Results of Orthogonal Polynomials

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18.1.E1 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 (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 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 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 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 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 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 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 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 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 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 (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 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 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 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 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 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 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 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 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 ChebyshevT(0, x)= 1 ChebyshevT[0, x]= 1 Successful Successful - -
18.5#Ex2 ChebyshevT(1, x)= x ChebyshevT[1, x]= x Successful Successful - -
18.5#Ex3 ChebyshevT(2, x)= 2*(x)^(2)- 1 ChebyshevT[2, x]= 2*(x)^(2)- 1 Successful Successful - -
18.5#Ex4 ChebyshevT(3, x)= 4*(x)^(3)- 3*x ChebyshevT[3, x]= 4*(x)^(3)- 3*x Successful Successful - -
18.5#Ex5 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 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 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 ChebyshevU(0, x)= 1 ChebyshevU[0, x]= 1 Successful Successful - -
18.5#Ex9 ChebyshevU(1, x)= 2*x ChebyshevU[1, x]= 2*x Successful Successful - -
18.5#Ex10 ChebyshevU(2, x)= 4*(x)^(2)- 1 ChebyshevU[2, x]= 4*(x)^(2)- 1 Successful Successful - -
18.5#Ex11 ChebyshevU(3, x)= 8*(x)^(3)- 4*x ChebyshevU[3, x]= 8*(x)^(3)- 4*x Successful Successful - -
18.5#Ex12 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 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 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 LegendreP(0, x)= 1 LegendreP[0, x]= 1 Successful Successful - -
18.5#Ex16 LegendreP(1, x)= x LegendreP[1, x]= x Successful Successful - -
18.5#Ex17 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 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 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 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 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 LaguerreL(0, x)= 1 Error Successful Error - -
18.5#Ex23 LaguerreL(1, x)= - x + 1 Error Successful Error - -
18.5#Ex24 LaguerreL(2, x)=(1)/(2)*(x)^(2)- 2*x + 1 Error Successful Error - -
18.5#Ex25 LaguerreL(3, x)= -(1)/(6)*(x)^(3)+(3)/(2)*(x)^(2)- 3*x + 1 Error Successful Error - -
18.5#Ex26 LaguerreL(4, x)=(1)/(24)*(x)^(4)-(2)/(3)*(x)^(3)+ 3*(x)^(2)- 4*x + 1 Error Successful Error - -
18.5#Ex27 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 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 HermiteH(0, x)= 1 HermiteH[0, x]= 1 Successful Successful - -
18.5#Ex30 HermiteH(1, x)= 2*x HermiteH[1, x]= 2*x Successful Successful - -
18.5#Ex31 HermiteH(2, x)= 4*(x)^(2)- 2 HermiteH[2, x]= 4*(x)^(2)- 2 Successful Successful - -
18.5#Ex32 HermiteH(3, x)= 8*(x)^(3)- 12*x HermiteH[3, x]= 8*(x)^(3)- 12*x Successful Successful - -
18.5#Ex33 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 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 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 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 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 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 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 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 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 ChebyshevU(n, x)= GegenbauerC(n, 1, x) ChebyshevU[n, x]= GegenbauerC[n, 1, x] Successful Successful - -
18.7.E4 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 LegendreP(n, x)= GegenbauerC(n, (1)/(2), x) LegendreP[n, x]= GegenbauerC[n, Divide[1,2], x] Successful Successful - -
18.7.E9 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 LegendreP(n, 2*(x) - 1)= LegendreP(n, 2*x - 1) Error Successful Error - -
18.7.E13 (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 (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 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 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 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 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 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 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 (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 (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 (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 (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 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 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 (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 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 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 (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 (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 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 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 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 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 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 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 (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 (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 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 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 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 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 ((- 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 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 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 (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 ((- 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 (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 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 (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 (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 (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 (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 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 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 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 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 ((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 (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 (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 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 (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 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 (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 (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 - 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 (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 (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 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 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 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 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 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 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 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 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 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 (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 (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 (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 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 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 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 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 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 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 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 (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 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 (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 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 x[n , 2]> .. Subscript[x, n , 2]> ... Failure Failure Skip Successful
18.17.E1 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 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 int(LaguerreL(m + n, y), y = 0..x)= LaguerreL(m + n, x)- LaguerreL(m + n + 1, x) Error Successful Error - -
18.17.E3 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 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 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 (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 (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 ((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 ((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 (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 (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 ((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 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 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 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 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 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 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 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 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 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 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 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 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 (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 (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 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 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 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 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 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 (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 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 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 (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 (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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 (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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 0 < q 0 < q Failure Failure Successful Successful
18.28.E11 q < 1 , a , b in real , a*b q < 1 , a , b \[Element]*Reals , a*b Failure Failure Successful Error
18.28.E11 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 1 , (a)^(- 1)* b < (q)^(- 1) 1 , (a)^(- 1)* b < (q)^(- 1) Failure Failure Skip -
18.28.E12 0 < q 0 < q Failure Failure Successful Successful
18.28.E12 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 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 0 , (a)^(- 1)* b < (q)^(- 1) 0 , (a)^(- 1)* b < (q)^(- 1) Failure Failure Skip -
18.28.E20 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 (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 (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 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 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 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 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 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 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 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 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 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 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 (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 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 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 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