip | Error |
| 18.18.E31 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{n}\ChebyshevpolyT{\ell}@{x}x^{n-\ell} = \ChebyshevpolyU{n}@{x}} | sum(ChebyshevT(ell, x)*(x)^(n - ell), ell = 0..n)= ChebyshevU(n, x) |
Sum[ChebyshevT[\[ScriptL], x]*(x)^(n - \[ScriptL]), {\[ScriptL], 0, n}]= ChebyshevU[n, x] |
Failure | Failure | Skip | Error |
| 18.18.E32 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2\sum_{\ell=0}^{n}\ChebyshevpolyT{2\ell}@{x} = 1+\ChebyshevpolyU{2n}@{x}} | 2*sum(ChebyshevT(2*ell, x), ell = 0..n)= 1 + ChebyshevU(2*n, x) |
2*Sum[ChebyshevT[2*\[ScriptL], x], {\[ScriptL], 0, n}]= 1 + ChebyshevU[2*n, x] |
Failure | Successful | Skip | - |
| 18.18.E33 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2\sum_{\ell=0}^{n}\ChebyshevpolyT{2\ell+1}@{x} = \ChebyshevpolyU{2n+1}@{x}} | 2*sum(ChebyshevT(2*ell + 1, x), ell = 0..n)= ChebyshevU(2*n + 1, x) |
2*Sum[ChebyshevT[2*\[ScriptL]+ 1, x], {\[ScriptL], 0, n}]= ChebyshevU[2*n + 1, x] |
Failure | Successful | Skip | - |
| 18.18.E34 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2(1-x^{2})\sum_{\ell=0}^{n}\ChebyshevpolyU{2\ell}@{x} = 1-\ChebyshevpolyT{2n+2}@{x}} | 2*(1 - (x)^(2))* sum(ChebyshevU(2*ell, x), ell = 0..n)= 1 - ChebyshevT(2*n + 2, x) |
2*(1 - (x)^(2))* Sum[ChebyshevU[2*\[ScriptL], x], {\[ScriptL], 0, n}]= 1 - ChebyshevT[2*n + 2, x] |
Failure | Successful | Skip | - |
| 18.18.E35 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2(1-x^{2})\sum_{\ell=0}^{n}\ChebyshevpolyU{2\ell+1}@{x} = x-\ChebyshevpolyT{2n+3}@{x}} | 2*(1 - (x)^(2))* sum(ChebyshevU(2*ell + 1, x), ell = 0..n)= x - ChebyshevT(2*n + 3, x) |
2*(1 - (x)^(2))* Sum[ChebyshevU[2*\[ScriptL]+ 1, x], {\[ScriptL], 0, n}]= x - ChebyshevT[2*n + 3, x] |
Failure | Successful | Skip | - |
| 18.18.E36 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{n}\LegendrepolyP{\ell}@{x}\LegendrepolyP{n-\ell}@{x} = \ChebyshevpolyU{n}@{x}} | sum(LegendreP(ell, x)*LegendreP(n - ell, x), ell = 0..n)= ChebyshevU(n, x) |
Sum[LegendreP[\[ScriptL], x]*LegendreP[n - \[ScriptL], x], {\[ScriptL], 0, n}]= ChebyshevU[n, x] |
Failure | Successful | Skip | - |
| 18.18.E40 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{n}\binom{n}{\ell}\HermitepolyH{2\ell}@{x}\HermitepolyH{2n-2\ell}@{y} = (-1)^{n}2^{2n}n!\LaguerrepolyL[]{n}@{x^{2}+y^{2}}} | sum(binomial(n,ell)*HermiteH(2*ell, x)*HermiteH(2*n - 2*ell, y), ell = 0..n)=(- 1)^(n)* (2)^(2*n)* factorial(n)*LaguerreL(n, (x)^(2)+ (y)^(2)) |
Error |
Failure | Error | - | - |
| 18.19.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h_{n} = \frac{2\pi\EulerGamma@{n+a+\conj{a}}\EulerGamma@{n+b+\conj{b}}|\EulerGamma@{n+a+\conj{b}}|^{2}}{\left(2n+2\realpart@@{(a+b)}-1\right)\EulerGamma@{n+2\realpart@@{(a+b)}-1}n!}} | h[n]=(2*Pi*GAMMA(n + a + conjugate(a))*GAMMA(n + b + conjugate(b))*(abs(GAMMA(n + a + conjugate(b))))^(2))/((2*n + 2*Re(a + b)- 1)* GAMMA(n + 2*Re(a + b)- 1)*factorial(n)) |
Subscript[h, n]=Divide[2*Pi*Gamma[n + a + Conjugate[a]]*Gamma[n + b + Conjugate[b]]*(Abs[Gamma[n + a + Conjugate[b]]])^(2),(2*n + 2*Re[a + b]- 1)* Gamma[n + 2*Re[a + b]- 1]*(n)!] |
Failure | Failure | Fail -6.363350679+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)+I*2^(1/2), n = 1}-112.1467589+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)+I*2^(1/2), n = 2}-2509.272955+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)+I*2^(1/2), n = 3}-6.363350679-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Skip |
| 18.19.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k_{n} = \frac{\Pochhammersym{n+2\realpart@@{(a+b)}-1}{n}}{n!}} | k[n]=(pochhammer(n + 2*Re(a + b)- 1, n))/(factorial(n)) |
Subscript[k, n]=Divide[Pochhammer[n + 2*Re[a + b]- 1, n],(n)!] |
Failure | Failure | Fail -4.242640686+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), k[n] = 2^(1/2)+I*2^(1/2), n = 1}-24.07106780+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), k[n] = 2^(1/2)+I*2^(1/2), n = 2}-105.2687108+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), k[n] = 2^(1/2)+I*2^(1/2), n = 3}-4.242640686-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), k[n] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Successful |
| 18.19#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h_{n} = \frac{2\pi\EulerGamma@{n+2\lambda}}{(2\sin@@{\phi})^{2\lambda}n!}} | h[n]=(2*Pi*GAMMA(n + 2*lambda))/((2*sin(phi))^(2*lambda)* factorial(n)) |
Subscript[h, n]=Divide[2*Pi*Gamma[n + 2*\[Lambda]],(2*Sin[\[Phi]])^(2*\[Lambda])* (n)!] |
Failure | Failure | Fail 1.256815617+1.596021436*I <- {lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)+I*2^(1/2), n = 1}.8558051203+1.539638353*I <- {lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)+I*2^(1/2), n = 2}.397217113+1.089609188*I <- {lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)+I*2^(1/2), n = 3}1.256815617-1.232405688*I <- {lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Skip |
| 18.19#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k_{n} = \frac{(2\sin@@{\phi})^{n}}{n!}} | k[n]=((2*sin(phi))^(n))/(factorial(n)) |
Subscript[k, n]=Divide[(2*Sin[\[Phi]])^(n),(n)!] |
Failure | Failure | Fail -2.888857518+.8106906210*I <- {phi = 2^(1/2)+I*2^(1/2), k[n] = 2^(1/2)+I*2^(1/2), n = 1}-7.661876828-1.182788552*I <- {phi = 2^(1/2)+I*2^(1/2), k[n] = 2^(1/2)+I*2^(1/2), n = 2}-11.08169034-4.136690922*I <- {phi = 2^(1/2)+I*2^(1/2), k[n] = 2^(1/2)+I*2^(1/2), n = 3}-2.888857518-2.017736503*I <- {phi = 2^(1/2)+I*2^(1/2), k[n] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Successful |
| 18.20#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \kappa_{n} = \Pochhammersym{-N}{n}\Pochhammersym{\alpha+1}{n}} | kappa[n]= pochhammer(- N, n)*pochhammer(alpha + 1, n) |
Subscript[\[Kappa], n]= Pochhammer[- N, n]*Pochhammer[\[Alpha]+ 1, n] |
Failure | Failure | Fail 2.828427124+6.828427122*I <- {N = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), kappa[n] = 2^(1/2)+I*2^(1/2), n = 1}31.55634916-3.071067807*I <- {N = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), kappa[n] = 2^(1/2)+I*2^(1/2), n = 2}115.3553390-182.3502881*I <- {N = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), kappa[n] = 2^(1/2)+I*2^(1/2), n = 3}2.828427124+3.999999998*I <- {N = 2^(1/2)+I*2^(1/2), alpha = 2^(1/2)+I*2^(1/2), kappa[n] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Successful |
| 18.22.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (n+1)p_{n+1}(x) = 2\left(x\sin@@{\phi}+(n+\lambda)\cos@@{\phi}\right)p_{n}(x)-(n+2\lambda-1)p_{n-1}(x)} | (n + 1)* p[n + 1]*(x)= 2*(x*sin(phi)+(n + lambda)*cos(phi))* p[n]*(x)-(n + 2*lambda - 1)* p[n - 1]*(x) |
(n + 1)* Subscript[p, n + 1]*(x)= 2*(x*Sin[\[Phi]]+(n + \[Lambda])*Cos[\[Phi]])* Subscript[p, n]*(x)-(n + 2*\[Lambda]- 1)* Subscript[p, n - 1]*(x) |
Failure | Failure | Skip | Skip |
| 18.22.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+n(n+2\realpart@@{(a+b)}-1)p_{n}(x) = 0} | A*(x)* p[n]*(x + I)-(A*(x)+ C*(x))* p[n]*(x)+ C*(x)* p[n]*(x - I)+ n*(n + 2*Re(a + b)- 1)* p[n]*(x)= 0 |
A*(x)* Subscript[p, n]*(x + I)-(A*(x)+ C*(x))* Subscript[p, n]*(x)+ C*(x)* Subscript[p, n]*(x - I)+ n*(n + 2*Re[a + b]- 1)* Subscript[p, n]*(x)= 0 |
Failure | Failure | Skip | Skip |
| 18.22#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A(x) = (x+\iunit\conj{a})(x+\iunit\conj{b})} | A*(x)=(x + I*conjugate(a))*(x + I*conjugate(b)) |
A*(x)=(x + I*Conjugate[a])*(x + I*Conjugate[b]) |
Failure | Failure | Fail -2.414213562-5.414213560*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), x = 1}-6.828427124-6.828427122*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), x = 2}-13.24264068-8.242640684*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), x = 3}4.414213560-1.414213562*I <- {A = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2), x = 1}... skip entries to safe data |
Fail
Complex[-2.414213562373094, -5.414213562373095] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}Complex[-6.82842712474619, -6.82842712474619] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}Complex[-13.242640687119282, -8.242640687119286] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}Complex[4.414213562373096, -1.4142135623730947] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}... skip entries to safe data |
| 18.22#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle C(x) = (x-\iunit a)(x-\iunit b)} | C*(x)=(x - I*a)*(x - I*b) |
C*(x)=(x - I*a)*(x - I*b) |
Failure | Failure | Fail -2.414213562+8.242640684*I <- {C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), x = 1}-6.828427124+12.48528137*I <- {C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), x = 2}-13.24264068+16.72792206*I <- {C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), x = 3}4.414213560+4.242640686*I <- {C = 2^(1/2)+I*2^(1/2), a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2), x = 1}... skip entries to safe data |
Fail
Complex[-2.414213562373094, 8.242640687119286] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}Complex[-6.82842712474619, 12.48528137423857] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}Complex[-13.242640687119282, 16.72792206135786] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}Complex[-2.414213562373094, 5.414213562373095] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}... skip entries to safe data |
| 18.22.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+2n\sin@@{\phi}\,p_{n}(x) = 0} | A*(x)* p[n]*(x + I)-(A*(x)+ C*(x))* p[n]*(x)+ C*(x)* p[n]*(x - I)+ 2*n*sin(phi)*p[n]*(x)= 0 |
A*(x)* Subscript[p, n]*(x + I)-(A*(x)+ C*(x))* Subscript[p, n]*(x)+ C*(x)* Subscript[p, n]*(x - I)+ 2*n*Sin[\[Phi]]*Subscript[p, n]*(x)= 0 |
Failure | Failure | Skip | Skip |
| 18.22#Ex9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A(x) = e^{\iunit\phi}(x+\iunit\lambda)} | A*(x)= exp(I*phi)*(x + I*lambda) |
A*(x)= Exp[I*\[Phi]]*(x + I*\[Lambda]) |
Failure | Failure | Fail 1.769530126+1.460067411*I <- {A = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 1}3.145831166+2.634138542*I <- {A = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 2}4.522132206+3.808209673*I <- {A = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 3}7.425753628+2.190006980*I <- {A = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), x = 1}... skip entries to safe data |
Fail
Complex[1.7695301261666299, 1.4600674117156012] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[3.145831166712761, 2.6341385429136204] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[4.522132207258893, 3.808209674111639] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[7.425753631849798, 2.190006983651361] <- {Rule[A, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 18.22#Ex10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle C(x) = e^{-\iunit\phi}(x-\iunit\lambda)} | C*(x)= exp(- I*phi)*(x - I*lambda) |
C*(x)= Exp[- I*\[Phi]]*(x - I*\[Lambda]) |
Failure | Failure | Fail 5.611500167+12.13011774*I <- {C = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 1}6.384278266+17.60725995*I <- {C = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 2}7.157056365+23.08440217*I <- {C = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), x = 3}1.662297320+2.047585079*I <- {C = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)-I*2^(1/2), x = 1}... skip entries to safe data |
Fail
Complex[5.611500173592372, 12.13011774491578] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[6.384278274402982, 17.607259958792373] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[7.1570563752135925, 23.084402172668966] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.662297321063714, 2.0475850791686696] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϕ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 18.25.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}p_{n}(x)p_{m}(x)w(x)\diff{x} = h_{n}\Kroneckerdelta{n}{m}} | int(p[n]*(x)* p[m]*(x)* w*(x), x = 0..infinity)= h[n]*KroneckerDelta[n, m] |
Integrate[Subscript[p, n]*(x)* Subscript[p, m]*(x)* w*(x), {x, 0, Infinity}]= Subscript[h, n]*KroneckerDelta[n, m] |
Failure | Failure | Skip | Skip |
| 18.25.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(y^{2}) = \frac{1}{2y}\left|\frac{\prod_{j}\EulerGamma@{a_{j}+iy}}{\EulerGamma@{2iy}}\right|^{2}} | w*((y)^(2))=(1)/(2*y)*(abs((product(GAMMA(a[j]+ I*y), j = - infinity..infinity))/(GAMMA(2*I*y))))^(2) |
w*((y)^(2))=Divide[1,2*y]*(Abs[Divide[Product[Gamma[Subscript[a, j]+ I*y], {j, - Infinity, Infinity}],Gamma[2*I*y]]])^(2) |
Failure | Failure | Skip | Skip |
| 18.25.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h_{n} = \frac{n!\,2\pi\prod_{j<\ell}\EulerGamma@{n+a_{j}+a_{\ell}}}{(2n-1+\sum_{j}a_{j})\EulerGamma@{n-1+\sum_{j}a_{j}}}} | h[n]=(factorial(n)*2*Pi*product(GAMMA(n + a[j]+ a[ell]), j = - infinity..ell - 1))/((2*n - 1 + sum(a[j], j = - infinity..infinity))* GAMMA(n - 1 + sum(a[j], j = - infinity..infinity))) |
Subscript[h, n]=Divide[(n)!*2*Pi*Product[Gamma[n + Subscript[a, j]+ Subscript[a, \[ScriptL]]], {j, - Infinity, \[ScriptL] - 1}],(2*n - 1 + Sum[Subscript[a, j], {j, - Infinity, Infinity}])* Gamma[n - 1 + Sum[Subscript[a, j], {j, - Infinity, Infinity}]]] |
Error | Error | - | - |
| 18.25.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(y^{2}) = \frac{1}{2y}\left|\frac{\prod_{j}\EulerGamma@{a_{j}+iy}}{\EulerGamma@{2iy}}\right|^{2}} | w*((y)^(2))=(1)/(2*y)*(abs((product(GAMMA(a[j]+ I*y), j = - infinity..infinity))/(GAMMA(2*I*y))))^(2) |
w*((y)^(2))=Divide[1,2*y]*(Abs[Divide[Product[Gamma[Subscript[a, j]+ I*y], {j, - Infinity, Infinity}],Gamma[2*I*y]]])^(2) |
Failure | Failure | Skip | Skip |
| 18.25.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h_{n} = n!\,2\pi\prod_{j<\ell}\EulerGamma@{n+a_{j}+a_{\ell}}} | h[n]= factorial(n)*2*Pi*product(GAMMA(n + a[j]+ a[ell]), j = - infinity..ell - 1) |
Subscript[h, n]= (n)!*2*Pi*Product[Gamma[n + Subscript[a, j]+ Subscript[a, \[ScriptL]]], {j, - Infinity, \[ScriptL] - 1}] |
Error | Error | - | - |
| 18.25.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{y=0}^{N}p_{n}(y(y+\gamma+\delta+1))p_{m}(y(y+\gamma+\delta+1))\*\frac{\gamma+\delta+1+2y}{\gamma+\delta+1+y}\omega_{y} = h_{n}\Kroneckerdelta{n}{m}} | sum(p[n]*(y*(y + gamma + delta + 1))* p[m]*(y*(y + gamma + delta + 1))*(gamma + delta + 1 + 2*y)/(gamma + delta + 1 + y)*omega[y], y = 0..N)= h[n]*KroneckerDelta[n, m] |
Sum[Subscript[p, n]*(y*(y + \[Gamma]+ \[Delta]+ 1))* Subscript[p, m]*(y*(y + \[Gamma]+ \[Delta]+ 1))*Divide[\[Gamma]+ \[Delta]+ 1 + 2*y,\[Gamma]+ \[Delta]+ 1 + y]*Subscript[\[Omega], y], {y, 0, N}]= Subscript[h, n]*KroneckerDelta[n, m] |
Failure | Failure | Skip | Skip |
| 18.25.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \omega_{y} = \frac{\Pochhammersym{\alpha+1}{y}\Pochhammersym{\beta+\delta+1}{y}\Pochhammersym{\gamma+1}{y}\Pochhammersym{\gamma+\delta+2}{y}}{\Pochhammersym{-\alpha+\gamma+\delta+1}{y}\Pochhammersym{-\beta+\gamma+1}{y}\Pochhammersym{\delta+1}{y}y!}} | omega[y]=(pochhammer(alpha + 1, y)*pochhammer(beta + delta + 1, y)*pochhammer(gamma + 1, y)*pochhammer(gamma + delta + 2, y))/(pochhammer(- alpha + gamma + delta + 1, y)*pochhammer(- beta + gamma + 1, y)*pochhammer(delta + 1, y)*factorial(y)) |
Subscript[\[Omega], y]=Divide[Pochhammer[\[Alpha]+ 1, y]*Pochhammer[\[Beta]+ \[Delta]+ 1, y]*Pochhammer[\[Gamma]+ 1, y]*Pochhammer[\[Gamma]+ \[Delta]+ 2, y],Pochhammer[- \[Alpha]+ \[Gamma]+ \[Delta]+ 1, y]*Pochhammer[- \[Beta]+ \[Gamma]+ 1, y]*Pochhammer[\[Delta]+ 1, y]*(y)!] |
Failure | Failure | Fail 12.16329441-7.801522345*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), omega[y] = 2^(1/2)+I*2^(1/2), y = 1}63.95461927+94.62189224*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), omega[y] = 2^(1/2)+I*2^(1/2), y = 2}-357.9031707+453.3958173*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), omega[y] = 2^(1/2)+I*2^(1/2), y = 3}12.16329441-10.62994947*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), omega[y] = 2^(1/2)-I*2^(1/2), y = 1}... skip entries to safe data |
Skip |
| 18.25.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h_{n} = \frac{\Pochhammersym{-\beta}{N}\Pochhammersym{\gamma+\delta+2}{N}}{\Pochhammersym{-\beta+\gamma+1}{N}\Pochhammersym{\delta+1}{N}}\frac{\Pochhammersym{n+\alpha+\beta+1}{n}n!}{\Pochhammersym{\alpha+\beta+2}{2n}}\*\frac{\Pochhammersym{\alpha+\beta-\gamma+1}{n}\Pochhammersym{\alpha-\delta+1}{n}\Pochhammersym{\beta+1}{n}}{\Pochhammersym{\alpha+1}{n}\Pochhammersym{\beta+\delta+1}{n}\Pochhammersym{\gamma+1}{n}}} | h[n]=(pochhammer(- beta, N)*pochhammer(gamma + delta + 2, N))/(pochhammer(- beta + gamma + 1, N)*pochhammer(delta + 1, N))*(pochhammer(n + alpha + beta + 1, n)*factorial(n))/(pochhammer(alpha + beta + 2, 2*n))*(pochhammer(alpha + beta - gamma + 1, n)*pochhammer(alpha - delta + 1, n)*pochhammer(beta + 1, n))/(pochhammer(alpha + 1, n)*pochhammer(beta + delta + 1, n)*pochhammer(gamma + 1, n)) |
Subscript[h, n]=Divide[Pochhammer[- \[Beta], N]*Pochhammer[\[Gamma]+ \[Delta]+ 2, N],Pochhammer[- \[Beta]+ \[Gamma]+ 1, N]*Pochhammer[\[Delta]+ 1, N]]*Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, n]*(n)!,Pochhammer[\[Alpha]+ \[Beta]+ 2, 2*n]]*Divide[Pochhammer[\[Alpha]+ \[Beta]- \[Gamma]+ 1, n]*Pochhammer[\[Alpha]- \[Delta]+ 1, n]*Pochhammer[\[Beta]+ 1, n],Pochhammer[\[Alpha]+ 1, n]*Pochhammer[\[Beta]+ \[Delta]+ 1, n]*Pochhammer[\[Gamma]+ 1, n]] |
Failure | Failure | Skip | Skip |
| 18.25.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \omega_{y} = \frac{(-1)^{y}\Pochhammersym{-N}{y}\Pochhammersym{\gamma+1}{y}\Pochhammersym{\gamma+\delta+1}{2}}{\Pochhammersym{N+\gamma+\delta+2}{y}\Pochhammersym{\delta+1}{y}y!}} | omega[y]=((- 1)^(y)* pochhammer(- N, y)*pochhammer(gamma + 1, y)*pochhammer(gamma + delta + 1, 2))/(pochhammer(N + gamma + delta + 2, y)*pochhammer(delta + 1, y)*factorial(y)) |
Subscript[\[Omega], y]=Divide[(- 1)^(y)* Pochhammer[- N, y]*Pochhammer[\[Gamma]+ 1, y]*Pochhammer[\[Gamma]+ \[Delta]+ 1, 2],Pochhammer[N + \[Gamma]+ \[Delta]+ 2, y]*Pochhammer[\[Delta]+ 1, y]*(y)!] |
Failure | Failure | Fail -.785651684+.48581315e-1*I <- {N = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), omega[y] = 2^(1/2)+I*2^(1/2), y = 1}1.316871147+1.251035041*I <- {N = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), omega[y] = 2^(1/2)+I*2^(1/2), y = 2}1.420652225+1.407312490*I <- {N = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), omega[y] = 2^(1/2)+I*2^(1/2), y = 3}-.785651684-2.779845809*I <- {N = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), omega[y] = 2^(1/2)-I*2^(1/2), y = 1}... skip entries to safe data |
Skip |
| 18.25.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h_{n} = \frac{n!\,(N-n)!\,\Pochhammersym{\gamma+\delta+2}{N}}{N!\,\Pochhammersym{\gamma+1}{n}\Pochhammersym{\delta+1}{N-n}}} | h[n]=(factorial(n)*factorial(N - n)*pochhammer(gamma + delta + 2, N))/(factorial(N)*pochhammer(gamma + 1, n)*pochhammer(delta + 1, N - n)) |
Subscript[h, n]=Divide[(n)!*(N - n)!*Pochhammer[\[Gamma]+ \[Delta]+ 2, N],(N)!*Pochhammer[\[Gamma]+ 1, n]*Pochhammer[\[Delta]+ 1, N - n]] |
Failure | Failure | Fail -1.311749265+.9066027273*I <- {N = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)+I*2^(1/2), n = 1}-3.476693885+1.928884423*I <- {N = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)+I*2^(1/2), n = 2}-4.217343867+7.012858818*I <- {N = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)+I*2^(1/2), n = 3}-1.311749265-1.921824397*I <- {N = 2^(1/2)+I*2^(1/2), delta = 2^(1/2)+I*2^(1/2), h[n] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Skip |
| 18.28.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)\diff{x} = h_{n}\Kroneckerdelta{n}{m}} | int(p[n]*(x)* p[m]*(x)* w*(x), x = - 1..1)= h[n]*KroneckerDelta[n, m] |
Integrate[Subscript[p, n]*(x)* Subscript[p, m]*(x)* w*(x), {x, - 1, 1}]= Subscript[h, n]*KroneckerDelta[n, m] |
Failure | Failure | Skip | Successful |
| 18.28.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-1}^{1}p_{n}(x)p_{m}(x)w(x)\diff{x}+\sum_{\ell}p_{n}(x_{\ell})p_{m}(x_{\ell})\omega_{\ell} = h_{n}\Kroneckerdelta{n}{m}} | int(p[n]*(x)* p[m]*(x)* w*(x), x = - 1..1)+ sum(p[n]*(x[ell])* p[m]*(x[ell])* omega[ell], ell = - infinity..infinity)= h[n]*KroneckerDelta[n, m] |
Integrate[Subscript[p, n]*(x)* Subscript[p, m]*(x)* w*(x), {x, - 1, 1}]+ Sum[Subscript[p, n]*(Subscript[x, \[ScriptL]])* Subscript[p, m]*(Subscript[x, \[ScriptL]])* Subscript[\[Omega], \[ScriptL]], {\[ScriptL], - Infinity, Infinity}]= Subscript[h, n]*KroneckerDelta[n, m] |
Failure | Failure | Skip | Error |
| 18.28.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < q} | 0 < q |
0 < q |
Failure | Failure | Successful | Successful |
| 18.28.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q < 1,a,b\in\Reals,ab} | q < 1 , a , b in real , a*b |
q < 1 , a , b \[Element]*Reals , a*b |
Failure | Failure | Successful | Error |
| 18.28.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1,a,b\in\Reals,ab > 1,a^{-1}b} | 1 , a , b in real , a*b > 1 , (a)^(- 1)* b |
1 , a , b \[Element]*Reals , a*b > 1 , (a)^(- 1)* b |
Error | Failure | - | Error |
| 18.28.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1,a^{-1}b < q^{-1}} | 1 , (a)^(- 1)* b < (q)^(- 1) |
1 , (a)^(- 1)* b < (q)^(- 1) |
Failure | Failure | Skip | - |
| 18.28.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < q} | 0 < q |
0 < q |
Failure | Failure | Successful | Successful |
| 18.28.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q < 1,\ifrac{a}{\iunit},\ifrac{b}{\iunit}\in\Reals,(\imagpart@@{a})(\imagpart@@{b})} | q < 1 ,(a)/(I),(b)/(I)in real ,(Im(a))*(Im(b)) |
q < 1 ,Divide[a,I],Divide[b,I]\[Element]*Reals ,(Im[a])*(Im[b]) |
Failure | Failure | Successful | Error |
| 18.28.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1,\ifrac{a}{\iunit},\ifrac{b}{\iunit}\in\Reals,(\imagpart@@{a})(\imagpart@@{b}) > 0,a^{-1}b} | 1 ,(a)/(I),(b)/(I)in real ,(Im(a))*(Im(b))> 0 , (a)^(- 1)* b |
1 ,Divide[a,I],Divide[b,I]\[Element]*Reals ,(Im[a])*(Im[b])> 0 , (a)^(- 1)* b |
Error | Failure | - | Error |
| 18.28.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0,a^{-1}b < q^{-1}} | 0 , (a)^(- 1)* b < (q)^(- 1) |
0 , (a)^(- 1)* b < (q)^(- 1) |
Failure | Failure | Skip | - |
| 18.28.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{y=0}^{N}R_{n}(q^{-y}+\gamma\delta q^{y+1})R_{m}(q^{-y}+\gamma\delta q^{y+1})\omega_{y} = h_{n}\Kroneckerdelta{n}{m}} | sum(R[n]*((q)^(- y)+ gamma*delta*(q)^(y + 1))* R[m]*((q)^(- y)+ gamma*delta*(q)^(y + 1))* omega[y], y = 0..N)= h[n]*KroneckerDelta[n, m] |
Sum[Subscript[R, n]*((q)^(- y)+ \[Gamma]*\[Delta]*(q)^(y + 1))* Subscript[R, m]*((q)^(- y)+ \[Gamma]*\[Delta]*(q)^(y + 1))* Subscript[\[Omega], y], {y, 0, N}]= Subscript[h, n]*KroneckerDelta[n, m] |
Failure | Failure | Skip | Skip |
| 18.33.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi_{n}^{*}(z) = z^{n}\conj{\phi_{n}(\conj{z}^{-1})}} | (phi[n])^(*)*(z)= (z)^(n)* conjugate(phi[n]*((conjugate(z))^(- 1))) |
(Subscript[\[Phi], n])^(*)*(z)= (z)^(n)* Conjugate[Subscript[\[Phi], n]*((Conjugate[z])^(- 1))] |
Error | Failure | - | Error |
| 18.33.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z^{n}\conj{\phi_{n}(\conj{z}^{-1})} = {\kappa_{n}}+\sum_{\ell=1}^{n}\conj{\kappa}_{n,n-\ell}z^{\ell}} | (z)^(n)* conjugate(phi[n]*((conjugate(z))^(- 1)))=kappa[n]+ sum(conjugate(kappa)[n , n - ell]*(z)^(ell), ell = 1..n) |
(z)^(n)* Conjugate[Subscript[\[Phi], n]*((Conjugate[z])^(- 1))]=Subscript[\[Kappa], n]+ Sum[Subscript[Conjugate[\[Kappa]], n , n - \[ScriptL]]*(z)^(\[ScriptL]), {\[ScriptL], 1, n}] |
Failure | Failure | Skip | Skip |
| 18.33#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w_{1}(x) = (1-x^{2})^{-\frac{1}{2}}w\left(x+\iunit(1-x^{2})^{\frac{1}{2}}\right)} | w[1]*(x)=(1 - (x)^(2))^(-(1)/(2))* w*(x + I*(1 - (x)^(2))^((1)/(2))) |
Subscript[w, 1]*(x)=(1 - (x)^(2))^(-Divide[1,2])* w*(x + I*(1 - (x)^(2))^(Divide[1,2])) |
Failure | Failure | Fail Float(infinity)+Float(infinity)*I <- {w = 2^(1/2)+I*2^(1/2), w[1] = 2^(1/2)+I*2^(1/2), x = 1}2.609647525+3.047206723*I <- {w = 2^(1/2)+I*2^(1/2), w[1] = 2^(1/2)+I*2^(1/2), x = 2}4.156854249+4.328427123*I <- {w = 2^(1/2)+I*2^(1/2), w[1] = 2^(1/2)+I*2^(1/2), x = 3}Float(infinity)+Float(infinity)*I <- {w = 2^(1/2)+I*2^(1/2), w[1] = 2^(1/2)-I*2^(1/2), x = 1}... skip entries to safe data |
Fail
DirectedInfinity[] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}DirectedInfinity[] <- {Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}DirectedInfinity[] <- {Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}DirectedInfinity[] <- {Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]} |
| 18.33#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w_{2}(x) = (1-x^{2})^{\frac{1}{2}}w\left(x+\iunit(1-x^{2})^{\frac{1}{2}}\right)} | w[2]*(x)=(1 - (x)^(2))^((1)/(2))* w*(x + I*(1 - (x)^(2))^((1)/(2))) |
Subscript[w, 2]*(x)=(1 - (x)^(2))^(Divide[1,2])* w*(x + I*(1 - (x)^(2))^(Divide[1,2])) |
Failure | Failure | Fail 1.414213562+1.414213562*I <- {w = 2^(1/2)+I*2^(1/2), w[2] = 2^(1/2)+I*2^(1/2), x = 1}3.484765921+2.172088327*I <- {w = 2^(1/2)+I*2^(1/2), w[2] = 2^(1/2)+I*2^(1/2), x = 2}4.928932186+3.556349186*I <- {w = 2^(1/2)+I*2^(1/2), w[2] = 2^(1/2)+I*2^(1/2), x = 3}1.414213562-1.414213562*I <- {w = 2^(1/2)+I*2^(1/2), w[2] = 2^(1/2)-I*2^(1/2), x = 1}... skip entries to safe data |
Successful |
| 18.33.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi_{n}(z) = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+1}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\,(n-\ell)!}\,z^{\ell}} | phi[n]*(z)= sum((pochhammer(lambda + 1, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*(z)^(ell), ell = 0..n) |
Subscript[\[Phi], n]*(z)= Sum[Divide[Pochhammer[\[Lambda]+ 1, \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(z)^(\[ScriptL]), {\[ScriptL], 0, n}] |
Failure | Failure | Skip | Skip |
| 18.33.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+1}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\,(n-\ell)!}\,z^{\ell} = \frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,\lambda+1}{-\lambda-n+1}{z}} | sum((pochhammer(lambda + 1, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*(z)^(ell), ell = 0..n)=(pochhammer(lambda, n))/(factorial(n))*hypergeom([- n , lambda + 1], [- lambda - n + 1], z) |
Sum[Divide[Pochhammer[\[Lambda]+ 1, \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(z)^(\[ScriptL]), {\[ScriptL], 0, n}]=Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , \[Lambda]+ 1}, {- \[Lambda]- n + 1}, z] |
Successful | Successful | - | - |
| 18.34.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Besselpolyy{n}@{x}{a} = \genhyperF{2}{0}@@{-n,n+a-1}{-}{-\frac{x}{2}}} | Error |
Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]= HypergeometricPFQ[{- n , n + a - 1}, {-}, -Divide[x,2]] |
Error | Failure | - | Error |
| 18.34.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \genhyperF{2}{0}@@{-n,n+a-1}{-}{-\frac{x}{2}} = \Pochhammersym{n+a-1}{n}\left(\frac{x}{2}\right)^{n}\genhyperF{1}{1}@@{-n}{-2n-a+2}{\frac{2}{x}}} | hypergeom([- n , n + a - 1], [-], -(x)/(2))= pochhammer(n + a - 1, n)*((x)/(2))^(n)* hypergeom([- n], [- 2*n - a + 2], (2)/(x)) |
HypergeometricPFQ[{- n , n + a - 1}, {-}, -Divide[x,2]]= Pochhammer[n + a - 1, n]*(Divide[x,2])^(n)* HypergeometricPFQ[{- n}, {- 2*n - a + 2}, Divide[2,x]] |
Error | Failure | - | Error |
| 18.34#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y_{n}(x) = \Besselpolyy{n}@{x}{2}} | Error |
Subscript[y, n]*(x)= Pochhammer[n + 2 - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - 2 + 2, 2/x] |
Error | Failure | - | Successful |
| 18.34.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Besselpolyy{n+1}@{x}{a} = (A_{n}x+B_{n})\Besselpolyy{n}@{x}{a}-C_{n}\Besselpolyy{n-1}@{x}{a}} | Error |
Pochhammer[n + 1 + a - 1, n + 1] (x/2)^n + 1 Hypergeometric1F1[-n + 1, -2 n + 1 - a + 2, 2/x]=(Subscript[A, n]*x + Subscript[B, n])* Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]- Subscript[C, n]*Pochhammer[n - 1 + a - 1, n - 1] (x/2)^n - 1 Hypergeometric1F1[-n - 1, -2 n - 1 - a + 2, 2/x] |
Error | Failure | - | Skip |
| 18.34.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x^{2}\Besselpolyy{n}''@{x}{a}+(ax+2)\Besselpolyy{n}'@{x}{a}-n(n+a-1)\Besselpolyy{n}@{x}{a} = 0} | Error |
(x)^(2)* (D[Pochhammer[n + a - 1, n] (temp/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/temp], {temp, 2}]/.temp-> x)+(a*x + 2)* (D[Pochhammer[n + a - 1, n] (temp/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/temp], {temp, 1}]/.temp-> x)- n*(n + a - 1)* Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]= 0 |
Error | Successful | - | - |
| 18.34.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{\alpha\to\infty}\frac{\JacobipolyP{\alpha}{a-\alpha-2}{n}@{1+\alpha x}}{\JacobipolyP{\alpha}{a-\alpha-2}{n}@{1}} = \Besselpolyy{n}@{x}{a}} | Error |
Limit[Divide[JacobiP[n, \[Alpha], a - \[Alpha]- 2, 1 + \[Alpha]*x],JacobiP[n, \[Alpha], a - \[Alpha]- 2, 1]], \[Alpha] -> Infinity]= Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x] |
Error | Failure | - | Error |
| 18.35.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n}}{n!}e^{\iunit n\theta}\*\genhyperF{2}{1}@@{-n,\lambda+\iunit\tau_{a,b}(\theta)}{-n-\lambda+1+\iunit\tau_{a,b}(\theta)}{e^{-2\iunit\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+\iunit\tau_{a,b}(\theta)}{\ell}}{\ell!}\frac{\Pochhammersym{\lambda-\iunit\tau_{a,b}(\theta)}{n-\ell}}{(n-\ell)!}e^{\iunit(n-2\ell)\theta}} | (pochhammer(lambda - I*tau[a , b]*(theta), n))/(factorial(n))*exp(I*n*theta)* hypergeom([- n , lambda + I*tau[a , b]*(theta)], [- n - lambda + 1 + I*tau[a , b]*(theta)], exp(- 2*I*theta))= sum((pochhammer(lambda + I*tau[a , b]*(theta), ell))/(factorial(ell))*(pochhammer(lambda - I*tau[a , b]*(theta), n - ell))/(factorial(n - ell))*exp(I*(n - 2*ell)* theta), ell = 0..n) |
Divide[Pochhammer[\[Lambda]- I*Subscript[\[Tau], a , b]*(\[Theta]), n],(n)!]*Exp[I*n*\[Theta]]* HypergeometricPFQ[{- n , \[Lambda]+ I*Subscript[\[Tau], a , b]*(\[Theta])}, {- n - \[Lambda]+ 1 + I*Subscript[\[Tau], a , b]*(\[Theta])}, Exp[- 2*I*\[Theta]]]= Sum[Divide[Pochhammer[\[Lambda]+ I*Subscript[\[Tau], a , b]*(\[Theta]), \[ScriptL]],(\[ScriptL])!]*Divide[Pochhammer[\[Lambda]- I*Subscript[\[Tau], a , b]*(\[Theta]), n - \[ScriptL]],(n - \[ScriptL])!]*Exp[I*(n - 2*\[ScriptL])* \[Theta]], {\[ScriptL], 0, n}] |
Successful | Successful | - | - |
| 18.38.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle V_{n}(x) = \ifrac{2n\HermitepolyH{n+1}@{x}\HermitepolyH{n-1}@{x}}{(\HermitepolyH{n}@{x})^{2}}} | V[n]*(x)=(2*n*HermiteH(n + 1, x)*HermiteH(n - 1, x))/((HermiteH(n, x))^(2)) |
Subscript[V, n]*(x)=Divide[2*n*HermiteH[n + 1, x]*HermiteH[n - 1, x],(HermiteH[n, x])^(2)] |
Failure | Failure | Fail .414213562+1.414213562*I <- {V[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 1}1.078427124+2.828427124*I <- {V[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 2}2.353751797+4.242640686*I <- {V[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 3}9.414213562+1.414213562*I <- {V[n] = 2^(1/2)+I*2^(1/2), n = 2, x = 1}... skip entries to safe data |
Skip |
| 18.38.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{m=0}^{n}\JacobipolyP{\alpha}{0}{m}@{x} >= 0} | sum(JacobiP(m, alpha, 0, x), m = 0..n)> = 0 |
Sum[JacobiP[m, \[Alpha], 0, x], {m, 0, n}]> = 0 |
Failure | Failure | Skip | Successful |
| 18.39.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \eta_{n}(x) = \pi^{-\frac{1}{4}}2^{-\frac{1}{2}n}(n!\,b)^{-\frac{1}{2}}\HermitepolyH{n}@{x/b}e^{-x^{2}/2b^{2}}} | eta[n]*(x)= (Pi)^(-(1)/(4))* (2)^(-(1)/(2)*n)*(factorial(n)*b)^(-(1)/(2))* HermiteH(n, x/ b)*exp(- (x)^(2)/ 2*(b)^(2)) |
Subscript[\[Eta], n]*(x)= (Pi)^(-Divide[1,4])* (2)^(-Divide[1,2]*n)*((n)!*b)^(-Divide[1,2])* HermiteH[n, x/ b]*Exp[- (x)^(2)/ 2*(b)^(2)] |
Failure | Failure | Fail 1.789526128+1.400506842*I <- {b = 2^(1/2)+I*2^(1/2), eta[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 1}3.556814811+3.011841849*I <- {b = 2^(1/2)+I*2^(1/2), eta[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 2}3.176214405+4.606180986*I <- {b = 2^(1/2)+I*2^(1/2), eta[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 3}1.266982360+1.020980529*I <- {b = 2^(1/2)+I*2^(1/2), eta[n] = 2^(1/2)+I*2^(1/2), n = 2, x = 1}... skip entries to safe data |
Successful |