Results of Zeta and Related Functions
Jump to navigation
Jump to search
| DLMF | Formula | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|
| 25.2.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \sum_{n=1}^{\infty}\frac{1}{n^{s}}} | Zeta(s)= sum((1)/((n)^(s)), n = 1..infinity) |
Zeta[s]= Sum[Divide[1,(n)^(s)], {n, 1, Infinity}] |
Failure | Successful | Skip | - |
| 25.2.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}} | Zeta(s)=(1)/(1 - (2)^(- s))*sum((1)/((2*n + 1)^(s)), n = 0..infinity) |
Zeta[s]=Divide[1,1 - (2)^(- s)]*Sum[Divide[1,(2*n + 1)^(s)], {n, 0, Infinity}] |
Successful | Successful | - | - |
| 25.2.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}} | Zeta(s)=(1)/(1 - (2)^(1 - s))*sum(((- 1)^(n - 1))/((n)^(s)), n = 1..infinity) |
Zeta[s]=Divide[1,1 - (2)^(1 - s)]*Sum[Divide[(- 1)^(n - 1),(n)^(s)], {n, 1, Infinity}] |
Failure | Successful | Skip | - |
| 25.2.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma_{n}(s-1)^{n}} | Zeta(s)=(1)/(s - 1)+ sum(((- 1)^(n))/(factorial(n))*gamma[n]*(s - 1)^(n), n = 0..infinity) |
Zeta[s]=Divide[1,s - 1]+ Sum[Divide[(- 1)^(n),(n)!]*Subscript[\[Gamma], n]*(s - 1)^(n), {n, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 25.2.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta'@{s} = -\sum_{n=2}^{\infty}(\ln@@{n})n^{-s}} | subs( temp=s, diff( Zeta(temp), temp$(1) ) )= - sum((ln(n))* (n)^(- s), n = 2..infinity) |
(D[Zeta[temp], {temp, 1}]/.temp-> s)= - Sum[(Log[n])* (n)^(- s), {n, 2, Infinity}] |
Successful | Successful | - | - |
| 25.2.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta^{(k)}@{s} = (-1)^{k}\sum_{n=2}^{\infty}(\ln@@{n})^{k}n^{-s}} | subs( temp=s, diff( Zeta(temp), temp$(k) ) )=(- 1)^(k)* sum((ln(n))^(k)* (n)^(- s), n = 2..infinity) |
(D[Zeta[temp], {temp, k}]/.temp-> s)=(- 1)^(k)* Sum[(Log[n])^(k)* (n)^(- s), {n, 2, Infinity}] |
Failure | Failure | Skip | Successful |
| 25.2.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{x^{s+1}}\diff{x}} | Zeta(s)= sum((1)/((k)^(s)), k = 1..N)+((N)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x)^(s + 1)), x = N..infinity) |
Zeta[s]= Sum[Divide[1,(k)^(s)], {k, 1, N}]+Divide[(N)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x)^(s + 1)], {x, N, Infinity}] |
Failure | Failure | Skip | Successful |
| 25.2.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \prod_{p}(1-p^{-s})^{-1}} | Zeta(s)= product((1 - (p)^(- s))^(- 1), p = - infinity..infinity) |
Zeta[s]= Product[(1 - (p)^(- s))^(- 1), {p, - Infinity, Infinity}] |
Failure | Failure | Skip | - |
| 25.2.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{(2\pi)^{s}e^{-s-(\EulerConstant s/2)}}{2(s-1)\EulerGamma@{\tfrac{1}{2}s+1}}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho}} | Zeta(s)=((2*Pi)^(s)* exp(- s -(gamma*s/ 2)))/(2*(s - 1)* GAMMA((1)/(2)*s + 1))*product((1 -(s)/(rho))* exp(s/ rho), rho = - infinity..infinity) |
Zeta[s]=Divide[(2*Pi)^(s)* Exp[- s -(EulerGamma*s/ 2)],2*(s - 1)* Gamma[Divide[1,2]*s + 1]]*Product[(1 -Divide[s,\[Rho]])* Exp[s/ \[Rho]], {\[Rho], - Infinity, Infinity}] |
Failure | Failure | Skip | Skip |
| 25.4.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{1-s} = 2(2\pi)^{-s}\cos@{\tfrac{1}{2}\pi s}\EulerGamma@{s}\Riemannzeta@{s}} | Zeta(1 - s)= 2*(2*Pi)^(- s)* cos((1)/(2)*Pi*s)*GAMMA(s)*Zeta(s) |
Zeta[1 - s]= 2*(2*Pi)^(- s)* Cos[Divide[1,2]*Pi*s]*Gamma[s]*Zeta[s] |
Failure | Successful | Successful | - |
| 25.4.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = 2(2\pi)^{s-1}\sin@{\tfrac{1}{2}\pi s}\EulerGamma@{1-s}\Riemannzeta@{1-s}} | Zeta(s)= 2*(2*Pi)^(s - 1)* sin((1)/(2)*Pi*s)*GAMMA(1 - s)*Zeta(1 - s) |
Zeta[s]= 2*(2*Pi)^(s - 1)* Sin[Divide[1,2]*Pi*s]*Gamma[1 - s]*Zeta[1 - s] |
Failure | Successful | Successful | - |
| 25.4.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannxi@{s} = \Riemannxi@{1-s}} | (s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 = (1 - s)*(1 - s-1)*GAMMA((1 - s)/2)*Pi^(-(1 - s)/2)*Zeta(1 - s)/2 |
Error |
Failure | Error | Successful | - |
| 25.4.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannxi@{s} = \tfrac{1}{2}s(s-1)\EulerGamma@{\tfrac{1}{2}s}\pi^{-s/2}\Riemannzeta@{s}} | (s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 =(1)/(2)*s*(s - 1)* GAMMA((1)/(2)*s)*(Pi)^(- s/ 2)* Zeta(s) |
Error |
Successful | Error | - | - |
| 25.4.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{k}\Riemannzeta^{(k)}@{1-s} = \frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\left(\realpart@@{(c^{k-m})}\cos@{\tfrac{1}{2}\pi s}+\imagpart@@{(c^{k-m})}\sin@{\tfrac{1}{2}\pi s}\right)\EulerGamma^{(r)}@{s}\Riemannzeta^{(m-r)}@{s}} | (- 1)^(k)* subs( temp=1 - s, diff( Zeta(temp), temp$(k) ) )=(2)/((2*Pi)^(s))*sum(sum(binomial(k,m)*binomial(m,r)*(Re((c)^(k - m))*cos((1)/(2)*Pi*s)+ Im((c)^(k - m))*sin((1)/(2)*Pi*s))* subs( temp=s, diff( GAMMA(temp), temp$(r) ) )*subs( temp=s, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k) |
(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - s)=Divide[2,(2*Pi)^(s)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*(Re[(c)^(k - m)]*Cos[Divide[1,2]*Pi*s]+ Im[(c)^(k - m)]*Sin[Divide[1,2]*Pi*s])* (D[Gamma[temp], {temp, r}]/.temp-> s)*(D[Zeta[temp], {temp, m - r}]/.temp-> s), {r, 0, m}], {m, 0, k}] |
Failure | Failure | Skip | Skip |
| 25.4.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c = -\ln@{2\pi}-\tfrac{1}{2}\pi\iunit} | c = - ln(2*Pi)-(1)/(2)*Pi*I |
c = - Log[2*Pi]-Divide[1,2]*Pi*I |
Failure | Failure | Fail 3.252090629+2.985009889*I <- {c = 2^(1/2)+I*2^(1/2)}3.252090629+.156582765*I <- {c = 2^(1/2)-I*2^(1/2)}.423663505+.156582765*I <- {c = -2^(1/2)-I*2^(1/2)}.423663505+2.985009889*I <- {c = -2^(1/2)+I*2^(1/2)} |
Fail
Complex[3.2520906287824403, 2.9850098891679915] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[3.2520906287824403, 0.1565827644218014] <- {Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.4236635040362502, 0.1565827644218014] <- {Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[0.4236635040362502, 2.9850098891679915] <- {Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 25.5.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\diff{x}} | Zeta(s)=(1)/(GAMMA(s))*int(((x)^(s - 1))/(exp(x)- 1), x = 0..infinity) |
Zeta[s]=Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]- 1], {x, 0, Infinity}] |
Failure | Successful | Skip | - |
| 25.5.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}-1)^{2}}\diff{x}} | Zeta(s)=(1)/(GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)- 1)^(2)), x = 0..infinity) |
Zeta[s]=Divide[1,Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]- 1)^(2)], {x, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 25.5.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{(1-2^{1-s})\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}+1}\diff{x}} | Zeta(s)=(1)/((1 - (2)^(1 - s))* GAMMA(s))*int(((x)^(s - 1))/(exp(x)+ 1), x = 0..infinity) |
Zeta[s]=Divide[1,(1 - (2)^(1 - s))* Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]+ 1], {x, 0, Infinity}] |
Failure | Successful | Skip | - |
| 25.5.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{(1-2^{1-s})\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\diff{x}} | Zeta(s)=(1)/((1 - (2)^(1 - s))* GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)+ 1)^(2)), x = 0..infinity) |
Zeta[s]=Divide[1,(1 - (2)^(1 - s))* Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]+ 1)^(2)], {x, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 25.5.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = -s\int_{0}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{x^{s+1}}\diff{x}} | Zeta(s)= - s*int((x - floor(x)-(1)/(2))/((x)^(s + 1)), x = 0..infinity) |
Zeta[s]= - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x)^(s + 1)], {x, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 25.5.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{2}+\frac{1}{s-1}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{x}}\diff{x}} | Zeta(s)=(1)/(2)+(1)/(s - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2))*((x)^(s - 1))/(exp(x)), x = 0..infinity) |
Zeta[s]=Divide[1,2]+Divide[1,s - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 25.5.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{\BernoullinumberB{2m}}{(2m)!}\Pochhammersym{s}{2m-1}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{\BernoullinumberB{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\diff{x}} | Zeta(s)=(1)/(2)+(1)/(s - 1)+ sum((bernoulli(2*m))/(factorial(2*m))*pochhammer(s, 2*m - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2)- sum((bernoulli(2*m))/(factorial(2*m))*(x)^(2*m - 1), m = 1..n))*((x)^(s - 1))/(exp(x)), x = 0..infinity), m = 1..n) |
Zeta[s]=Divide[1,2]+Divide[1,s - 1]+ Sum[Divide[BernoulliB[2*m],(2*m)!]*Pochhammer[s, 2*m - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2]- Sum[Divide[BernoulliB[2*m],(2*m)!]*(x)^(2*m - 1), {m, 1, n}])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}], {m, 1, n}] |
Failure | Failure | Skip | Error |
| 25.5.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{2(1-2^{-s})\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh@@{x}}\diff{x}} | Zeta(s)=(1)/(2*(1 - (2)^(- s))* GAMMA(s))*int(((x)^(s - 1))/(sinh(x)), x = 0..infinity) |
Zeta[s]=Divide[1,2*(1 - (2)^(- s))* Gamma[s]]*Integrate[Divide[(x)^(s - 1),Sinh[x]], {x, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 25.5.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{2^{s-1}}{\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{x^{s}}{(\sinh@@{x})^{2}}\diff{x}} | Zeta(s)=((2)^(s - 1))/(GAMMA(s + 1))*int(((x)^(s))/((sinh(x))^(2)), x = 0..infinity) |
Zeta[s]=Divide[(2)^(s - 1),Gamma[s + 1]]*Integrate[Divide[(x)^(s),(Sinh[x])^(2)], {x, 0, Infinity}] |
Failure | Failure | Skip | - |
| 25.5.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos@{s\atan@@{x}}}{(1+x^{2})^{s/2}\cosh@{\frac{1}{2}\pi x}}\diff{x}} | Zeta(s)=((2)^(s - 1))/(1 - (2)^(1 - s))*int((cos(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)* cosh((1)/(2)*Pi*x)), x = 0..infinity) |
Zeta[s]=Divide[(2)^(s - 1),1 - (2)^(1 - s)]*Integrate[Divide[Cos[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)* Cosh[Divide[1,2]*Pi*x]], {x, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 25.5.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{2}+\frac{1}{s-1}+2\int_{0}^{\infty}\frac{\sin@{s\atan@@{x}}}{(1+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}} | Zeta(s)=(1)/(2)+(1)/(s - 1)+ 2*int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity) |
Zeta[s]=Divide[1,2]+Divide[1,s - 1]+ 2*Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}] |
Failure | Successful | Skip | - |
| 25.5.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty}\frac{\sin@{s\atan@@{x}}}{(1+x^{2})^{s/2}(e^{\pi x}+1)}\diff{x}} | Zeta(s)=((2)^(s - 1))/(s - 1)- (2)^(s)* int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)*(exp(Pi*x)+ 1)), x = 0..infinity) |
Zeta[s]=Divide[(2)^(s - 1),s - 1]- (2)^(s)* Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)*(Exp[Pi*x]+ 1)], {x, 0, Infinity}] |
Failure | Successful | Skip | - |
| 25.5.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{\pi^{s/2}}{s(s-1)\EulerGamma@{\frac{1}{2}s}}+\frac{\pi^{s/2}}{\EulerGamma@{\frac{1}{2}s}}\*\int_{1}^{\infty}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega(x)}{x}\diff{x}} | Zeta(s)=((Pi)^(s/ 2))/(s*(s - 1)* GAMMA((1)/(2)*s))+((Pi)^(s/ 2))/(GAMMA((1)/(2)*s))* int(((x)^(s/ 2)+ (x)^((1 - s)/ 2))*(omega*(x))/(x), x = 1..infinity) |
Zeta[s]=Divide[(Pi)^(s/ 2),s*(s - 1)* Gamma[Divide[1,2]*s]]+Divide[(Pi)^(s/ 2),Gamma[Divide[1,2]*s]]* Integrate[((x)^(s/ 2)+ (x)^((1 - s)/ 2))*Divide[\[Omega]*(x),x], {x, 1, Infinity}] |
Failure | Failure | Skip | Skip |
| 25.5.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \omega(x) = \sum_{n=1}^{\infty}e^{-n^{2}\pi x}} | omega*(x)= sum(exp(- (n)^(2)* Pi*x), n = 1..infinity) |
\[Omega]*(x)= Sum[Exp[- (n)^(2)* Pi*x], {n, 1, Infinity}] |
Failure | Failure | Skip | Fail
Complex[1.370996156766441, 1.4142135623730951] <- {Rule[x, 1], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[2.826559682002321, 2.8284271247461903] <- {Rule[x, 2], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[4.242559987601715, 4.242640687119286] <- {Rule[x, 3], Rule[ω, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.370996156766441, -1.4142135623730951] <- {Rule[x, 1], Rule[ω, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 25.5.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}e^{-n^{2}\pi x} = \frac{1}{2}\left(\Jacobithetatau{3}@{0}{ix}-1\right)} | sum(exp(- (n)^(2)* Pi*x), n = 1..infinity)=(1)/(2)*(JacobiTheta3(0,exp(I*Pi*I*x))- 1) |
Sum[Exp[- (n)^(2)* Pi*x], {n, 1, Infinity}]=Divide[1,2]*(EllipticTheta[3, 0, I*x]- 1) |
Failure | Failure | Skip | Successful |
| 25.5.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{s-1}+\frac{\sin@{\pi s}}{\pi}\*\int_{0}^{\infty}(\ln@{1+x}-\digamma@{1+x})x^{-s}\diff{x}} | Zeta(s)=(1)/(s - 1)+(sin(Pi*s))/(Pi)* int((ln(1 + x)- Psi(1 + x))* (x)^(- s), x = 0..infinity) |
Zeta[s]=Divide[1,s - 1]+Divide[Sin[Pi*s],Pi]* Integrate[(Log[1 + x]- PolyGamma[1 + x])* (x)^(- s), {x, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 25.5.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{s-1}+\frac{\sin@{\pi s}}{\pi(s-1)}\*\int_{0}^{\infty}\left(\frac{1}{1+x}-\digamma'@{1+x}\right)x^{1-s}\diff{x}} | Zeta(s)=(1)/(s - 1)+(sin(Pi*s))/(Pi*(s - 1))* int(((1)/(1 + x)- subs( temp=1 + x, diff( Psi(temp), temp$(1) ) ))* (x)^(1 - s), x = 0..infinity) |
Zeta[s]=Divide[1,s - 1]+Divide[Sin[Pi*s],Pi*(s - 1)]* Integrate[(Divide[1,1 + x]- (D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x))* (x)^(1 - s), {x, 0, Infinity}] |
Failure | Failure | Skip | - |
| 25.5.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{1+s} = \frac{\sin@{\pi s}}{\pi}\int_{0}^{\infty}\left(\EulerConstant+\digamma@{1+x}\right)x^{-s-1}\diff{x}} | Zeta(1 + s)=(sin(Pi*s))/(Pi)*int((gamma + Psi(1 + x))* (x)^(- s - 1), x = 0..infinity) |
Zeta[1 + s]=Divide[Sin[Pi*s],Pi]*Integrate[(EulerGamma + PolyGamma[1 + x])* (x)^(- s - 1), {x, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 25.5.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{1+s} = \frac{\sin@{\pi s}}{\pi s}\int_{0}^{\infty}\digamma'@{1+x}x^{-s}\diff{x}} | Zeta(1 + s)=(sin(Pi*s))/(Pi*s)*int(subs( temp=1 + x, diff( Psi(temp), temp$(1) ) )*(x)^(- s), x = 0..infinity) |
Zeta[1 + s]=Divide[Sin[Pi*s],Pi*s]*Integrate[(D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 25.5.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{m+s} = (-1)^{m-1}\frac{\EulerGamma@{s}\sin@{\pi s}}{\pi\EulerGamma@{m+s}}\*\int_{0}^{\infty}\digamma^{(m)}@{1+x}x^{-s}\diff{x}} | Zeta(m + s)=(- 1)^(m - 1)*(GAMMA(s)*sin(Pi*s))/(Pi*GAMMA(m + s))* int(subs( temp=1 + x, diff( Psi(temp), temp$(m) ) )*(x)^(- s), x = 0..infinity) |
Zeta[m + s]=(- 1)^(m - 1)*Divide[Gamma[s]*Sin[Pi*s],Pi*Gamma[m + s]]* Integrate[(D[PolyGamma[temp], {temp, m}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 25.5.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{\EulerGamma@{1-s}}{2\pi i}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}-1}\diff{z}} | Zeta(s)=(GAMMA(1 - s))/(2*Pi*I)*int(((z)^(s - 1))/(exp(- z)- 1), z = - infinity..(0 +)) |
Zeta[s]=Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[(z)^(s - 1),Exp[- z]- 1], {z, - Infinity, (0 +)}] |
Error | Failure | - | Error |
| 25.5.E21 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{\EulerGamma@{1-s}}{2\pi i(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}+1}\diff{z}} | Zeta(s)=(GAMMA(1 - s))/(2*Pi*I*(1 - (2)^(1 - s)))* int(((z)^(s - 1))/(exp(- z)+ 1), z = - infinity..(0 +)) |
Zeta[s]=Divide[Gamma[1 - s],2*Pi*I*(1 - (2)^(1 - s))]* Integrate[Divide[(z)^(s - 1),Exp[- z]+ 1], {z, - Infinity, (0 +)}] |
Error | Failure | - | Error |
| 25.6#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{0} = -\frac{1}{2}} | Zeta(0)= -(1)/(2) |
Zeta[0]= -Divide[1,2] |
Successful | Successful | - | - |
| 25.6#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{2} = \frac{\pi^{2}}{6}} | Zeta(2)=((Pi)^(2))/(6) |
Zeta[2]=Divide[(Pi)^(2),6] |
Successful | Successful | - | - |
| 25.6#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{4} = \frac{\pi^{4}}{90}} | Zeta(4)=((Pi)^(4))/(90) |
Zeta[4]=Divide[(Pi)^(4),90] |
Successful | Successful | - | - |
| 25.6#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{6} = \frac{\pi^{6}}{945}} | Zeta(6)=((Pi)^(6))/(945) |
Zeta[6]=Divide[(Pi)^(6),945] |
Successful | Successful | - | - |
| 25.6.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{2n} = \frac{(2\pi)^{2n}}{2(2n)!}\left|\BernoullinumberB{2n}\right|} | Zeta(2*n)=((2*Pi)^(2*n))/(2*factorial(2*n))*abs(bernoulli(2*n)) |
Zeta[2*n]=Divide[(2*Pi)^(2*n),2*(2*n)!]*Abs[BernoulliB[2*n]] |
Failure | Failure | Successful | Successful |
| 25.6.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{-n} = -\frac{\BernoullinumberB{n+1}}{n+1}} | Zeta(- n)= -(bernoulli(n + 1))/(n + 1) |
Zeta[- n]= -Divide[BernoulliB[n + 1],n + 1] |
Failure | Failure | Successful | Successful |
| 25.6.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{-2n} = 0} | Zeta(- 2*n)= 0 |
Zeta[- 2*n]= 0 |
Failure | Failure | Successful | Successful |
| 25.6.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{2k+1} = \frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}\BernoullipolyB{2k+1}@{t}\cot@{\pi t}\diff{t}} | Zeta(2*k + 1)=((- 1)^(k + 1)*(2*Pi)^(2*k + 1))/(2*factorial(2*k + 1))*int(bernoulli(2*k + 1, t)*cot(Pi*t), t = 0..1) |
Zeta[2*k + 1]=Divide[(- 1)^(k + 1)*(2*Pi)^(2*k + 1),2*(2*k + 1)!]*Integrate[BernoulliB[2*k + 1, t]*Cot[Pi*t], {t, 0, 1}] |
Failure | Failure | Skip | Successful |
| 25.6.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{2} = \int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\diff{x}\diff{y}} | Zeta(2)= int(int((1)/(1 - x*y), x = 0..1), y = 0..1) |
Zeta[2]= Integrate[Integrate[Divide[1,1 - x*y], {x, 0, 1}], {y, 0, 1}] |
Successful | Successful | - | - |
| 25.6.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{2} = 3\sum_{k=1}^{\infty}\frac{1}{k^{2}\binom{2k}{k}}} | Zeta(2)= 3*sum((1)/((k)^(2)*binomial(2*k,k)), k = 1..infinity) |
Zeta[2]= 3*Sum[Divide[1,(k)^(2)*Binomial[2*k,k]], {k, 1, Infinity}] |
Successful | Successful | - | - |
| 25.6.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{3} = \frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\binom{2k}{k}}} | Zeta(3)=(5)/(2)*sum(((- 1)^(k - 1))/((k)^(3)*binomial(2*k,k)), k = 1..infinity) |
Zeta[3]=Divide[5,2]*Sum[Divide[(- 1)^(k - 1),(k)^(3)*Binomial[2*k,k]], {k, 1, Infinity}] |
Failure | Successful | Skip | - |
| 25.6.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{4} = \frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\binom{2k}{k}}} | Zeta(4)=(36)/(17)*sum((1)/((k)^(4)*binomial(2*k,k)), k = 1..infinity) |
Zeta[4]=Divide[36,17]*Sum[Divide[1,(k)^(4)*Binomial[2*k,k]], {k, 1, Infinity}] |
Failure | Successful | Skip | - |
| 25.6.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta'@{0} = -\tfrac{1}{2}\ln@{2\pi}} | subs( temp=0, diff( Zeta(temp), temp$(1) ) )= -(1)/(2)*ln(2*Pi) |
(D[Zeta[temp], {temp, 1}]/.temp-> 0)= -Divide[1,2]*Log[2*Pi] |
Successful | Successful | - | - |
| 25.6.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta''@{0} = -\tfrac{1}{2}(\ln@{2\pi})^{2}+\tfrac{1}{2}\EulerConstant^{2}-\tfrac{1}{24}\pi^{2}+\gamma_{1}} | subs( temp=0, diff( Zeta(temp), temp$(2) ) )= -(1)/(2)*(ln(2*Pi))^(2)+(1)/(2)*(gamma)^(2)-(1)/(24)*(Pi)^(2)+ gamma[1] |
(D[Zeta[temp], {temp, 2}]/.temp-> 0)= -Divide[1,2]*(Log[2*Pi])^(2)+Divide[1,2]*(EulerGamma)^(2)-Divide[1,24]*(Pi)^(2)+ Subscript[\[Gamma], 1] |
Failure | Failure | Fail -1.487029407-1.414213562*I <- {gamma[1] = 2^(1/2)+I*2^(1/2)}-1.487029407+1.414213562*I <- {gamma[1] = 2^(1/2)-I*2^(1/2)}1.341397717+1.414213562*I <- {gamma[1] = -2^(1/2)-I*2^(1/2)}1.341397717-1.414213562*I <- {gamma[1] = -2^(1/2)+I*2^(1/2)} |
Fail
Complex[-1.487029407856772, -1.4142135623730951] <- {Rule[Subscript[γ, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-1.487029407856772, 1.4142135623730951] <- {Rule[Subscript[γ, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[1.3413977168894184, 1.4142135623730951] <- {Rule[Subscript[γ, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[1.3413977168894184, -1.4142135623730951] <- {Rule[Subscript[γ, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 25.6.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{k}\Riemannzeta^{(k)}@{-2n} = \frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\imagpart@@{(c^{k-m})}\*\EulerGamma^{(r)}@{2n+1}\Riemannzeta^{(m-r)}@{2n+1}} | (- 1)^(k)* subs( temp=- 2*n, diff( Zeta(temp), temp$(k) ) )=(2*(- 1)^(n))/((2*Pi)^(2*n + 1))*sum(sum(binomial(k,m)*binomial(m,r)*Im((c)^(k - m))* subs( temp=2*n + 1, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n + 1, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k) |
(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> - 2*n)=Divide[2*(- 1)^(n),(2*Pi)^(2*n + 1)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Im[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n + 1)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n + 1), {r, 0, m}], {m, 0, k}] |
Failure | Failure | Skip | Skip |
| 25.6.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{k}\Riemannzeta^{(k)}@{1-2n} = \frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\realpart@@{(c^{k-m})}\*\EulerGamma^{(r)}@{2n}\Riemannzeta^{(m-r)}@{2n}} | (- 1)^(k)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(k) ) )=(2*(- 1)^(n))/((2*Pi)^(2*n))*sum(sum(binomial(k,m)*binomial(m,r)*Re((c)^(k - m))* subs( temp=2*n, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k) |
(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - 2*n)=Divide[2*(- 1)^(n),(2*Pi)^(2*n)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Re[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n), {r, 0, m}], {m, 0, k}] |
Failure | Failure | Skip | Skip |
| 25.6.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta'@{2n} = \frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\Riemannzeta'@{1-2n}-(\digamma@{2n}-\ln@{2\pi})\BernoullinumberB{2n}\right)} | subs( temp=2*n, diff( Zeta(temp), temp$(1) ) )=((- 1)^(n + 1)*(2*Pi)^(2*n))/(2*factorial(2*n))*(2*n*subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) )-(Psi(2*n)- ln(2*Pi))*bernoulli(2*n)) |
(D[Zeta[temp], {temp, 1}]/.temp-> 2*n)=Divide[(- 1)^(n + 1)*(2*Pi)^(2*n),2*(2*n)!]*(2*n*(D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n)-(PolyGamma[2*n]- Log[2*Pi])*BernoulliB[2*n]) |
Failure | Failure | Successful | Successful |
| 25.6.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(n+\tfrac{1}{2}\right)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}\Riemannzeta@{2k}\Riemannzeta@{2n-2k}} | (n +(1)/(2))* Zeta(2*n)= sum(Zeta(2*k)*Zeta(2*n - 2*k), k = 1..n - 1) |
(n +Divide[1,2])* Zeta[2*n]= Sum[Zeta[2*k]*Zeta[2*n - 2*k], {k, 1, n - 1}] |
Failure | Failure | Skip | Successful |
| 25.6.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(n+\tfrac{3}{4}\right)\Riemannzeta@{4n+2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n+2-2k}} | (n +(3)/(4))* Zeta(4*n + 2)= sum(Zeta(2*k)*Zeta(4*n + 2 - 2*k), k = 1..n) |
(n +Divide[3,4])* Zeta[4*n + 2]= Sum[Zeta[2*k]*Zeta[4*n + 2 - 2*k], {k, 1, n}] |
Failure | Failure | Skip | Successful |
| 25.6.E20 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \tfrac{1}{2}(2^{2n}-1)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}(2^{2n-2k}-1)\Riemannzeta@{2n-2k}\Riemannzeta@{2k}} | (1)/(2)*((2)^(2*n)- 1)* Zeta(2*n)= sum(((2)^(2*n - 2*k)- 1)* Zeta(2*n - 2*k)*Zeta(2*k), k = 1..n - 1) |
Divide[1,2]*((2)^(2*n)- 1)* Zeta[2*n]= Sum[((2)^(2*n - 2*k)- 1)* Zeta[2*n - 2*k]*Zeta[2*k], {k, 1, n - 1}] |
Failure | Failure | Skip | Successful |
| 25.8.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=2}^{\infty}\left(\Riemannzeta@{k}-1\right) = 1} | sum(Zeta(k)- 1, k = 2..infinity)= 1 |
Sum[Zeta[k]- 1, {k, 2, Infinity}]= 1 |
Failure | Successful | Skip | - |
| 25.8.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{\infty}\frac{\EulerGamma@{s+k}}{(k+1)!}\left(\Riemannzeta@{s+k}-1\right) = \EulerGamma@{s-1}} | sum((GAMMA(s + k))/(factorial(k + 1))*(Zeta(s + k)- 1), k = 0..infinity)= GAMMA(s - 1) |
Sum[Divide[Gamma[s + k],(k + 1)!]*(Zeta[s + k]- 1), {k, 0, Infinity}]= Gamma[s - 1] |
Failure | Failure | Skip | Error |
| 25.8.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{\infty}\frac{\Pochhammersym{s}{k}\Riemannzeta@{s+k}}{k!2^{s+k}} = (1-2^{-s})\Riemannzeta@{s}} | sum((pochhammer(s, k)*Zeta(s + k))/(factorial(k)*(2)^(s + k)), k = 0..infinity)=(1 - (2)^(- s))* Zeta(s) |
Sum[Divide[Pochhammer[s, k]*Zeta[s + k],(k)!*(2)^(s + k)], {k, 0, Infinity}]=(1 - (2)^(- s))* Zeta[s] |
Failure | Failure | Skip | Successful |
| 25.8.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\Riemannzeta@{nk}-1) = \ln@{\prod_{j=0}^{n-1}\EulerGamma@{2-e^{(2j+1)\pi i/n}}}} | sum(((- 1)^(k))/(k)*(Zeta(n*k)- 1), k = 1..infinity)= ln(product(GAMMA(2 - exp((2*j + 1)* Pi*I/ n)), j = 0..n - 1)) |
Sum[Divide[(- 1)^(k),k]*(Zeta[n*k]- 1), {k, 1, Infinity}]= Log[Product[Gamma[2 - Exp[(2*j + 1)* Pi*I/ n]], {j, 0, n - 1}]] |
Failure | Failure | Skip | Fail
Complex[0.7210663818131499, 0.6288153989756469] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.7210663818131499, -2.199611725770543] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.1073607429330403, -2.199611725770543] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.1073607429330403, 0.6288153989756469] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 25.8.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=2}^{\infty}\Riemannzeta@{k}z^{k} = -\EulerConstant z-z\digamma@{1-z}} | sum(Zeta(k)*(z)^(k), k = 2..infinity)= - gamma*z - z*Psi(1 - z) |
Sum[Zeta[k]*(z)^(k), {k, 2, Infinity}]= - EulerGamma*z - z*PolyGamma[1 - z] |
Failure | Successful | Skip | - |
| 25.8.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{\infty}\Riemannzeta@{2k}z^{2k} = -\tfrac{1}{2}\pi z\cot@{\pi z}} | sum(Zeta(2*k)*(z)^(2*k), k = 0..infinity)= -(1)/(2)*Pi*z*cot(Pi*z) |
Sum[Zeta[2*k]*(z)^(2*k), {k, 0, Infinity}]= -Divide[1,2]*Pi*z*Cot[Pi*z] |
Failure | Failure | Skip | Skip |
| 25.8.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=2}^{\infty}\frac{\Riemannzeta@{k}}{k}z^{k} = -\EulerConstant z+\ln@@{\EulerGamma@{1-z}}} | sum((Zeta(k))/(k)*(z)^(k), k = 2..infinity)= - gamma*z + ln(GAMMA(1 - z)) |
Sum[Divide[Zeta[k],k]*(z)^(k), {k, 2, Infinity}]= - EulerGamma*z + Log[Gamma[1 - z]] |
Failure | Successful | Skip | - |
| 25.8.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{k}z^{2k} = \ln@{\frac{\pi z}{\sin@{\pi z}}}} | sum((Zeta(2*k))/(k)*(z)^(2*k), k = 1..infinity)= ln((Pi*z)/(sin(Pi*z))) |
Sum[Divide[Zeta[2*k],k]*(z)^(2*k), {k, 1, Infinity}]= Log[Divide[Pi*z,Sin[Pi*z]]] |
Failure | Successful | Skip | - |
| 25.8.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{(2k+1)2^{2k}} = \frac{1}{2}-\frac{1}{2}\ln@@{2}} | sum((Zeta(2*k))/((2*k + 1)* (2)^(2*k)), k = 1..infinity)=(1)/(2)-(1)/(2)*ln(2) |
Sum[Divide[Zeta[2*k],(2*k + 1)* (2)^(2*k)], {k, 1, Infinity}]=Divide[1,2]-Divide[1,2]*Log[2] |
Failure | Successful | Skip | - |
| 25.8.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{(2k+1)(2k+2)2^{2k}} = \frac{1}{4}-\frac{7}{4\pi^{2}}\Riemannzeta@{3}} | sum((Zeta(2*k))/((2*k + 1)*(2*k + 2)* (2)^(2*k)), k = 1..infinity)=(1)/(4)-(7)/(4*(Pi)^(2))*Zeta(3) |
Sum[Divide[Zeta[2*k],(2*k + 1)*(2*k + 2)* (2)^(2*k)], {k, 1, Infinity}]=Divide[1,4]-Divide[7,4*(Pi)^(2)]*Zeta[3] |
Failure | Successful | Skip | - |
| 25.9.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \chi(s) = \pi^{s-\frac{1}{2}}\EulerGamma@{\tfrac{1}{2}-\tfrac{1}{2}s}/\EulerGamma@{\tfrac{1}{2}s}} | chi*(s)= (Pi)^(s -(1)/(2))* GAMMA((1)/(2)-(1)/(2)*s)/ GAMMA((1)/(2)*s) |
\[Chi]*(s)= (Pi)^(s -Divide[1,2])* Gamma[Divide[1,2]-Divide[1,2]*s]/ Gamma[Divide[1,2]*s] |
Failure | Failure | Fail .5066144201+7.721862512*I <- {chi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2)}4.506614418-3.721862514*I <- {chi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)-I*2^(1/2)}-.5006270982e-1-4.069033292*I <- {chi = 2^(1/2)+I*2^(1/2), s = -2^(1/2)-I*2^(1/2)}-4.050062708+.6903329420e-1*I <- {chi = 2^(1/2)+I*2^(1/2), s = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Fail
Complex[0.5066144187413095, 7.721862514810475] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[4.506614418741309, 3.721862514810475] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.5066144187413095, -0.2781374851895251] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-3.4933855812586905, 3.721862514810475] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 25.10.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle Z(t) = \exp@{i\vartheta(t)}\Riemannzeta@{\tfrac{1}{2}+it}} | Z*(t)= exp(I*vartheta*(t))*Zeta((1)/(2)+ I*t) |
Z*(t)= Exp[I*\[CurlyTheta]*(t)]*Zeta[Divide[1,2]+ I*t] |
Failure | Failure | Fail -.1598353599e-2+4.002319388*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2)}.1528788606+3.983270213*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2)}-4.764624907+10.91400505*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2)}-.3879562929e-1+3.851182221*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Fail
Complex[-0.0015983535965552907, 4.002319390307897] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϑ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.15287886062247902, 3.9832702156526483] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϑ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-4.764624919768366, 10.914005063393518] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϑ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-0.03879562949747604, 3.8511822226969143] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ϑ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 25.10.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle Z(t) = 2\sum_{n=1}^{m}\frac{\cos@{\vartheta(t)-t\ln@@{n}}}{n^{1/2}}+R(t)} | Z*(t)= 2*sum((cos(vartheta*(t)- t*ln(n)))/((n)^(1/ 2)), n = 1..m)+ R*(t) |
Z*(t)= 2*Sum[Divide[Cos[\[CurlyTheta]*(t)- t*Log[n]],(n)^(1/ 2)], {n, 1, m}]+ R*(t) |
Failure | Failure | Skip | Skip |
| 25.11.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}} | Zeta(0, s, a)= sum((1)/((n + a)^(s)), n = 0..infinity) |
HurwitzZeta[s, a]= Sum[Divide[1,(n + a)^(s)], {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 25.11.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{1} = \Riemannzeta@{s}} | Zeta(0, s, 1)= Zeta(s) |
HurwitzZeta[s, 1]= Zeta[s] |
Successful | Successful | - | - |
| 25.11.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \Hurwitzzeta@{s}{a+1}+a^{-s}} | Zeta(0, s, a)= Zeta(0, s, a + 1)+ (a)^(- s) |
HurwitzZeta[s, a]= HurwitzZeta[s, a + 1]+ (a)^(- s) |
Failure | Successful | Successful | - |
| 25.11.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \Hurwitzzeta@{s}{a+m}+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}}} | Zeta(0, s, a)= Zeta(0, s, a + m)+ sum((1)/((n + a)^(s)), n = 0..m - 1) |
HurwitzZeta[s, a]= HurwitzZeta[s, a + m]+ Sum[Divide[1,(n + a)^(s)], {n, 0, m - 1}] |
Failure | Successful | Skip | - |
| 25.11.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{(x+a)^{s+1}}\diff{x}} | Zeta(0, s, a)= sum((1)/((n + a)^(s)), n = 0..N)+((N + a)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x + a)^(s + 1)), x = N..infinity) |
HurwitzZeta[s, a]= Sum[Divide[1,(n + a)^(s)], {n, 0, N}]+Divide[(N + a)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x + a)^(s + 1)], {x, N, Infinity}] |
Failure | Failure | Skip | Error |
| 25.11.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{\tfrac{1}{2}a} = \Hurwitzzeta@{s}{\tfrac{1}{2}a+\tfrac{1}{2}}+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}} | Zeta(0, s, (1)/(2)*a)= Zeta(0, s, (1)/(2)*a +(1)/(2))+ (2)^(s)* sum(((- 1)^(n))/((n + a)^(s)), n = 0..infinity) |
HurwitzZeta[s, Divide[1,2]*a]= HurwitzZeta[s, Divide[1,2]*a +Divide[1,2]]+ (2)^(s)* Sum[Divide[(- 1)^(n),(n + a)^(s)], {n, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 25.11.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{1-s}{a} = \frac{2\EulerGamma@{s}}{(2\pi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos@{\tfrac{1}{2}\pi s-2n\pi a}} | Zeta(0, 1 - s, a)=(2*GAMMA(s))/((2*Pi)^(s))* sum((1)/((n)^(s))*cos((1)/(2)*Pi*s - 2*n*Pi*a), n = 1..infinity) |
HurwitzZeta[1 - s, a]=Divide[2*Gamma[s],(2*Pi)^(s)]* Sum[Divide[1,(n)^(s)]*Cos[Divide[1,2]*Pi*s - 2*n*Pi*a], {n, 1, Infinity}] |
Failure | Failure | Skip | Error |
| 25.11.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \sum_{n=0}^{\infty}\frac{\Pochhammersym{s}{n}}{n!}\Riemannzeta@{n+s}(1-a)^{n}} | Zeta(0, s, a)= sum((pochhammer(s, n))/(factorial(n))*Zeta(n + s)*(1 - a)^(n), n = 0..infinity) |
HurwitzZeta[s, a]= Sum[Divide[Pochhammer[s, n],(n)!]*Zeta[n + s]*(1 - a)^(n), {n, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 25.11.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{\tfrac{1}{2}} = (2^{s}-1)\Riemannzeta@{s}} | Zeta(0, s, (1)/(2))=((2)^(s)- 1)* Zeta(s) |
HurwitzZeta[s, Divide[1,2]]=((2)^(s)- 1)* Zeta[s] |
Successful | Failure | - | Successful |
| 25.11.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{n+1}{a} = \frac{(-1)^{n+1}\digamma^{(n)}@{a}}{n!}} | Zeta(0, n + 1, a)=((- 1)^(n + 1)* subs( temp=a, diff( Psi(temp), temp$(n) ) ))/(factorial(n)) |
HurwitzZeta[n + 1, a]=Divide[(- 1)^(n + 1)* (D[PolyGamma[temp], {temp, n}]/.temp-> a),(n)!] |
Failure | Failure | Successful | Successful |
| 25.11.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{0}{a} = \tfrac{1}{2}-a} | Zeta(0, 0, a)=(1)/(2)- a |
HurwitzZeta[0, a]=Divide[1,2]- a |
Successful | Successful | - | - |
| 25.11.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{-n}{a} = -\frac{\BernoullipolyB{n+1}@{a}}{n+1}} | Zeta(0, - n, a)= -(bernoulli(n + 1, a))/(n + 1) |
HurwitzZeta[- n, a]= -Divide[BernoulliB[n + 1, a],n + 1] |
Failure | Failure | Successful | Successful |
| 25.11.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{ka} = k^{-s}\*\sum_{n=0}^{k-1}\Hurwitzzeta@{s}{a+\frac{n}{k}}} | Zeta(0, s, k*a)= (k)^(- s)* sum(Zeta(0, s, a +(n)/(k)), n = 0..k - 1) |
HurwitzZeta[s, k*a]= (k)^(- s)* Sum[HurwitzZeta[s, a +Divide[n,k]], {n, 0, k - 1}] |
Failure | Failure | Skip | Error |
| 25.11.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{1-s}{\frac{h}{k}} = \frac{2\EulerGamma@{s}}{(2\pi k)^{s}}\*\sum_{r=1}^{k}\cos@{\frac{\pi s}{2}-\frac{2\pi rh}{k}}\Hurwitzzeta@{s}{\frac{r}{k}}} | Zeta(0, 1 - s, (h)/(k))=(2*GAMMA(s))/((2*Pi*k)^(s))* sum(cos((Pi*s)/(2)-(2*Pi*r*h)/(k))*Zeta(0, s, (r)/(k)), r = 1..k) |
HurwitzZeta[1 - s, Divide[h,k]]=Divide[2*Gamma[s],(2*Pi*k)^(s)]* Sum[Cos[Divide[Pi*s,2]-Divide[2*Pi*r*h,k]]*HurwitzZeta[s, Divide[r,k]], {r, 1, k}] |
Failure | Failure | Skip | Error |
| 25.11.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \pderiv{}{a}\Hurwitzzeta@{s}{a} = -s\Hurwitzzeta@{s+1}{a}} | diff(Zeta(0, s, a), a)= - s*Zeta(0, s + 1, a) |
D[HurwitzZeta[s, a], a]= - s*HurwitzZeta[s + 1, a] |
Successful | Successful | - | - |
| 25.11.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta'@{0}{a} = \ln@@{\EulerGamma@{a}}-\tfrac{1}{2}\ln@{2\pi}} | subs( temp=0, diff( Zeta(0, temp, a), temp$(1) ) )= ln(GAMMA(a))-(1)/(2)*ln(2*Pi) |
(D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> 0)= Log[Gamma[a]]-Divide[1,2]*Log[2*Pi] |
Failure | Failure | Successful | Successful |
| 25.11.E21 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta'@{1-2n}{\frac{h}{k}} = \frac{(\digamma@{2n}-\ln@{2\pi k})\BernoullipolyB{2n}@{h/k}}{2n}-\frac{(\digamma@{2n}-\ln@{2\pi})\BernoullinumberB{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\sin@{\frac{2\pi rh}{k}}\digamma^{(2n-1)}@{\frac{r}{k}}+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\cos@{\frac{2\pi rh}{k}}\Hurwitzzeta'@{2n}{\frac{r}{k}}+\frac{\Riemannzeta'@{1-2n}}{k^{2n}}} | subs( temp=1 - 2*n, diff( Zeta(0, temp, (h)/(k)), temp$(1) ) )=((Psi(2*n)- ln(2*Pi*k))* bernoulli(2*n, h/ k))/(2*n)-((Psi(2*n)- ln(2*Pi))* bernoulli(2*n))/(2*n*(k)^(2*n))+((- 1)^(n + 1)* Pi)/((2*Pi*k)^(2*n))*sum(sin((2*Pi*r*h)/(k))*subs( temp=(r)/(k), diff( Psi(temp), temp$(2*n - 1) ) ), r = 1..k - 1)+((- 1)^(n + 1)* 2 *factorial(2*n - 1))/((2*Pi*k)^(2*n))*sum(cos((2*Pi*r*h)/(k))*subs( temp=2*n, diff( Zeta(0, temp, (r)/(k)), temp$(1) ) ), r = 1..k - 1)+(subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/((k)^(2*n)) |
(D[HurwitzZeta[temp, Divide[h,k]], {temp, 1}]/.temp-> 1 - 2*n)=Divide[(PolyGamma[2*n]- Log[2*Pi*k])* BernoulliB[2*n, h/ k],2*n]-Divide[(PolyGamma[2*n]- Log[2*Pi])* BernoulliB[2*n],2*n*(k)^(2*n)]+Divide[(- 1)^(n + 1)* Pi,(2*Pi*k)^(2*n)]*Sum[Sin[Divide[2*Pi*r*h,k]]*(D[PolyGamma[temp], {temp, 2*n - 1}]/.temp-> Divide[r,k]), {r, 1, k - 1}]+Divide[(- 1)^(n + 1)* 2 *(2*n - 1)!,(2*Pi*k)^(2*n)]*Sum[Cos[Divide[2*Pi*r*h,k]]*(D[HurwitzZeta[temp, Divide[r,k]], {temp, 1}]/.temp-> 2*n), {r, 1, k - 1}]+Divide[D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n,(k)^(2*n)] |
Failure | Failure | Skip | Error |
| 25.11.E22 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta'@{1-2n}{\tfrac{1}{2}} = -\frac{\BernoullinumberB{2n}\ln@@{2}}{n\cdot 4^{n}}-\frac{(2^{2n-1}-1)\Riemannzeta'@{1-2n}}{2^{2n-1}}} | subs( temp=1 - 2*n, diff( Zeta(0, temp, (1)/(2)), temp$(1) ) )= -(bernoulli(2*n)*ln(2))/(n * (4)^(n))-(((2)^(2*n - 1)- 1)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/((2)^(2*n - 1)) |
(D[HurwitzZeta[temp, Divide[1,2]], {temp, 1}]/.temp-> 1 - 2*n)= -Divide[BernoulliB[2*n]*Log[2],n * (4)^(n)]-Divide[((2)^(2*n - 1)- 1)* (D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n),(2)^(2*n - 1)] |
Failure | Failure | Successful | Successful |
| 25.11.E23 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta'@{1-2n}{\tfrac{1}{3}} = -\frac{\pi(9^{n}-1)\BernoullinumberB{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{\BernoullinumberB{2n}\ln@@{3}}{4n\cdot 3^{2n-1}}-\frac{(-1)^{n}\digamma^{(2n-1)}@{\frac{1}{3}}}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right)\Riemannzeta'@{1-2n}}{2\cdot 3^{2n-1}}} | subs( temp=1 - 2*n, diff( Zeta(0, temp, (1)/(3)), temp$(1) ) )= -(Pi*((9)^(n)- 1)* bernoulli(2*n))/(8*n*sqrt(3)*((3)^(2*n - 1)- 1))-(bernoulli(2*n)*ln(3))/(4*n * (3)^(2*n - 1))-((- 1)^(n)* subs( temp=(1)/(3), diff( Psi(temp), temp$(2*n - 1) ) ))/(2*sqrt(3)*(6*Pi)^(2*n - 1))-(((3)^(2*n - 1)- 1)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/(2 * (3)^(2*n - 1)) |
(D[HurwitzZeta[temp, Divide[1,3]], {temp, 1}]/.temp-> 1 - 2*n)= -Divide[Pi*((9)^(n)- 1)* BernoulliB[2*n],8*n*Sqrt[3]*((3)^(2*n - 1)- 1)]-Divide[BernoulliB[2*n]*Log[3],4*n * (3)^(2*n - 1)]-Divide[(- 1)^(n)* (D[PolyGamma[temp], {temp, 2*n - 1}]/.temp-> Divide[1,3]),2*Sqrt[3]*(6*Pi)^(2*n - 1)]-Divide[((3)^(2*n - 1)- 1)* (D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n),2 * (3)^(2*n - 1)] |
Failure | Failure | Successful | Successful |
| 25.11.E24 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{r=1}^{k-1}\Hurwitzzeta'@{s}{\frac{r}{k}} = (k^{s}-1)\Riemannzeta'@{s}+k^{s}\Riemannzeta@{s}\ln@@{k}} | sum(subs( temp=s, diff( Zeta(0, temp, (r)/(k)), temp$(1) ) ), r = 1..k - 1)=((k)^(s)- 1)* subs( temp=s, diff( Zeta(temp), temp$(1) ) )+ (k)^(s)* Zeta(s)*ln(k) |
Sum[D[HurwitzZeta[temp, Divide[r,k]], {temp, 1}]/.temp-> s, {r, 1, k - 1}]=((k)^(s)- 1)* (D[Zeta[temp], {temp, 1}]/.temp-> s)+ (k)^(s)* Zeta[s]*Log[k] |
Failure | Failure | Skip | Successful |
| 25.11.E25 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-e^{-x}}\diff{x}} | Zeta(0, s, a)=(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 - exp(- x)), x = 0..infinity) |
HurwitzZeta[s, a]=Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 - Exp[- x]], {x, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 25.11.E26 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = -s\int_{-a}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{(x+a)^{s+1}}\diff{x}} | Zeta(0, s, a)= - s*int((x - floor(x)-(1)/(2))/((x + a)^(s + 1)), x = - a..infinity) |
HurwitzZeta[s, a]= - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x + a)^(s + 1)], {x, - a, Infinity}] |
Failure | Failure | Skip | Error |
| 25.11.E27 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{ax}}\diff{x}} | Zeta(0, s, a)=(1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2))*((x)^(s - 1))/(exp(a*x)), x = 0..infinity) |
HurwitzZeta[s, a]=Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2])*Divide[(x)^(s - 1),Exp[a*x]], {x, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 25.11.E28 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}\frac{\BernoullinumberB{2k}}{(2k)!}\Pochhammersym{s}{2k-1}a^{1-s-2k}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{\BernoullinumberB{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-ax}\diff{x}} | Zeta(0, s, a)=(1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+ sum((bernoulli(2*k))/(factorial(2*k))*pochhammer(s, 2*k - 1)*(a)^(1 - s - 2*k)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2)- sum((bernoulli(2*k))/(factorial(2*k))*(x)^(2*k - 1), k = 1..n))* (x)^(s - 1)* exp(- a*x), x = 0..infinity), k = 1..n) |
HurwitzZeta[s, a]=Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+ Sum[Divide[BernoulliB[2*k],(2*k)!]*Pochhammer[s, 2*k - 1]*(a)^(1 - s - 2*k)+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2]- Sum[Divide[BernoulliB[2*k],(2*k)!]*(x)^(2*k - 1), {k, 1, n}])* (x)^(s - 1)* Exp[- a*x], {x, 0, Infinity}], {k, 1, n}] |
Failure | Failure | Skip | Error |
| 25.11.E29 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}\frac{\sin@{s\atan@{x/a}}}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}} | Zeta(0, s, a)=(1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+ 2*int((sin(s*arctan(x/ a)))/(((a)^(2)+ (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity) |
HurwitzZeta[s, a]=Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+ 2*Integrate[Divide[Sin[s*ArcTan[x/ a]],((a)^(2)+ (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 25.11.E30 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \frac{\EulerGamma@{1-s}}{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}\diff{z}} | Zeta(0, s, a)=(GAMMA(1 - s))/(2*Pi*I)*int((exp(a*z)*(z)^(s - 1))/(1 - exp(z)), z = - infinity..(0 +)) |
HurwitzZeta[s, a]=Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[Exp[a*z]*(z)^(s - 1),1 - Exp[z]], {z, - Infinity, (0 +)}] |
Error | Failure | - | Error |
| 25.11.E31 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{2\cosh@@{x}}\diff{x} = 4^{-s}\left(\Hurwitzzeta@{s}{\tfrac{1}{4}+\tfrac{1}{4}a}-\Hurwitzzeta@{s}{\tfrac{3}{4}+\tfrac{1}{4}a}\right)} | (1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(2*cosh(x)), x = 0..infinity)= (4)^(- s)*(Zeta(0, s, (1)/(4)+(1)/(4)*a)- Zeta(0, s, (3)/(4)+(1)/(4)*a)) |
Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],2*Cosh[x]], {x, 0, Infinity}]= (4)^(- s)*(HurwitzZeta[s, Divide[1,4]+Divide[1,4]*a]- HurwitzZeta[s, Divide[3,4]+Divide[1,4]*a]) |
Failure | Failure | Skip | Successful |
| 25.11.E32 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{a}x^{n}\digamma@{x}\diff{x} = (-1)^{n-1}\Riemannzeta'@{-n}+(-1)^{n}h(n)\frac{\BernoullinumberB{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}h(k)\frac{\BernoullinumberB{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\Hurwitzzeta'@{-k}{a}a^{n-k}} | int((x)^(n)* Psi(x), x = 0..a)=(- 1)^(n - 1)* subs( temp=- n, diff( Zeta(temp), temp$(1) ) )+(- 1)^(n)* h*(n)*(bernoulli(n + 1))/(n + 1)- sum((- 1)^(k)*binomial(n,k)*h*(k)*(bernoulli(k + 1)*(a))/(k + 1)*(a)^(n - k), k = 0..n)+ sum((- 1)^(k)*binomial(n,k)*subs( temp=- k, diff( Zeta(0, temp, a), temp$(1) ) )*(a)^(n - k), k = 0..n) |
Integrate[(x)^(n)* PolyGamma[x], {x, 0, a}]=(- 1)^(n - 1)* (D[Zeta[temp], {temp, 1}]/.temp-> - n)+(- 1)^(n)* h*(n)*Divide[BernoulliB[n + 1],n + 1]- Sum[(- 1)^(k)*Binomial[n,k]*h*(k)*Divide[BernoulliB[k + 1]*(a),k + 1]*(a)^(n - k), {k, 0, n}]+ Sum[(- 1)^(k)*Binomial[n,k]*(D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> - k)*(a)^(n - k), {k, 0, n}] |
Failure | Failure | Skip | Error |
| 25.11.E34 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n\int_{0}^{a}\Hurwitzzeta'@{1-n}{x}\diff{x} = \Hurwitzzeta'@{-n}{a}-\Riemannzeta'@{-n}+\frac{\BernoullinumberB{n+1}-\BernoullipolyB{n+1}@{a}}{n(n+1)}} | n*int(subs( temp=1 - n, diff( Zeta(0, temp, x), temp$(1) ) ), x = 0..a)= subs( temp=- n, diff( Zeta(0, temp, a), temp$(1) ) )- subs( temp=- n, diff( Zeta(temp), temp$(1) ) )+(bernoulli(n + 1)- bernoulli(n + 1, a))/(n*(n + 1)) |
n*Integrate[D[HurwitzZeta[temp, x], {temp, 1}]/.temp-> 1 - n, {x, 0, a}]= (D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> - n)- (D[Zeta[temp], {temp, 1}]/.temp-> - n)+Divide[BernoulliB[n + 1]- BernoulliB[n + 1, a],n*(n + 1)] |
Failure | Failure | Skip | Successful |
| 25.11.E35 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\diff{x}} | sum(((- 1)^(n))/((n + a)^(s)), n = 0..infinity)=(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 + exp(- x)), x = 0..infinity) |
Sum[Divide[(- 1)^(n),(n + a)^(s)], {n, 0, Infinity}]=Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 + Exp[- x]], {x, 0, Infinity}] |
Error | Failure | - | Skip |
| 25.11.E35 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\diff{x} = 2^{-s}\left(\Hurwitzzeta@{s}{\tfrac{1}{2}a}-\Hurwitzzeta@{s}{\tfrac{1}{2}(1+a)}\right)} | (1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 + exp(- x)), x = 0..infinity)= (2)^(- s)*(Zeta(0, s, (1)/(2)*a)- Zeta(0, s, (1)/(2)*(1 + a))) |
Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 + Exp[- x]], {x, 0, Infinity}]= (2)^(- s)*(HurwitzZeta[s, Divide[1,2]*a]- HurwitzZeta[s, Divide[1,2]*(1 + a)]) |
Error | Failure | - | Skip |
| 25.11.E36 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}} = k^{-s}\sum_{r=1}^{k}\chi(r)\Hurwitzzeta@{s}{\frac{r}{k}}} | sum((chi*(n))/((n)^(s)), n = 1..infinity)= (k)^(- s)* sum(chi*(r)* Zeta(0, s, (r)/(k)), r = 1..k) |
Sum[Divide[\[Chi]*(n),(n)^(s)], {n, 1, Infinity}]= (k)^(- s)* Sum[\[Chi]*(r)* HurwitzZeta[s, Divide[r,k]], {r, 1, k}] |
Failure | Failure | Skip | Successful |
| 25.11.E37 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\Hurwitzzeta@{nk}{a} = -n\ln@@{\EulerGamma@{a}}+\ln@{\prod_{j=0}^{n-1}\EulerGamma@{a-e^{(2j+1)\pi i/n}}}} | sum(((- 1)^(k))/(k)*Zeta(0, n*k, a), k = 1..infinity)= - n*ln(GAMMA(a))+ ln(product(GAMMA(a - exp((2*j + 1)* Pi*I/ n)), j = 0..n - 1)) |
Sum[Divide[(- 1)^(k),k]*HurwitzZeta[n*k, a], {k, 1, Infinity}]= - n*Log[Gamma[a]]+ Log[Product[Gamma[a - Exp[(2*j + 1)* Pi*I/ n]], {j, 0, n - 1}]] |
Failure | Failure | Skip | Skip |
| 25.11.E38 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{\infty}\binom{n+k}{k}\Hurwitzzeta@{n+k+1}{a}z^{k} = \frac{(-1)^{n}}{n!}\left(\digamma^{(n)}@{a}-\digamma^{(n)}@{a-z}\right)} | sum(binomial(n + k,k)*Zeta(0, n + k + 1, a)*(z)^(k), k = 1..infinity)=((- 1)^(n))/(factorial(n))*(subs( temp=a, diff( Psi(temp), temp$(n) ) )- subs( temp=a - z, diff( Psi(temp), temp$(n) ) )) |
Sum[Binomial[n + k,k]*HurwitzZeta[n + k + 1, a]*(z)^(k), {k, 1, Infinity}]=Divide[(- 1)^(n),(n)!]*((D[PolyGamma[temp], {temp, n}]/.temp-> a)- (D[PolyGamma[temp], {temp, n}]/.temp-> a - z)) |
Failure | Failure | Skip | Successful |
| 25.11.E39 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=2}^{\infty}\frac{k}{2^{k}}\Hurwitzzeta@{k+1}{\tfrac{3}{4}} = 8G} | sum((k)/((2)^(k))*Zeta(0, k + 1, (3)/(4)), k = 2..infinity)= 8*G |
Sum[Divide[k,(2)^(k)]*HurwitzZeta[k + 1, Divide[3,4]], {k, 2, Infinity}]= 8*G |
Failure | Failure | Skip | Fail
Complex[-3.985983745567009, -11.313708498984761] <- {Rule[G, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-3.985983745567009, 11.313708498984761] <- {Rule[G, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[18.641433252402514, 11.313708498984761] <- {Rule[G, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[18.641433252402514, -11.313708498984761] <- {Rule[G, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 25.12.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{z} = \sum_{n=1}^{\infty}\frac{z^{n}}{n^{2}}} | dilog(z)= sum(((z)^(n))/((n)^(2)), n = 1..infinity) |
PolyLog[2, z]= Sum[Divide[(z)^(n),(n)^(2)], {n, 1, Infinity}] |
Failure | Successful | Skip | - |
| 25.12.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{z} = -\int_{0}^{z}t^{-1}\ln@{1-t}\diff{t}} | dilog(z)= - int((t)^(- 1)* ln(1 - t), t = 0..z) |
PolyLog[2, z]= - Integrate[(t)^(- 1)* Log[1 - t], {t, 0, z}] |
Failure | Failure | Skip | Error |
| 25.12.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{z}+\dilog@{\frac{z}{z-1}} = -\frac{1}{2}(\ln@{1-z})^{2}} | dilog(z)+ dilog((z)/(z - 1))= -(1)/(2)*(ln(1 - z))^(2) |
PolyLog[2, z]+ PolyLog[2, Divide[z,z - 1]]= -Divide[1,2]*(Log[1 - z])^(2) |
Failure | Failure | Fail 3.289868134-2.177586090*I <- {z = 1/2} |
Successful |
| 25.12.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{z}+\dilog@{\frac{1}{z}} = -\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln@{-z})^{2}} | dilog(z)+ dilog((1)/(z))= -(1)/(6)*(Pi)^(2)-(1)/(2)*(ln(- z))^(2) |
PolyLog[2, z]+ PolyLog[2, Divide[1,z]]= -Divide[1,6]*(Pi)^(2)-Divide[1,2]*(Log[- z])^(2) |
Failure | Failure | Fail 6.579736268-4.725198502*I <- {z = -1/2} |
Successful |
| 25.12.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{z^{m}} = m\sum_{k=0}^{m-1}\dilog@{ze^{2\pi ik/m}}} | dilog((z)^(m))= m*sum(dilog(z*exp(2*Pi*I*k/ m)), k = 0..m - 1) |
PolyLog[2, (z)^(m)]= m*Sum[PolyLog[2, z*Exp[2*Pi*I*k/ m]], {k, 0, m - 1}] |
Failure | Failure | Skip | Successful |
| 25.12.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{x}+\dilog@{1-x} = \frac{1}{6}\pi^{2}-(\ln@@{x})\ln@{1-x}} | dilog(x)+ dilog(1 - x)=(1)/(6)*(Pi)^(2)-(ln(x))* ln(1 - x) |
PolyLog[2, x]+ PolyLog[2, 1 - x]=Divide[1,6]*(Pi)^(2)-(Log[x])* Log[1 - x] |
Successful | Successful | - | - |
| 25.12.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{e^{i\theta}} = \sum_{n=1}^{\infty}\frac{\cos@{n\theta}}{n^{2}}+i\sum_{n=1}^{\infty}\frac{\sin@{n\theta}}{n^{2}}} | dilog(exp(I*theta))= sum((cos(n*theta))/((n)^(2)), n = 1..infinity)+ I*sum((sin(n*theta))/((n)^(2)), n = 1..infinity) |
PolyLog[2, Exp[I*\[Theta]]]= Sum[Divide[Cos[n*\[Theta]],(n)^(2)], {n, 1, Infinity}]+ I*Sum[Divide[Sin[n*\[Theta]],(n)^(2)], {n, 1, Infinity}] |
Failure | Successful | Skip | - |
| 25.12.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}\frac{\cos@{n\theta}}{n^{2}} = \frac{\pi^{2}}{6}-\frac{\pi\theta}{2}+\frac{\theta^{2}}{4}} | sum((cos(n*theta))/((n)^(2)), n = 1..infinity)=((Pi)^(2))/(6)-(Pi*theta)/(2)+((theta)^(2))/(4) |
Sum[Divide[Cos[n*\[Theta]],(n)^(2)], {n, 1, Infinity}]=Divide[(Pi)^(2),6]-Divide[Pi*\[Theta],2]+Divide[(\[Theta])^(2),4] |
Failure | Failure | Skip | Fail
Complex[-4.442882938158366, -4.442882938158366] <- {Rule[θ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-4.442882938158366, 4.442882938158366] <- {Rule[θ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 25.12.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}\frac{\sin@{n\theta}}{n^{2}} = -\int_{0}^{\theta}\ln@{2\sin@{\tfrac{1}{2}x}}\diff{x}} | sum((sin(n*theta))/((n)^(2)), n = 1..infinity)= - int(ln(2*sin((1)/(2)*x)), x = 0..theta) |
Sum[Divide[Sin[n*\[Theta]],(n)^(2)], {n, 1, Infinity}]= - Integrate[Log[2*Sin[Divide[1,2]*x]], {x, 0, \[Theta]}] |
Failure | Failure | Skip | Error |
| 25.12.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polylog{s}@{z} = \sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}} | polylog(s, z)= sum(((z)^(n))/((n)^(s)), n = 1..infinity) |
PolyLog[s, z]= Sum[Divide[(z)^(n),(n)^(s)], {n, 1, Infinity}] |
Failure | Successful | Skip | - |
| 25.12.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polylog{s}@{z} = \frac{z}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-z}\diff{x}} | polylog(s, z)=(z)/(GAMMA(s))*int(((x)^(s - 1))/(exp(x)- z), x = 0..infinity) |
PolyLog[s, z]=Divide[z,Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]- z], {x, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 25.12.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polylog{s}@{z} = \EulerGamma@{1-s}\left(\ln@@{\frac{1}{z}}\right)^{s-1}+\sum_{n=0}^{\infty}\Riemannzeta@{s-n}\frac{(\ln@@{z})^{n}}{n!}} | polylog(s, z)= GAMMA(1 - s)*(ln((1)/(z)))^(s - 1)+ sum(Zeta(s - n)*((ln(z))^(n))/(factorial(n)), n = 0..infinity) |
PolyLog[s, z]= Gamma[1 - s]*(Log[Divide[1,z]])^(s - 1)+ Sum[Zeta[s - n]*Divide[(Log[z])^(n),(n)!], {n, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 25.12.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polylog{s}@{e^{2\pi ia}}+e^{\pi is}\polylog{s}@{e^{-2\pi ia}} = \frac{(2\pi)^{s}e^{\pi is/2}}{\EulerGamma@{s}}\Hurwitzzeta@{1-s}{a}} | polylog(s, exp(2*Pi*I*a))+ exp(Pi*I*s)*polylog(s, exp(- 2*Pi*I*a))=((2*Pi)^(s)* exp(Pi*I*s/ 2))/(GAMMA(s))*Zeta(0, 1 - s, a) |
PolyLog[s, Exp[2*Pi*I*a]]+ Exp[Pi*I*s]*PolyLog[s, Exp[- 2*Pi*I*a]]=Divide[(2*Pi)^(s)* Exp[Pi*I*s/ 2],Gamma[s]]*HurwitzZeta[1 - s, a] |
Failure | Failure | Fail .5737863933-.4240983936*I <- {a = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2)}2281.720763-318.166068*I <- {a = 2^(1/2)+I*2^(1/2), s = 2^(1/2)-I*2^(1/2)}11.12441999-13.46800186*I <- {a = 2^(1/2)+I*2^(1/2), s = -2^(1/2)-I*2^(1/2)}-.5088647019e-2+.1836981228e-2*I <- {a = 2^(1/2)+I*2^(1/2), s = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Fail
Complex[0.5737863944300513, -0.4240983930049895] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[2281.720765767148, -318.1660682691354] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[11.124419974397522, -13.468001871634662] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-0.00508864702414036, 0.0018369812232921614] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[s, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 25.12#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle F_{s}(x) = -\polylog{s+1}@{-e^{x}}} | F[s]*(x)= - polylog(s + 1, - exp(x)) |
Subscript[F, s]*(x)= - PolyLog[s + 1, - Exp[x]] |
Failure | Failure | Fail -.701287556+.9004371571*I <- {s = 2^(1/2)+I*2^(1/2), F[s] = 2^(1/2)+I*2^(1/2), x = 1}-1.490772176+.967006968*I <- {s = 2^(1/2)+I*2^(1/2), F[s] = 2^(1/2)+I*2^(1/2), x = 2}-3.225675211-.894244793*I <- {s = 2^(1/2)+I*2^(1/2), F[s] = 2^(1/2)+I*2^(1/2), x = 3}-.701287556-1.927989967*I <- {s = 2^(1/2)+I*2^(1/2), F[s] = 2^(1/2)-I*2^(1/2), x = 1}... skip entries to safe data |
Successful |
| 25.12#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{s}(x) = \polylog{s+1}@{e^{x}}} | G[s]*(x)= polylog(s + 1, exp(x)) |
Subscript[G, s]*(x)= PolyLog[s + 1, Exp[x]] |
Failure | Failure | Fail -.592976910+2.518819642*I <- {s = 2^(1/2)+I*2^(1/2), G[s] = 2^(1/2)+I*2^(1/2), x = 1}-.593582812+5.469840344*I <- {s = 2^(1/2)+I*2^(1/2), G[s] = 2^(1/2)+I*2^(1/2), x = 2}-1.275024344+9.341921066*I <- {s = 2^(1/2)+I*2^(1/2), G[s] = 2^(1/2)+I*2^(1/2), x = 3}-.592976910-.309607482*I <- {s = 2^(1/2)+I*2^(1/2), G[s] = 2^(1/2)-I*2^(1/2), x = 1}... skip entries to safe data |
Successful |
| 25.14.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \LerchPhi@{1}{s}{a}} | Zeta(0, s, a)= LerchPhi(1, s, a) |
HurwitzZeta[s, a]= LerchPhi[1, s, a] |
Successful | Failure | - | Successful |
| 25.14.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polylog{s}@{z} = z\LerchPhi@{z}{s}{1}} | polylog(s, z)= z*LerchPhi(z, s, 1) |
PolyLog[s, z]= z*LerchPhi[z, s, 1] |
Successful | Successful | - | - |
| 25.14.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LerchPhi@{z}{s}{a} = z^{m}\LerchPhi@{z}{s}{a+m}+\sum_{n=0}^{m-1}\frac{z^{n}}{(a+n)^{s}}} | LerchPhi(z, s, a)= (z)^(m)* LerchPhi(z, s, a + m)+ sum(((z)^(n))/((a + n)^(s)), n = 0..m - 1) |
LerchPhi[z, s, a]= (z)^(m)* LerchPhi[z, s, a + m]+ Sum[Divide[(z)^(n),(a + n)^(s)], {n, 0, m - 1}] |
Failure | Successful | Skip | - |
| 25.14.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LerchPhi@{z}{s}{a} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-ze^{-x}}\diff{x}} | LerchPhi(z, s, a)=(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 - z*exp(- x)), x = 0..infinity) |
LerchPhi[z, s, a]=Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 - z*Exp[- x]], {x, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 25.14.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LerchPhi@{z}{s}{a} = \frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{s}}\diff{x}-2\int_{0}^{\infty}\frac{\sin@{x\ln@@{z}-s\atan@{x/a}}}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}} | LerchPhi(z, s, a)=(1)/(2)*(a)^(- s)+ int(((z)^(x))/((a + x)^(s)), x = 0..infinity)- 2*int((sin(x*ln(z)- s*arctan(x/ a)))/(((a)^(2)+ (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity) |
LerchPhi[z, s, a]=Divide[1,2]*(a)^(- s)+ Integrate[Divide[(z)^(x),(a + x)^(s)], {x, 0, Infinity}]- 2*Integrate[Divide[Sin[x*Log[z]- s*ArcTan[x/ a]],((a)^(2)+ (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 25.16.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{2}\Riemannzeta@{1-2a} = -\frac{\BernoullinumberB{2a}}{4a}} | (1)/(2)*Zeta(1 - 2*a)= -(bernoulli(2*a))/(4*a) |
Divide[1,2]*Zeta[1 - 2*a]= -Divide[BernoulliB[2*a],4*a] |
Failure | Failure | Successful | Successful |
| 25.16.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2} = \frac{17}{4}\Riemannzeta@{4}} | sum(((h*(n))/(n))^(2), n = 1..infinity)=(17)/(4)*Zeta(4) |
Sum[(Divide[h*(n),n])^(2), {n, 1, Infinity}]=Divide[17,4]*Zeta[4] |
Failure | Failure | Skip | Fail
Complex[-3.1856601808992417, 1.4142135623730951] <- {Rule[Sum[Power[h, 2], {n, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-3.1856601808992417, -1.4142135623730951] <- {Rule[Sum[Power[h, 2], {n, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-6.014087305645432, -1.4142135623730951] <- {Rule[Sum[Power[h, 2], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-6.014087305645432, 1.4142135623730951] <- {Rule[Sum[Power[h, 2], {n, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 25.16.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{r^{2}(r+k)} = \frac{3}{4}\Riemannzeta@{3}} | sum(sum((1)/((r)^(2)*(r + k)), k = 1..r), r = 1..infinity)=(3)/(4)*Zeta(3) |
Sum[Sum[Divide[1,(r)^(2)*(r + k)], {k, 1, r}], {r, 1, Infinity}]=Divide[3,4]*Zeta[3] |
Failure | Failure | Skip | Error |