Results of Mathieu Functions and Hill’s Equation

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
28.1#Ex15 Se n ⁢ ( s , z ) = ce n ⁡ ( z , q ) ce n ⁡ ( 0 , q ) subscript Se 𝑛 𝑠 𝑧 Mathieu-ce 𝑛 𝑧 𝑞 Mathieu-ce 𝑛 0 𝑞 {\displaystyle{\displaystyle\mathrm{Se}_{n}(s,z)=\dfrac{\mathrm{ce}_{n}\left(z% ,q\right)}{\mathrm{ce}_{n}\left(0,q\right)}}} S*exp(1)[n]*(s , z)=(MathieuCE(n, q, z))/(MathieuCE(n, q, 0)) S*Subscript[E*, n]*(s , z)=Divide[MathieuC[n, q, z],MathieuC[n, q, 0]] Failure Failure Error Error
28.1#Ex16 So n ⁢ ( s , z ) = se n ⁡ ( z , q ) se n ′ ⁡ ( 0 , q ) subscript So 𝑛 𝑠 𝑧 Mathieu-se 𝑛 𝑧 𝑞 diffop Mathieu-se 𝑛 1 0 𝑞 {\displaystyle{\displaystyle\mathrm{So}_{n}(s,z)=\dfrac{\mathrm{se}_{n}\left(z% ,q\right)}{\mathrm{se}_{n}'\left(0,q\right)}}} S*o[n]*(s , z)=(MathieuSE(n, q, z))/(subs( temp=0, diff( MathieuSE(n, q, temp), temp$(1) ) )) S*Subscript[o, n]*(s , z)=Divide[MathieuS[n, q, z],D[MathieuS[n, q, temp], {temp, 1}]/.temp-> 0] Failure Failure Skip Error
28.1#Ex17 Se n ⁢ ( c , z ) = ce n ⁡ ( z , q ) ce n ⁡ ( 0 , q ) subscript Se 𝑛 𝑐 𝑧 Mathieu-ce 𝑛 𝑧 𝑞 Mathieu-ce 𝑛 0 𝑞 {\displaystyle{\displaystyle\mathrm{Se}_{n}(c,z)=\dfrac{\mathrm{ce}_{n}\left(z% ,q\right)}{\mathrm{ce}_{n}\left(0,q\right)}}} S*exp(1)[n]*(c , z)=(MathieuCE(n, q, z))/(MathieuCE(n, q, 0)) S*Subscript[E*, n]*(c , z)=Divide[MathieuC[n, q, z],MathieuC[n, q, 0]] Failure Failure Error Error
28.1#Ex18 So n ⁢ ( c , z ) = se n ⁡ ( z , q ) se n ′ ⁡ ( 0 , q ) subscript So 𝑛 𝑐 𝑧 Mathieu-se 𝑛 𝑧 𝑞 diffop Mathieu-se 𝑛 1 0 𝑞 {\displaystyle{\displaystyle\mathrm{So}_{n}(c,z)=\dfrac{\mathrm{se}_{n}\left(z% ,q\right)}{\mathrm{se}_{n}'\left(0,q\right)}}} S*o[n]*(c , z)=(MathieuSE(n, q, z))/(subs( temp=0, diff( MathieuSE(n, q, temp), temp$(1) ) )) S*Subscript[o, n]*(c , z)=Divide[MathieuS[n, q, z],D[MathieuS[n, q, temp], {temp, 1}]/.temp-> 0] Failure Failure Skip Error
28.2.E14 w ⁢ ( z + π ) = e π ⁢ i ⁢ ν ⁢ w ⁢ ( z ) 𝑤 𝑧 𝜋 superscript 𝑒 𝜋 imaginary-unit 𝜈 𝑤 𝑧 {\displaystyle{\displaystyle w(z+\pi)=e^{\pi\mathrm{i}\nu}w(z)}} w*(z + Pi)= exp(Pi*I*nu)*w*(z) w*(z + Pi)= Exp[Pi*I*\[Nu]]*w*(z) Failure Failure
Fail
4.397533327+8.455409696*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
8.455409696+4.488232549*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
4.488232549+.4303561791*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
.4303561791+4.397533327*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[4.397533326856219, 8.455409698756243] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-323.35991533293776, 98.99067506580616] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[332.24568120925454, 98.99067506580616] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[4.4882325494605135, 8.455409698756243] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
28.2.E17 w ⁢ ( z + π ) + w ⁢ ( z - π ) = 2 ⁢ cos ⁡ ( π ⁢ ν ) ⁢ w ⁢ ( z ) 𝑤 𝑧 𝜋 𝑤 𝑧 𝜋 2 𝜋 𝜈 𝑤 𝑧 {\displaystyle{\displaystyle w(z+\pi)+w(z-\pi)=2\cos\left(\pi\nu\right)w(z)}} w*(z + Pi)+ w*(z - Pi)= 2*cos(Pi*nu)*w*(z) w*(z + Pi)+ w*(z - Pi)= 2*Cos[Pi*\[Nu]]*w*(z) Failure Failure
Fail
327.7574484+98.56031892*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
98.56031892-327.7574484*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-327.7574484-98.56031892*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-98.56031892+327.7574484*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[327.757448659794, 98.56031888824567] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-327.757448659794, 98.56031888824567] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[327.757448659794, 98.56031888824567] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-327.757448659794, 98.56031888824567] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
28.2.E18 w ⁢ ( z ) = ∑ n = - ∞ ∞ c 2 ⁢ n ⁢ e i ⁢ ( ν + 2 ⁢ n ) ⁢ z 𝑤 𝑧 superscript subscript 𝑛 subscript 𝑐 2 𝑛 superscript 𝑒 imaginary-unit 𝜈 2 𝑛 𝑧 {\displaystyle{\displaystyle w(z)=\sum_{n=-\infty}^{\infty}c_{2n}e^{\mathrm{i}% (\nu+2n)z}}} w*(z)= sum(c[2*n]*exp(I*(nu + 2*n)* z), n = - infinity..infinity) w*(z)= Sum[Subscript[c, 2*n]*Exp[I*(\[Nu]+ 2*n)* z], {n, - Infinity, Infinity}] Failure Failure Skip Skip
28.2.E21 0 , & ⁢ a = a 2 ⁢ n ⁡ ( q ) , w I ⁢ ( 1 2 ⁢ π ; a , q ) formulae-sequence 0 & 𝑎 Mathieu-eigenvalue-a 2 𝑛 𝑞 subscript 𝑤 I 1 2 𝜋 𝑎 𝑞 {\displaystyle{\displaystyle 0,&a=a_{2n}\left(q\right),\\ w_{\mbox{\tiny I}}(\tfrac{1}{2}\pi;a,q)}} 0 , 0 , Error Error - -
28.2.E22 0 , & ⁢ a = b 2 ⁢ n + 1 ⁡ ( q ) , w II ⁢ ( 1 2 ⁢ π ; a , q ) formulae-sequence 0 & 𝑎 Mathieu-eigenvalue-b 2 𝑛 1 𝑞 subscript 𝑤 II 1 2 𝜋 𝑎 𝑞 {\displaystyle{\displaystyle 0,&a=b_{2n+1}\left(q\right),\\ w_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a,q)}} 0 , 0 , Error Error - -
28.2.E23 a n ⁡ ( 0 ) = n 2 Mathieu-eigenvalue-a 𝑛 0 superscript 𝑛 2 {\displaystyle{\displaystyle a_{n}\left(0\right)=n^{2}}} MathieuA(n, 0)= (n)^(2) MathieuCharacteristicA[n, 0]= (n)^(2) Successful Successful - -
28.2.E24 b n ⁡ ( 0 ) = n 2 Mathieu-eigenvalue-b 𝑛 0 superscript 𝑛 2 {\displaystyle{\displaystyle b_{n}\left(0\right)=n^{2}}} MathieuB(n, 0)= (n)^(2) MathieuCharacteristicB[n, 0]= (n)^(2) Successful Successful - -
28.2.E25 b 1 ⁡ < a 1 ⁡ Mathieu-eigenvalue-b 1 𝑞 Mathieu-eigenvalue-a 1 𝑞 {\displaystyle{\displaystyle b_{1}<a_{1}}} MathieuB(1, q)< MathieuA(1, q) MathieuCharacteristicB[1, q]< MathieuCharacteristicA[1, q] Failure Failure Successful Successful
28.2.E25 a 1 ⁡ < b 2 ⁡ Mathieu-eigenvalue-a 1 𝑞 Mathieu-eigenvalue-b 2 𝑞 {\displaystyle{\displaystyle a_{1}<b_{2}}} MathieuA(1, q)< MathieuB(2, q) MathieuCharacteristicA[1, q]< MathieuCharacteristicB[2, q] Failure Failure Skip Successful
28.2.E25 b 2 ⁡ < a 2 ⁡ Mathieu-eigenvalue-b 2 𝑞 Mathieu-eigenvalue-a 2 𝑞 {\displaystyle{\displaystyle b_{2}<a_{2}}} MathieuB(2, q)< MathieuA(2, q) MathieuCharacteristicB[2, q]< MathieuCharacteristicA[2, q] Failure Failure Skip Successful
28.2.E25 a 2 ⁡ < b 3 ⁡ Mathieu-eigenvalue-a 2 𝑞 Mathieu-eigenvalue-b 3 𝑞 {\displaystyle{\displaystyle a_{2}<b_{3}}} MathieuA(2, q)< MathieuB(3, q) MathieuCharacteristicA[2, q]< MathieuCharacteristicB[3, q] Failure Failure Skip Successful
28.2.E25 a 1 ⁡ < b 1 ⁡ Mathieu-eigenvalue-a 1 𝑞 Mathieu-eigenvalue-b 1 𝑞 {\displaystyle{\displaystyle a_{1}<b_{1}}} MathieuA(1, q)< MathieuB(1, q) MathieuCharacteristicA[1, q]< MathieuCharacteristicB[1, q] Failure Failure Skip Successful
28.2.E26 a 2 ⁢ n ⁡ ( - q ) = a 2 ⁢ n ⁡ ( q ) Mathieu-eigenvalue-a 2 𝑛 𝑞 Mathieu-eigenvalue-a 2 𝑛 𝑞 {\displaystyle{\displaystyle a_{2n}\left(-q\right)=a_{2n}\left(q\right)}} MathieuA(2*n, - q)= MathieuA(2*n, q) MathieuCharacteristicA[2*n, - q]= MathieuCharacteristicA[2*n, q] Failure Failure Successful Successful
28.2.E27 a 2 ⁢ n + 1 ⁡ ( - q ) = b 2 ⁢ n + 1 ⁡ ( q ) Mathieu-eigenvalue-a 2 𝑛 1 𝑞 Mathieu-eigenvalue-b 2 𝑛 1 𝑞 {\displaystyle{\displaystyle a_{2n+1}\left(-q\right)=b_{2n+1}\left(q\right)}} MathieuA(2*n + 1, - q)= MathieuB(2*n + 1, q) MathieuCharacteristicA[2*n + 1, - q]= MathieuCharacteristicB[2*n + 1, q] Failure Failure Successful Successful
28.2.E28 b 2 ⁢ n + 2 ⁡ ( - q ) = b 2 ⁢ n + 2 ⁡ ( q ) Mathieu-eigenvalue-b 2 𝑛 2 𝑞 Mathieu-eigenvalue-b 2 𝑛 2 𝑞 {\displaystyle{\displaystyle b_{2n+2}\left(-q\right)=b_{2n+2}\left(q\right)}} MathieuB(2*n + 2, - q)= MathieuB(2*n + 2, q) MathieuCharacteristicB[2*n + 2, - q]= MathieuCharacteristicB[2*n + 2, q] Failure Failure Successful Successful
28.2#Ex4 ce 0 ⁡ ( z , 0 ) = 1 / 2 Mathieu-ce 0 𝑧 0 1 2 {\displaystyle{\displaystyle\mathrm{ce}_{0}\left(z,0\right)=1/\sqrt{2}}} MathieuCE(0, 0, z)= 1/sqrt(2) MathieuC[0, 0, z]= 1/Sqrt[2] Failure Successful - -
28.2#Ex5 ce n ⁡ ( z , 0 ) = cos ⁡ ( n ⁢ z ) Mathieu-ce 𝑛 𝑧 0 𝑛 𝑧 {\displaystyle{\displaystyle\mathrm{ce}_{n}\left(z,0\right)=\cos\left(nz\right% )}} MathieuCE(n, 0, z)= cos(n*z) MathieuC[n, 0, z]= Cos[n*z] Successful Failure -
Fail
Complex[6.510464566583027, -0.7008927266021505] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[11.262006072549275, -34.68845328285399] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[6.510464566583027, 0.7008927266021505] <- {Rule[n, 2], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[11.262006072549275, 34.68845328285399] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
28.2#Ex6 se n ⁡ ( z , 0 ) = sin ⁡ ( n ⁢ z ) Mathieu-se 𝑛 𝑧 0 𝑛 𝑧 {\displaystyle{\displaystyle\mathrm{se}_{n}\left(z,0\right)=\sin\left(nz\right% )}} MathieuSE(n, 0, z)= sin(n*z) MathieuS[n, 0, z]= Sin[n*z] Successful Failure -
Fail
Complex[0.8057439410809155, 6.510552888635757] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[34.75636341426024, 11.321973340262975] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8057439410809155, -6.510552888635757] <- {Rule[n, 2], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[34.75636341426024, -11.321973340262975] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
28.2#Ex7 ∫ 0 2 ⁢ π ( ce n ⁡ ( x , q ) ) 2 ⁢ d x = π superscript subscript 0 2 𝜋 superscript Mathieu-ce 𝑛 𝑥 𝑞 2 𝑥 𝜋 {\displaystyle{\displaystyle\int_{0}^{2\pi}\left(\mathrm{ce}_{n}\left(x,q% \right)\right)^{2}\mathrm{d}x=\pi}} int((MathieuCE(n, q, x))^(2), x = 0..2*Pi)= Pi Integrate[(MathieuC[n, q, x])^(2), {x, 0, 2*Pi}]= Pi Failure Failure Skip
Fail
Complex[-1.727379091216698, 1.4142135623730951] <- {Rule[Integrate[Power[MathieuC[n, q, x], 2], {x, 0, Times[2, Pi]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.727379091216698, -1.4142135623730951] <- {Rule[Integrate[Power[MathieuC[n, q, x], 2], {x, 0, Times[2, Pi]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-4.555806215962888, -1.4142135623730951] <- {Rule[Integrate[Power[MathieuC[n, q, x], 2], {x, 0, Times[2, Pi]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-4.555806215962888, 1.4142135623730951] <- {Rule[Integrate[Power[MathieuC[n, q, x], 2], {x, 0, Times[2, Pi]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
28.2#Ex8 ∫ 0 2 ⁢ π ( se n ⁡ ( x , q ) ) 2 ⁢ d x = π superscript subscript 0 2 𝜋 superscript Mathieu-se 𝑛 𝑥 𝑞 2 𝑥 𝜋 {\displaystyle{\displaystyle\int_{0}^{2\pi}\left(\mathrm{se}_{n}\left(x,q% \right)\right)^{2}\mathrm{d}x=\pi}} int((MathieuSE(n, q, x))^(2), x = 0..2*Pi)= Pi Integrate[(MathieuS[n, q, x])^(2), {x, 0, 2*Pi}]= Pi Failure Failure Skip
Fail
Complex[-1.727379091216698, 1.4142135623730951] <- {Rule[Integrate[Power[MathieuS[n, q, x], 2], {x, 0, Times[2, Pi]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.727379091216698, -1.4142135623730951] <- {Rule[Integrate[Power[MathieuS[n, q, x], 2], {x, 0, Times[2, Pi]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-4.555806215962888, -1.4142135623730951] <- {Rule[Integrate[Power[MathieuS[n, q, x], 2], {x, 0, Times[2, Pi]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-4.555806215962888, 1.4142135623730951] <- {Rule[Integrate[Power[MathieuS[n, q, x], 2], {x, 0, Times[2, Pi]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
28.2.E31 ∫ 0 2 ⁢ π ce m ⁡ ( x , q ) ⁢ ce n ⁡ ( x , q ) ⁢ d x = 0 superscript subscript 0 2 𝜋 Mathieu-ce 𝑚 𝑥 𝑞 Mathieu-ce 𝑛 𝑥 𝑞 𝑥 0 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathrm{ce}_{m}\left(x,q\right)% \mathrm{ce}_{n}\left(x,q\right)\mathrm{d}x=0}} int(MathieuCE(m, q, x)*MathieuCE(n, q, x), x = 0..2*Pi)= 0 Integrate[MathieuC[m, q, x]*MathieuC[n, q, x], {x, 0, 2*Pi}]= 0 Failure Failure Skip Successful
28.2.E32 ∫ 0 2 ⁢ π se m ⁡ ( x , q ) ⁢ se n ⁡ ( x , q ) ⁢ d x = 0 superscript subscript 0 2 𝜋 Mathieu-se 𝑚 𝑥 𝑞 Mathieu-se 𝑛 𝑥 𝑞 𝑥 0 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathrm{se}_{m}\left(x,q\right)% \mathrm{se}_{n}\left(x,q\right)\mathrm{d}x=0}} int(MathieuSE(m, q, x)*MathieuSE(n, q, x), x = 0..2*Pi)= 0 Integrate[MathieuS[m, q, x]*MathieuS[n, q, x], {x, 0, 2*Pi}]= 0 Failure Failure Skip Successful
28.2.E33 ∫ 0 2 ⁢ π ce m ⁡ ( x , q ) ⁢ se n ⁡ ( x , q ) ⁢ d x = 0 superscript subscript 0 2 𝜋 Mathieu-ce 𝑚 𝑥 𝑞 Mathieu-se 𝑛 𝑥 𝑞 𝑥 0 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathrm{ce}_{m}\left(x,q\right)% \mathrm{se}_{n}\left(x,q\right)\mathrm{d}x=0}} int(MathieuCE(m, q, x)*MathieuSE(n, q, x), x = 0..2*Pi)= 0 Integrate[MathieuC[m, q, x]*MathieuS[n, q, x], {x, 0, 2*Pi}]= 0 Failure Failure Skip
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Integrate[Times[MathieuC[m, q, x], MathieuS[n, q, x]], {x, 0, Times[2, Pi]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Integrate[Times[MathieuC[m, q, x], MathieuS[n, q, x]], {x, 0, Times[2, Pi]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Integrate[Times[MathieuC[m, q, x], MathieuS[n, q, x]], {x, 0, Times[2, Pi]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Integrate[Times[MathieuC[m, q, x], MathieuS[n, q, x]], {x, 0, Times[2, Pi]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
28.2.E34 ce 2 ⁢ n ⁡ ( z , - q ) = ( - 1 ) n ⁢ ce 2 ⁢ n ⁡ ( 1 2 ⁢ π - z , q ) Mathieu-ce 2 𝑛 𝑧 𝑞 superscript 1 𝑛 Mathieu-ce 2 𝑛 1 2 𝜋 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{2n}\left(z,-q\right)=(-1)^{n}\mathrm{% ce}_{2n}\left(\tfrac{1}{2}\pi-z,q\right)}} MathieuCE(2*n, - q, z)=(- 1)^(n)* MathieuCE(2*n, q, (1)/(2)*Pi - z) MathieuC[2*n, - q, z]=(- 1)^(n)* MathieuC[2*n, q, Divide[1,2]*Pi - z] Failure Failure Successful
Fail
Complex[-35.48938782449842, 4.362541390270897] <- {Rule[n, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[8.113008541051663, 1.5552980263308331] <- {Rule[n, 2], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[35.468444164343595, -60.307504750945526] <- {Rule[n, 3], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.4218686670701555, 4.355505323618388] <- {Rule[n, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
28.2.E35 ce 2 ⁢ n + 1 ⁡ ( z , - q ) = ( - 1 ) n ⁢ se 2 ⁢ n + 1 ⁡ ( 1 2 ⁢ π - z , q ) Mathieu-ce 2 𝑛 1 𝑧 𝑞 superscript 1 𝑛 Mathieu-se 2 𝑛 1 1 2 𝜋 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{2n+1}\left(z,-q\right)=(-1)^{n}% \mathrm{se}_{2n+1}\left(\tfrac{1}{2}\pi-z,q\right)}} MathieuCE(2*n + 1, - q, z)=(- 1)^(n)* MathieuSE(2*n + 1, q, (1)/(2)*Pi - z) MathieuC[2*n + 1, - q, z]=(- 1)^(n)* MathieuS[2*n + 1, q, Divide[1,2]*Pi - z] Failure Failure Successful
Fail
Complex[23.78813652597893, -3.3722327992542382] <- {Rule[n, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[44.411224266286126, -43.54652180033389] <- {Rule[n, 2], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[31.14066048310312, -63.12632131709884] <- {Rule[n, 3], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.8575002662710154, -10.873868123505298] <- {Rule[n, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
28.2.E36 se 2 ⁢ n + 1 ⁡ ( z , - q ) = ( - 1 ) n ⁢ ce 2 ⁢ n + 1 ⁡ ( 1 2 ⁢ π - z , q ) Mathieu-se 2 𝑛 1 𝑧 𝑞 superscript 1 𝑛 Mathieu-ce 2 𝑛 1 1 2 𝜋 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{2n+1}\left(z,-q\right)=(-1)^{n}% \mathrm{ce}_{2n+1}\left(\tfrac{1}{2}\pi-z,q\right)}} MathieuSE(2*n + 1, - q, z)=(- 1)^(n)* MathieuCE(2*n + 1, q, (1)/(2)*Pi - z) MathieuS[2*n + 1, - q, z]=(- 1)^(n)* MathieuC[2*n + 1, q, Divide[1,2]*Pi - z] Failure Failure Successful
Fail
Complex[-2.359402591966014, 27.60697614269746] <- {Rule[n, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[47.446183037810655, 46.79242964252569] <- {Rule[n, 2], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[63.57245907030711, 29.231688225822445] <- {Rule[n, 3], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-9.92152066646148, 3.1462660649328162] <- {Rule[n, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
28.2.E37 se 2 ⁢ n + 2 ⁡ ( z , - q ) = ( - 1 ) n ⁢ se 2 ⁢ n + 2 ⁡ ( 1 2 ⁢ π - z , q ) Mathieu-se 2 𝑛 2 𝑧 𝑞 superscript 1 𝑛 Mathieu-se 2 𝑛 2 1 2 𝜋 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{2n+2}\left(z,-q\right)=(-1)^{n}% \mathrm{se}_{2n+2}\left(\tfrac{1}{2}\pi-z,q\right)}} MathieuSE(2*n + 2, - q, z)=(- 1)^(n)* MathieuSE(2*n + 2, q, (1)/(2)*Pi - z) MathieuS[2*n + 2, - q, z]=(- 1)^(n)* MathieuS[2*n + 2, q, Divide[1,2]*Pi - z] Failure Failure
Fail
-33.89568074-158.8507182*I <- {q = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1}
-1663.167618+2486.508244*I <- {q = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2}
50746.06591-21466.91354*I <- {q = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3}
43.05202189+35.26439985*I <- {q = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
Fail
Complex[-13.013045011746364, -6.523555599290751] <- {Rule[n, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[60.43374347015696, 35.92848167143245] <- {Rule[n, 2], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[80.47427800717344, 35.59929164615286] <- {Rule[n, 3], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.5195173062774003, 0.020895058047888604] <- {Rule[n, 1], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
28.4.E1 ce 2 ⁢ n ⁡ ( z , q ) = ∑ m = 0 ∞ A 2 ⁢ m 2 ⁢ n ⁢ ( q ) ⁢ cos ⁡ 2 ⁢ m ⁢ z Mathieu-ce 2 𝑛 𝑧 𝑞 superscript subscript 𝑚 0 subscript superscript 𝐴 2 𝑛 2 𝑚 𝑞 2 𝑚 𝑧 {\displaystyle{\displaystyle\mathrm{ce}_{2n}\left(z,q\right)=\sum_{m=0}^{% \infty}A^{2n}_{2m}(q)\cos 2mz}} MathieuCE(2*n, q, z)sum(A(A[2*m])^(2*n)*(q)* cos(2*m*z), m = 0..infinity) MathieuC[2*n, q, z]Sum[A(Subscript[A, 2*m])^(2*n)*(q)* Cos[2*m*z], {m, 0, Infinity}] Failure Failure Skip Skip
28.4.E2 ce 2 ⁢ n + 1 ⁡ ( z , q ) = ∑ m = 0 ∞ A 2 ⁢ m + 1 2 ⁢ n + 1 ⁢ ( q ) ⁢ cos ⁡ ( 2 ⁢ m + 1 ) ⁢ z Mathieu-ce 2 𝑛 1 𝑧 𝑞 superscript subscript 𝑚 0 subscript superscript 𝐴 2 𝑛 1 2 𝑚 1 𝑞 2 𝑚 1 𝑧 {\displaystyle{\displaystyle\mathrm{ce}_{2n+1}\left(z,q\right)=\sum_{m=0}^{% \infty}A^{2n+1}_{2m+1}(q)\cos(2m+1)z}} MathieuCE(2*n + 1, q, z)sum(A(A[2*m + 1])^(2*n + 1)*(q)* cos((2*m + 1)* z), m = 0..infinity) MathieuC[2*n + 1, q, z]Sum[A(Subscript[A, 2*m + 1])^(2*n + 1)*(q)* Cos[(2*m + 1)* z], {m, 0, Infinity}] Failure Failure Skip Skip
28.4.E3 se 2 ⁢ n + 1 ⁡ ( z , q ) = ∑ m = 0 ∞ B 2 ⁢ m + 1 2 ⁢ n + 1 ⁢ ( q ) ⁢ sin ⁡ ( 2 ⁢ m + 1 ) ⁢ z Mathieu-se 2 𝑛 1 𝑧 𝑞 superscript subscript 𝑚 0 subscript superscript 𝐵 2 𝑛 1 2 𝑚 1 𝑞 2 𝑚 1 𝑧 {\displaystyle{\displaystyle\mathrm{se}_{2n+1}\left(z,q\right)=\sum_{m=0}^{% \infty}B^{2n+1}_{2m+1}(q)\sin(2m+1)z}} MathieuSE(2*n + 1, q, z)sum(B(B[2*m + 1])^(2*n + 1)*(q)* sin((2*m + 1)* z), m = 0..infinity) MathieuS[2*n + 1, q, z]Sum[B(Subscript[B, 2*m + 1])^(2*n + 1)*(q)* Sin[(2*m + 1)* z], {m, 0, Infinity}] Failure Failure Skip Skip
28.4.E4 se 2 ⁢ n + 2 ⁡ ( z , q ) = ∑ m = 0 ∞ B 2 ⁢ m + 2 2 ⁢ n + 2 ⁢ ( q ) ⁢ sin ⁡ ( 2 ⁢ m + 2 ) ⁢ z Mathieu-se 2 𝑛 2 𝑧 𝑞 superscript subscript 𝑚 0 subscript superscript 𝐵 2 𝑛 2 2 𝑚 2 𝑞 2 𝑚 2 𝑧 {\displaystyle{\displaystyle\mathrm{se}_{2n+2}\left(z,q\right)=\sum_{m=0}^{% \infty}B^{2n+2}_{2m+2}(q)\sin(2m+2)z}} MathieuSE(2*n + 2, q, z)sum(B(B[2*m + 2])^(2*n + 2)*(q)* sin((2*m + 2)* z), m = 0..infinity) MathieuS[2*n + 2, q, z]Sum[B(Subscript[B, 2*m + 2])^(2*n + 2)*(q)* Sin[(2*m + 2)* z], {m, 0, Infinity}] Failure Failure Skip Skip
28.5.E5 ( C n ⁢ ( q ) ) 2 ⁢ ∫ 0 2 ⁢ π ( f n ⁢ ( x , q ) ) 2 ⁢ d x = ( S n ⁢ ( q ) ) 2 ⁢ ∫ 0 2 ⁢ π ( g n ⁢ ( x , q ) ) 2 ⁢ d x superscript subscript 𝐶 𝑛 𝑞 2 superscript subscript 0 2 𝜋 superscript subscript 𝑓 𝑛 𝑥 𝑞 2 𝑥 superscript subscript 𝑆 𝑛 𝑞 2 superscript subscript 0 2 𝜋 superscript subscript 𝑔 𝑛 𝑥 𝑞 2 𝑥 {\displaystyle{\displaystyle(C_{n}(q))^{2}\int_{0}^{2\pi}(f_{n}(x,q))^{2}% \mathrm{d}x=(S_{n}(q))^{2}\int_{0}^{2\pi}(g_{n}(x,q))^{2}\mathrm{d}x}} (C[n]*(q))^(2)* int((f[n]*(x , q))^(2), x = 0..2*Pi)=(S[n]*(q))^(2)* int((g[n]*(x , q))^(2), x = 0..2*Pi) (Subscript[C, n]*(q))^(2)* Integrate[(Subscript[f, n]*(x , q))^(2), {x, 0, 2*Pi}]=(Subscript[S, n]*(q))^(2)* Integrate[(Subscript[g, n]*(x , q))^(2), {x, 0, 2*Pi}] Failure Failure Skip Error
28.5.E5 ( S n ⁢ ( q ) ) 2 ⁢ ∫ 0 2 ⁢ π ( g n ⁢ ( x , q ) ) 2 ⁢ d x = π superscript subscript 𝑆 𝑛 𝑞 2 superscript subscript 0 2 𝜋 superscript subscript 𝑔 𝑛 𝑥 𝑞 2 𝑥 𝜋 {\displaystyle{\displaystyle(S_{n}(q))^{2}\int_{0}^{2\pi}(g_{n}(x,q))^{2}% \mathrm{d}x=\pi}} (S[n]*(q))^(2)* int((g[n]*(x , q))^(2), x = 0..2*Pi)= Pi (Subscript[S, n]*(q))^(2)* Integrate[(Subscript[g, n]*(x , q))^(2), {x, 0, 2*Pi}]= Pi Failure Failure Skip Error
28.6.E20 lim inf n → ∞ ⁡ ρ n ( j ) n 2 ≥ k ⁢ k ′ ⁢ ( K ⁡ ( k ) ) 2 subscript limit-infimum → 𝑛 superscript subscript 𝜌 𝑛 𝑗 superscript 𝑛 2 𝑘 superscript 𝑘 ′ superscript complete-elliptic-integral-first-kind-K 𝑘 2 {\displaystyle{\displaystyle\liminf_{n\to\infty}\frac{\rho_{n}^{(j)}}{n^{2}}>=% kk^{\prime}(K\left(k\right))^{2}}} (rho(rho[n])^(j))/((n)^(2))> = k*sqrt(1 - (k)^(2))*(EllipticK(k))^(2) Divide[\[Rho](Subscript[\[Rho], n])^(j),(n)^(2)]> = k*Sqrt[1 - (k)^(2)]*(EllipticK[(k)^2])^(2) Failure Failure Skip Successful
28.7.E1 ∑ n = 0 ∞ ( a 2 ⁢ n ⁡ ( q ) - ( 2 ⁢ n ) 2 ) = 0 superscript subscript 𝑛 0 Mathieu-eigenvalue-a 2 𝑛 𝑞 superscript 2 𝑛 2 0 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\left(a_{2n}\left(q\right)-(2n)% ^{2}\right)=0}} sum(MathieuA(2*n, q)-(2*n)^(2), n = 0..infinity)= 0 Sum[MathieuCharacteristicA[2*n, q]-(2*n)^(2), {n, 0, Infinity}]= 0 Failure Failure Skip
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Sum[Plus[Times[-4, Power[n, 2]], MathieuCharacteristicA[Times[2, n], q]], {n, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Sum[Plus[Times[-4, Power[n, 2]], MathieuCharacteristicA[Times[2, n], q]], {n, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Sum[Plus[Times[-4, Power[n, 2]], MathieuCharacteristicA[Times[2, n], q]], {n, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Sum[Plus[Times[-4, Power[n, 2]], MathieuCharacteristicA[Times[2, n], q]], {n, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
28.7.E2 ∑ n = 0 ∞ ( a 2 ⁢ n + 1 ⁡ ( q ) - ( 2 ⁢ n + 1 ) 2 ) = q superscript subscript 𝑛 0 Mathieu-eigenvalue-a 2 𝑛 1 𝑞 superscript 2 𝑛 1 2 𝑞 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\left(a_{2n+1}\left(q\right)-(2% n+1)^{2}\right)=q}} sum(MathieuA(2*n + 1, q)-(2*n + 1)^(2), n = 0..infinity)= q Sum[MathieuCharacteristicA[2*n + 1, q]-(2*n + 1)^(2), {n, 0, Infinity}]= q Failure Failure Skip Skip
28.7.E3 ∑ n = 0 ∞ ( b 2 ⁢ n + 1 ⁡ ( q ) - ( 2 ⁢ n + 1 ) 2 ) = - q superscript subscript 𝑛 0 Mathieu-eigenvalue-b 2 𝑛 1 𝑞 superscript 2 𝑛 1 2 𝑞 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\left(b_{2n+1}\left(q\right)-(2% n+1)^{2}\right)=-q}} sum(MathieuB(2*n + 1, q)-(2*n + 1)^(2), n = 0..infinity)= - q Sum[MathieuCharacteristicB[2*n + 1, q]-(2*n + 1)^(2), {n, 0, Infinity}]= - q Failure Failure Skip Skip
28.7.E4 ∑ n = 0 ∞ ( b 2 ⁢ n + 2 ⁡ ( q ) - ( 2 ⁢ n + 2 ) 2 ) = 0 superscript subscript 𝑛 0 Mathieu-eigenvalue-b 2 𝑛 2 𝑞 superscript 2 𝑛 2 2 0 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\left(b_{2n+2}\left(q\right)-(2% n+2)^{2}\right)=0}} sum(MathieuB(2*n + 2, q)-(2*n + 2)^(2), n = 0..infinity)= 0 Sum[MathieuCharacteristicB[2*n + 2, q]-(2*n + 2)^(2), {n, 0, Infinity}]= 0 Failure Failure Skip
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Sum[Plus[Times[-1, Power[Plus[2, Times[2, n]], 2]], MathieuCharacteristicB[Plus[2, Times[2, n]], q]], {n, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Sum[Plus[Times[-1, Power[Plus[2, Times[2, n]], 2]], MathieuCharacteristicB[Plus[2, Times[2, n]], q]], {n, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Sum[Plus[Times[-1, Power[Plus[2, Times[2, n]], 2]], MathieuCharacteristicB[Plus[2, Times[2, n]], q]], {n, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Sum[Plus[Times[-1, Power[Plus[2, Times[2, n]], 2]], MathieuCharacteristicB[Plus[2, Times[2, n]], q]], {n, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
28.8#Ex3 ce m ⁡ ( x , h 2 ) ce m ⁡ ( 0 , h 2 ) = 2 m - ( 1 / 2 ) σ m ⁢ ( W m + ⁢ ( x ) ⁢ ( P m ⁢ ( x ) - Q m ⁢ ( x ) ) + W m - ⁢ ( x ) ⁢ ( P m ⁢ ( x ) + Q m ⁢ ( x ) ) ) Mathieu-ce 𝑚 𝑥 superscript ℎ 2 Mathieu-ce 𝑚 0 superscript ℎ 2 superscript 2 𝑚 1 2 subscript 𝜎 𝑚 superscript subscript 𝑊 𝑚 𝑥 subscript 𝑃 𝑚 𝑥 subscript 𝑄 𝑚 𝑥 superscript subscript 𝑊 𝑚 𝑥 subscript 𝑃 𝑚 𝑥 subscript 𝑄 𝑚 𝑥 {\displaystyle{\displaystyle\dfrac{\mathrm{ce}_{m}\left(x,h^{2}\right)}{% \mathrm{ce}_{m}\left(0,h^{2}\right)}=\dfrac{2^{m-(\ifrac{1}{2})}}{\sigma_{m}}% \left(W_{m}^{+}(x)(P_{m}(x)-Q_{m}(x))+W_{m}^{-}(x)(P_{m}(x)+Q_{m}(x))\right)}} (MathieuCE(m, (h)^(2), x))/(MathieuCE(m, (h)^(2), 0))(W(W[m])^(+)*(x)*(P[m]*(x)- Q[m]*(x))+ W(W[m])^(-)*(x)*(P[m]*(x)+ Q[m]*(x))) Divide[MathieuC[m, (h)^(2), x],MathieuC[m, (h)^(2), 0]](W(Subscript[W, m])^(+)*(x)*(Subscript[P, m]*(x)- Subscript[Q, m]*(x))+ W(Subscript[W, m])^(-)*(x)*(Subscript[P, m]*(x)+ Subscript[Q, m]*(x))) Error Failure - Error
28.8#Ex4 se m + 1 ⁡ ( x , h 2 ) se m + 1 ′ ⁡ ( 0 , h 2 ) = 2 m - ( 1 / 2 ) τ m + 1 ⁢ ( W m + ⁢ ( x ) ⁢ ( P m ⁢ ( x ) - Q m ⁢ ( x ) ) - W m - ⁢ ( x ) ⁢ ( P m ⁢ ( x ) + Q m ⁢ ( x ) ) ) Mathieu-se 𝑚 1 𝑥 superscript ℎ 2 diffop Mathieu-se 𝑚 1 1 0 superscript ℎ 2 superscript 2 𝑚 1 2 subscript 𝜏 𝑚 1 superscript subscript 𝑊 𝑚 𝑥 subscript 𝑃 𝑚 𝑥 subscript 𝑄 𝑚 𝑥 superscript subscript 𝑊 𝑚 𝑥 subscript 𝑃 𝑚 𝑥 subscript 𝑄 𝑚 𝑥 {\displaystyle{\displaystyle\dfrac{\mathrm{se}_{m+1}\left(x,h^{2}\right)}{% \mathrm{se}_{m+1}'\left(0,h^{2}\right)}=\dfrac{2^{m-(\ifrac{1}{2})}}{\tau_{m+1% }}\left(W_{m}^{+}(x)(P_{m}(x)-Q_{m}(x))-W_{m}^{-}(x)(P_{m}(x)+Q_{m}(x))\right)}} (MathieuSE(m + 1, (h)^(2), x))/(subs( temp=0, diff( MathieuSE(m + 1, (h)^(2), temp), temp$(1) ) ))(W(W[m])^(+)*(x)*(P[m]*(x)- Q[m]*(x))- W(W[m])^(-)*(x)*(P[m]*(x)+ Q[m]*(x))) Divide[MathieuS[m + 1, (h)^(2), x],D[MathieuS[m + 1, (h)^(2), temp], {temp, 1}]/.temp-> 0](W(Subscript[W, m])^(+)*(x)*(Subscript[P, m]*(x)- Subscript[Q, m]*(x))- W(Subscript[W, m])^(-)*(x)*(Subscript[P, m]*(x)+ Subscript[Q, m]*(x))) Error Failure - Error
28.10.E1 2 π ⁢ ∫ 0 π / 2 cos ⁡ ( 2 ⁢ h ⁢ cos ⁡ z ⁢ cos ⁡ t ) ⁢ ce 2 ⁢ n ⁡ ( t , h 2 ) ⁢ d t = A 0 2 ⁢ n ⁢ ( h 2 ) ce 2 ⁢ n ⁡ ( 1 2 ⁢ π , h 2 ) ⁢ ce 2 ⁢ n ⁡ ( z , h 2 ) 2 𝜋 superscript subscript 0 𝜋 2 2 ℎ 𝑧 𝑡 Mathieu-ce 2 𝑛 𝑡 superscript ℎ 2 𝑡 superscript subscript 𝐴 0 2 𝑛 superscript ℎ 2 Mathieu-ce 2 𝑛 1 2 𝜋 superscript ℎ 2 Mathieu-ce 2 𝑛 𝑧 superscript ℎ 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos\left(2h% \cos z\cos t\right)\mathrm{ce}_{2n}\left(t,h^{2}\right)\mathrm{d}t=\frac{A_{0}% ^{2n}(h^{2})}{\mathrm{ce}_{2n}\left(\frac{1}{2}\pi,h^{2}\right)}\mathrm{ce}_{2% n}\left(z,h^{2}\right)}} (2)/(Pi)*int(cos(2*h*cos(z)*cos(t))*MathieuCE(2*n, (h)^(2), t), t = 0..(Pi)/(2))(A(A[0])^(2*n)*((h)^(2)))/(MathieuCE(2*n, (h)^(2), (1)/(2)*Pi))*MathieuCE(2*n, (h)^(2), z) Divide[2,Pi]*Integrate[Cos[2*h*Cos[z]*Cos[t]]*MathieuC[2*n, (h)^(2), t], {t, 0, Divide[Pi,2]}]Divide[A(Subscript[A, 0])^(2*n)*((h)^(2)),MathieuC[2*n, (h)^(2), Divide[1,2]*Pi]]*MathieuC[2*n, (h)^(2), z] Failure Failure Skip Skip
28.10.E2 2 π ⁢ ∫ 0 π / 2 cosh ⁡ ( 2 ⁢ h ⁢ sin ⁡ z ⁢ sin ⁡ t ) ⁢ ce 2 ⁢ n ⁡ ( t , h 2 ) ⁢ d t = A 0 2 ⁢ n ⁢ ( h 2 ) ce 2 ⁢ n ⁡ ( 0 , h 2 ) ⁢ ce 2 ⁢ n ⁡ ( z , h 2 ) 2 𝜋 superscript subscript 0 𝜋 2 2 ℎ 𝑧 𝑡 Mathieu-ce 2 𝑛 𝑡 superscript ℎ 2 𝑡 superscript subscript 𝐴 0 2 𝑛 superscript ℎ 2 Mathieu-ce 2 𝑛 0 superscript ℎ 2 Mathieu-ce 2 𝑛 𝑧 superscript ℎ 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cosh\left(2% h\sin z\sin t\right)\mathrm{ce}_{2n}\left(t,h^{2}\right)\mathrm{d}t=\frac{A_{0% }^{2n}(h^{2})}{\mathrm{ce}_{2n}\left(0,h^{2}\right)}\mathrm{ce}_{2n}\left(z,h^% {2}\right)}} (2)/(Pi)*int(cosh(2*h*sin(z)*sin(t))*MathieuCE(2*n, (h)^(2), t), t = 0..(Pi)/(2))(A(A[0])^(2*n)*((h)^(2)))/(MathieuCE(2*n, (h)^(2), 0))*MathieuCE(2*n, (h)^(2), z) Divide[2,Pi]*Integrate[Cosh[2*h*Sin[z]*Sin[t]]*MathieuC[2*n, (h)^(2), t], {t, 0, Divide[Pi,2]}]Divide[A(Subscript[A, 0])^(2*n)*((h)^(2)),MathieuC[2*n, (h)^(2), 0]]*MathieuC[2*n, (h)^(2), z] Failure Failure Skip Skip
28.10.E3 2 π ⁢ ∫ 0 π / 2 sin ⁡ ( 2 ⁢ h ⁢ cos ⁡ z ⁢ cos ⁡ t ) ⁢ ce 2 ⁢ n + 1 ⁡ ( t , h 2 ) ⁢ d t = - h ⁢ A 1 2 ⁢ n + 1 ⁢ ( h 2 ) ce 2 ⁢ n + 1 ′ ⁡ ( 1 2 ⁢ π , h 2 ) ⁢ ce 2 ⁢ n + 1 ⁡ ( z , h 2 ) 2 𝜋 superscript subscript 0 𝜋 2 2 ℎ 𝑧 𝑡 Mathieu-ce 2 𝑛 1 𝑡 superscript ℎ 2 𝑡 ℎ superscript subscript 𝐴 1 2 𝑛 1 superscript ℎ 2 diffop Mathieu-ce 2 𝑛 1 1 1 2 𝜋 superscript ℎ 2 Mathieu-ce 2 𝑛 1 𝑧 superscript ℎ 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin\left(2h% \cos z\cos t\right)\mathrm{ce}_{2n+1}\left(t,h^{2}\right)\mathrm{d}t=-\frac{hA% _{1}^{2n+1}(h^{2})}{\mathrm{ce}_{2n+1}'\left(\frac{1}{2}\pi,h^{2}\right)}% \mathrm{ce}_{2n+1}\left(z,h^{2}\right)}} (2)/(Pi)*int(sin(2*h*cos(z)*cos(t))*MathieuCE(2*n + 1, (h)^(2), t), t = 0..(Pi)/(2))= (h*A(A[1])^(2*n + 1)*((h)^(2)))/(subs( temp=(1)/(2)*Pi, diff( MathieuCE(2*n + 1, (h)^(2), temp), temp$(1) ) ))*MathieuCE(2*n + 1, (h)^(2), z) Divide[2,Pi]*Integrate[Sin[2*h*Cos[z]*Cos[t]]*MathieuC[2*n + 1, (h)^(2), t], {t, 0, Divide[Pi,2]}]= Divide[h*A(Subscript[A, 1])^(2*n + 1)*((h)^(2)),D[MathieuC[2*n + 1, (h)^(2), temp], {temp, 1}]/.temp-> Divide[1,2]*Pi]*MathieuC[2*n + 1, (h)^(2), z] Failure Failure Skip Skip
28.10.E4 2 π ⁢ ∫ 0 π / 2 cos ⁡ z ⁢ cos ⁡ t ⁢ cosh ⁡ ( 2 ⁢ h ⁢ sin ⁡ z ⁢ sin ⁡ t ) ⁢ ce 2 ⁢ n + 1 ⁡ ( t , h 2 ) ⁢ d t = A 1 2 ⁢ n + 1 ⁢ ( h 2 ) 2 ⁢ ce 2 ⁢ n + 1 ⁡ ( 0 , h 2 ) ⁢ ce 2 ⁢ n + 1 ⁡ ( z , h 2 ) 2 𝜋 superscript subscript 0 𝜋 2 𝑧 𝑡 2 ℎ 𝑧 𝑡 Mathieu-ce 2 𝑛 1 𝑡 superscript ℎ 2 𝑡 superscript subscript 𝐴 1 2 𝑛 1 superscript ℎ 2 2 Mathieu-ce 2 𝑛 1 0 superscript ℎ 2 Mathieu-ce 2 𝑛 1 𝑧 superscript ℎ 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos z\cos t% \cosh\left(2h\sin z\sin t\right)\mathrm{ce}_{2n+1}\left(t,h^{2}\right)\mathrm{% d}t=\frac{A_{1}^{2n+1}(h^{2})}{2\mathrm{ce}_{2n+1}\left(0,h^{2}\right)}\mathrm% {ce}_{2n+1}\left(z,h^{2}\right)}} (2)/(Pi)*int(cos(z)*cos(t)*cosh(2*h*sin(z)*sin(t))*MathieuCE(2*n + 1, (h)^(2), t), t = 0..(Pi)/(2))(A(A[1])^(2*n + 1)*((h)^(2)))/(2*MathieuCE(2*n + 1, (h)^(2), 0))*MathieuCE(2*n + 1, (h)^(2), z) Divide[2,Pi]*Integrate[Cos[z]*Cos[t]*Cosh[2*h*Sin[z]*Sin[t]]*MathieuC[2*n + 1, (h)^(2), t], {t, 0, Divide[Pi,2]}]Divide[A(Subscript[A, 1])^(2*n + 1)*((h)^(2)),2*MathieuC[2*n + 1, (h)^(2), 0]]*MathieuC[2*n + 1, (h)^(2), z] Failure Failure Skip Error
28.10.E5 2 π ⁢ ∫ 0 π / 2 sinh ⁡ ( 2 ⁢ h ⁢ sin ⁡ z ⁢ sin ⁡ t ) ⁢ se 2 ⁢ n + 1 ⁡ ( t , h 2 ) ⁢ d t = h ⁢ B 1 2 ⁢ n + 1 ⁢ ( h 2 ) se 2 ⁢ n + 1 ′ ⁡ ( 0 , h 2 ) ⁢ se 2 ⁢ n + 1 ⁡ ( z , h 2 ) 2 𝜋 superscript subscript 0 𝜋 2 2 ℎ 𝑧 𝑡 Mathieu-se 2 𝑛 1 𝑡 superscript ℎ 2 𝑡 ℎ superscript subscript 𝐵 1 2 𝑛 1 superscript ℎ 2 diffop Mathieu-se 2 𝑛 1 1 0 superscript ℎ 2 Mathieu-se 2 𝑛 1 𝑧 superscript ℎ 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sinh\left(2% h\sin z\sin t\right)\mathrm{se}_{2n+1}\left(t,h^{2}\right)\mathrm{d}t=\frac{hB% _{1}^{2n+1}(h^{2})}{\mathrm{se}_{2n+1}'\left(0,h^{2}\right)}\mathrm{se}_{2n+1}% \left(z,h^{2}\right)}} (2)/(Pi)*int(sinh(2*h*sin(z)*sin(t))*MathieuSE(2*n + 1, (h)^(2), t), t = 0..(Pi)/(2))(h*B(B[1])^(2*n + 1)*((h)^(2)))/(subs( temp=0, diff( MathieuSE(2*n + 1, (h)^(2), temp), temp$(1) ) ))*MathieuSE(2*n + 1, (h)^(2), z) Divide[2,Pi]*Integrate[Sinh[2*h*Sin[z]*Sin[t]]*MathieuS[2*n + 1, (h)^(2), t], {t, 0, Divide[Pi,2]}]Divide[h*B(Subscript[B, 1])^(2*n + 1)*((h)^(2)),D[MathieuS[2*n + 1, (h)^(2), temp], {temp, 1}]/.temp-> 0]*MathieuS[2*n + 1, (h)^(2), z] Failure Failure Skip Successful
28.10.E6 2 π ⁢ ∫ 0 π / 2 sin ⁡ z ⁢ sin ⁡ t ⁢ cos ⁡ ( 2 ⁢ h ⁢ cos ⁡ z ⁢ cos ⁡ t ) ⁢ se 2 ⁢ n + 1 ⁡ ( t , h 2 ) ⁢ d t = B 1 2 ⁢ n + 1 ⁢ ( h 2 ) 2 ⁢ se 2 ⁢ n + 1 ⁡ ( 1 2 ⁢ π , h 2 ) ⁢ se 2 ⁢ n + 1 ⁡ ( z , h 2 ) 2 𝜋 superscript subscript 0 𝜋 2 𝑧 𝑡 2 ℎ 𝑧 𝑡 Mathieu-se 2 𝑛 1 𝑡 superscript ℎ 2 𝑡 superscript subscript 𝐵 1 2 𝑛 1 superscript ℎ 2 2 Mathieu-se 2 𝑛 1 1 2 𝜋 superscript ℎ 2 Mathieu-se 2 𝑛 1 𝑧 superscript ℎ 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin z\sin t% \cos\left(2h\cos z\cos t\right)\mathrm{se}_{2n+1}\left(t,h^{2}\right)\mathrm{d% }t=\frac{B_{1}^{2n+1}(h^{2})}{2\mathrm{se}_{2n+1}\left(\frac{1}{2}\pi,h^{2}% \right)}\mathrm{se}_{2n+1}\left(z,h^{2}\right)}} (2)/(Pi)*int(sin(z)*sin(t)*cos(2*h*cos(z)*cos(t))*MathieuSE(2*n + 1, (h)^(2), t), t = 0..(Pi)/(2))(B(B[1])^(2*n + 1)*((h)^(2)))/(2*MathieuSE(2*n + 1, (h)^(2), (1)/(2)*Pi))*MathieuSE(2*n + 1, (h)^(2), z) Divide[2,Pi]*Integrate[Sin[z]*Sin[t]*Cos[2*h*Cos[z]*Cos[t]]*MathieuS[2*n + 1, (h)^(2), t], {t, 0, Divide[Pi,2]}]Divide[B(Subscript[B, 1])^(2*n + 1)*((h)^(2)),2*MathieuS[2*n + 1, (h)^(2), Divide[1,2]*Pi]]*MathieuS[2*n + 1, (h)^(2), z] Failure Failure Skip Skip
28.10.E7 2 π ⁢ ∫ 0 π / 2 sin ⁡ z ⁢ sin ⁡ t ⁢ sin ⁡ ( 2 ⁢ h ⁢ cos ⁡ z ⁢ cos ⁡ t ) ⁢ se 2 ⁢ n + 2 ⁡ ( t , h 2 ) ⁢ d t = - h ⁢ B 2 2 ⁢ n + 2 ⁢ ( h 2 ) 2 ⁢ se 2 ⁢ n + 2 ′ ⁡ ( 1 2 ⁢ π , h 2 ) ⁢ se 2 ⁢ n + 2 ⁡ ( z , h 2 ) 2 𝜋 superscript subscript 0 𝜋 2 𝑧 𝑡 2 ℎ 𝑧 𝑡 Mathieu-se 2 𝑛 2 𝑡 superscript ℎ 2 𝑡 ℎ superscript subscript 𝐵 2 2 𝑛 2 superscript ℎ 2 2 diffop Mathieu-se 2 𝑛 2 1 1 2 𝜋 superscript ℎ 2 Mathieu-se 2 𝑛 2 𝑧 superscript ℎ 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin z\sin t% \sin\left(2h\cos z\cos t\right)\mathrm{se}_{2n+2}\left(t,h^{2}\right)\mathrm{d% }t=-\frac{hB_{2}^{2n+2}(h^{2})}{2\mathrm{se}_{2n+2}'\left(\frac{1}{2}\pi,h^{2}% \right)}\mathrm{se}_{2n+2}\left(z,h^{2}\right)}} (2)/(Pi)*int(sin(z)*sin(t)*sin(2*h*cos(z)*cos(t))*MathieuSE(2*n + 2, (h)^(2), t), t = 0..(Pi)/(2))= (h*B(B[2])^(2*n + 2)*((h)^(2)))/(2*subs( temp=(1)/(2)*Pi, diff( MathieuSE(2*n + 2, (h)^(2), temp), temp$(1) ) ))*MathieuSE(2*n + 2, (h)^(2), z) Divide[2,Pi]*Integrate[Sin[z]*Sin[t]*Sin[2*h*Cos[z]*Cos[t]]*MathieuS[2*n + 2, (h)^(2), t], {t, 0, Divide[Pi,2]}]= Divide[h*B(Subscript[B, 2])^(2*n + 2)*((h)^(2)),2*(D[MathieuS[2*n + 2, (h)^(2), temp], {temp, 1}]/.temp-> Divide[1,2]*Pi)]*MathieuS[2*n + 2, (h)^(2), z] Failure Failure Skip Error
28.10.E8 2 π ⁢ ∫ 0 π / 2 cos ⁡ z ⁢ cos ⁡ t ⁢ sinh ⁡ ( 2 ⁢ h ⁢ sin ⁡ z ⁢ sin ⁡ t ) ⁢ se 2 ⁢ n + 2 ⁡ ( t , h 2 ) ⁢ d t = h ⁢ B 2 2 ⁢ n + 2 ⁢ ( h 2 ) 2 ⁢ se 2 ⁢ n + 2 ′ ⁡ ( 0 , h 2 ) ⁢ se 2 ⁢ n + 2 ⁡ ( z , h 2 ) 2 𝜋 superscript subscript 0 𝜋 2 𝑧 𝑡 2 ℎ 𝑧 𝑡 Mathieu-se 2 𝑛 2 𝑡 superscript ℎ 2 𝑡 ℎ superscript subscript 𝐵 2 2 𝑛 2 superscript ℎ 2 2 diffop Mathieu-se 2 𝑛 2 1 0 superscript ℎ 2 Mathieu-se 2 𝑛 2 𝑧 superscript ℎ 2 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos z\cos t% \sinh\left(2h\sin z\sin t\right)\mathrm{se}_{2n+2}\left(t,h^{2}\right)\mathrm{% d}t=\frac{hB_{2}^{2n+2}(h^{2})}{2\mathrm{se}_{2n+2}'\left(0,h^{2}\right)}% \mathrm{se}_{2n+2}\left(z,h^{2}\right)}} (2)/(Pi)*int(cos(z)*cos(t)*sinh(2*h*sin(z)*sin(t))*MathieuSE(2*n + 2, (h)^(2), t), t = 0..(Pi)/(2))(h*B(B[2])^(2*n + 2)*((h)^(2)))/(2*subs( temp=0, diff( MathieuSE(2*n + 2, (h)^(2), temp), temp$(1) ) ))*MathieuSE(2*n + 2, (h)^(2), z) Divide[2,Pi]*Integrate[Cos[z]*Cos[t]*Sinh[2*h*Sin[z]*Sin[t]]*MathieuS[2*n + 2, (h)^(2), t], {t, 0, Divide[Pi,2]}]Divide[h*B(Subscript[B, 2])^(2*n + 2)*((h)^(2)),2*(D[MathieuS[2*n + 2, (h)^(2), temp], {temp, 1}]/.temp-> 0)]*MathieuS[2*n + 2, (h)^(2), z] Failure Failure Skip Error
28.11.E3 1 = 2 ⁢ ∑ n = 0 ∞ A 0 2 ⁢ n ⁢ ( q ) ⁢ ce 2 ⁢ n ⁡ ( z , q ) 1 2 superscript subscript 𝑛 0 superscript subscript 𝐴 0 2 𝑛 𝑞 Mathieu-ce 2 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle 1=2\sum_{n=0}^{\infty}A_{0}^{2n}(q)\mathrm{ce}_{2% n}\left(z,q\right)}} 1 = sum(A(A[0])^(2*n)*(q)* MathieuCE(2*n, q, z), n = 0..infinity) 1 = Sum[A(Subscript[A, 0])^(2*n)*(q)* MathieuC[2*n, q, z], {n, 0, Infinity}] Failure Failure Skip Error
28.11.E4 cos ⁡ 2 ⁢ m ⁢ z = ∑ n = 0 ∞ A 2 ⁢ m 2 ⁢ n ⁢ ( q ) ⁢ ce 2 ⁢ n ⁡ ( z , q ) 2 𝑚 𝑧 superscript subscript 𝑛 0 superscript subscript 𝐴 2 𝑚 2 𝑛 𝑞 Mathieu-ce 2 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\cos 2mz=\sum_{n=0}^{\infty}A_{2m}^{2n}(q)\mathrm{% ce}_{2n}\left(z,q\right)}} cos(2*m*z)sum(A(A[2*m])^(2*n)*(q)* MathieuCE(2*n, q, z), n = 0..infinity) Cos[2*m*z]Sum[A(Subscript[A, 2*m])^(2*n)*(q)* MathieuC[2*n, q, z], {n, 0, Infinity}] Failure Failure Skip Error
28.11.E5 cos ⁡ ( 2 ⁢ m + 1 ) ⁢ z = ∑ n = 0 ∞ A 2 ⁢ m + 1 2 ⁢ n + 1 ⁢ ( q ) ⁢ ce 2 ⁢ n + 1 ⁡ ( z , q ) 2 𝑚 1 𝑧 superscript subscript 𝑛 0 superscript subscript 𝐴 2 𝑚 1 2 𝑛 1 𝑞 Mathieu-ce 2 𝑛 1 𝑧 𝑞 {\displaystyle{\displaystyle\cos(2m+1)z=\sum_{n=0}^{\infty}A_{2m+1}^{2n+1}(q)% \mathrm{ce}_{2n+1}\left(z,q\right)}} cos((2*m + 1)* z)sum(A(A[2*m + 1])^(2*n + 1)*(q)* MathieuCE(2*n + 1, q, z), n = 0..infinity) Cos[(2*m + 1)* z]Sum[A(Subscript[A, 2*m + 1])^(2*n + 1)*(q)* MathieuC[2*n + 1, q, z], {n, 0, Infinity}] Failure Failure Skip Error
28.11.E6 sin ⁡ ( 2 ⁢ m + 1 ) ⁢ z = ∑ n = 0 ∞ B 2 ⁢ m + 1 2 ⁢ n + 1 ⁢ ( q ) ⁢ se 2 ⁢ n + 1 ⁡ ( z , q ) 2 𝑚 1 𝑧 superscript subscript 𝑛 0 superscript subscript 𝐵 2 𝑚 1 2 𝑛 1 𝑞 Mathieu-se 2 𝑛 1 𝑧 𝑞 {\displaystyle{\displaystyle\sin(2m+1)z=\sum_{n=0}^{\infty}B_{2m+1}^{2n+1}(q)% \mathrm{se}_{2n+1}\left(z,q\right)}} sin((2*m + 1)* z)sum(B(B[2*m + 1])^(2*n + 1)*(q)* MathieuSE(2*n + 1, q, z), n = 0..infinity) Sin[(2*m + 1)* z]Sum[B(Subscript[B, 2*m + 1])^(2*n + 1)*(q)* MathieuS[2*n + 1, q, z], {n, 0, Infinity}] Failure Failure Skip Error
28.11.E7 sin ⁡ ( 2 ⁢ m + 2 ) ⁢ z = ∑ n = 0 ∞ B 2 ⁢ m + 2 2 ⁢ n + 2 ⁢ ( q ) ⁢ se 2 ⁢ n + 2 ⁡ ( z , q ) 2 𝑚 2 𝑧 superscript subscript 𝑛 0 superscript subscript 𝐵 2 𝑚 2 2 𝑛 2 𝑞 Mathieu-se 2 𝑛 2 𝑧 𝑞 {\displaystyle{\displaystyle\sin(2m+2)z=\sum_{n=0}^{\infty}B_{2m+2}^{2n+2}(q)% \mathrm{se}_{2n+2}\left(z,q\right)}} sin((2*m + 2)* z)sum(B(B[2*m + 2])^(2*n + 2)*(q)* MathieuSE(2*n + 2, q, z), n = 0..infinity) Sin[(2*m + 2)* z]Sum[B(Subscript[B, 2*m + 2])^(2*n + 2)*(q)* MathieuS[2*n + 2, q, z], {n, 0, Infinity}] Failure Failure Skip Error
28.12.E4 me ν ⁡ ( z , 0 ) = e i ⁢ ν ⁢ z Mathieu-me 𝜈 𝑧 0 superscript 𝑒 imaginary-unit 𝜈 𝑧 {\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z,0\right)=e^{\mathrm{i}\nu z% }}} Error Sqrt[2]*MathieuC[\[Nu], 0, z]= Exp[I*\[Nu]*z] Error Failure - Error
28.12.E5 ∫ 0 π me ν ⁡ ( x , q ) ⁢ me ν ⁡ ( - x , q ) ⁢ d x = π superscript subscript 0 𝜋 Mathieu-me 𝜈 𝑥 𝑞 Mathieu-me 𝜈 𝑥 𝑞 𝑥 𝜋 {\displaystyle{\displaystyle\int_{0}^{\pi}\mathrm{me}_{\nu}\left(x,q\right)% \mathrm{me}_{\nu}\left(-x,q\right)\mathrm{d}x=\pi}} Error Integrate[Sqrt[2]*MathieuC[\[Nu], q, x]*Sqrt[2]*MathieuC[\[Nu], q, - x], {x, 0, Pi}]= Pi Error Failure - Error
28.12.E6 me ν ⁡ ( z + π , q ) = e π ⁢ i ⁢ ν ⁢ me ν ⁡ ( z , q ) Mathieu-me 𝜈 𝑧 𝜋 𝑞 superscript 𝑒 𝜋 imaginary-unit 𝜈 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z+\pi,q\right)=e^{\pi% \mathrm{i}\nu}\mathrm{me}_{\nu}\left(z,q\right)}} Error Sqrt[2]*MathieuC[\[Nu], q, z + Pi]= Exp[Pi*I*\[Nu]]*Sqrt[2]*MathieuC[\[Nu], q, z] Error Failure - Error
28.12.E8 me - ν ⁡ ( z , q ) = me ν ⁡ ( - z , q ) Mathieu-me 𝜈 𝑧 𝑞 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{-\nu}\left(z,q\right)=\mathrm{me}_{% \nu}\left(-z,q\right)}} Error Sqrt[2]*MathieuC[- \[Nu], q, z]= Sqrt[2]*MathieuC[\[Nu], q, - z] Error Failure - Error
28.12.E9 me ν ⁡ ( z , - q ) = e i ⁢ ν ⁢ π / 2 ⁢ me ν ⁡ ( z - 1 2 ⁢ π , q ) Mathieu-me 𝜈 𝑧 𝑞 superscript 𝑒 imaginary-unit 𝜈 𝜋 2 Mathieu-me 𝜈 𝑧 1 2 𝜋 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z,-q\right)=e^{\mathrm{i}% \nu\pi/2}\mathrm{me}_{\nu}\left(z-\tfrac{1}{2}\pi,q\right)}} Error Sqrt[2]*MathieuC[\[Nu], - q, z]= Exp[I*\[Nu]*Pi/ 2]*Sqrt[2]*MathieuC[\[Nu], q, z -Divide[1,2]*Pi] Error Failure - Error
28.12.E10 me ν ⁡ ( z , q ) ¯ = me ν ¯ ⁡ ( - z ¯ , q ¯ ) Mathieu-me 𝜈 𝑧 𝑞 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\overline{\mathrm{me}_{\nu}\left(z,q\right)}=% \mathrm{me}_{\overline{\nu}}\left(-\overline{z},\overline{q}\right)}} Error Conjugate[Sqrt[2]*MathieuC[\[Nu], q, z]]= Sqrt[2]*MathieuC[Conjugate[\[Nu]], Conjugate[q], - Conjugate[z]] Error Failure - Error
28.12#Ex1 me n ⁡ ( z , q ) = 2 ⁢ ce n ⁡ ( z , q ) Mathieu-me 𝑛 𝑧 𝑞 2 Mathieu-ce 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{n}\left(z,q\right)=\sqrt{2}\mathrm{ce% }_{n}\left(z,q\right)}} Error Sqrt[2]*MathieuC[n, q, z]=Sqrt[2]*MathieuC[n, q, z] Error Successful - -
28.12#Ex2 me - n ⁡ ( z , q ) = - 2 ⁢ i ⁢ se n ⁡ ( z , q ) Mathieu-me 𝑛 𝑧 𝑞 2 imaginary-unit Mathieu-se 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{-n}\left(z,q\right)=-\sqrt{2}\mathrm{% i}\mathrm{se}_{n}\left(z,q\right)}} Error Sqrt[2]*MathieuC[- n, q, z]= -Sqrt[2]*I*MathieuS[n, q, z] Error Failure - Error
28.12.E12 ce ν ⁡ ( z , q ) = 1 2 ⁢ ( me ν ⁡ ( z , q ) + me ν ⁡ ( - z , q ) ) Mathieu-ce 𝜈 𝑧 𝑞 1 2 Mathieu-me 𝜈 𝑧 𝑞 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(z,q\right)=\tfrac{1}{2}% \left(\mathrm{me}_{\nu}\left(z,q\right)+\mathrm{me}_{\nu}\left(-z,q\right)% \right)}} Error MathieuC[\[Nu], q, z]=Divide[1,2]*(Sqrt[2]*MathieuC[\[Nu], q, z]+ Sqrt[2]*MathieuC[\[Nu], q, - z]) Error Failure - Error
28.12.E13 se ν ⁡ ( z , q ) = - 1 2 ⁢ i ⁢ ( me ν ⁡ ( z , q ) - me ν ⁡ ( - z , q ) ) Mathieu-se 𝜈 𝑧 𝑞 1 2 imaginary-unit Mathieu-me 𝜈 𝑧 𝑞 Mathieu-me 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{\nu}\left(z,q\right)=-\tfrac{1}{2}% \mathrm{i}\left(\mathrm{me}_{\nu}\left(z,q\right)-\mathrm{me}_{\nu}\left(-z,q% \right)\right)}} Error MathieuS[\[Nu], q, z]= -Divide[1,2]*I*(Sqrt[2]*MathieuC[\[Nu], q, z]- Sqrt[2]*MathieuC[\[Nu], q, - z]) Error Failure - Error
28.12.E14 ce ν ⁡ ( z , q ) = ce ν ⁡ ( - z , q ) Mathieu-ce 𝜈 𝑧 𝑞 Mathieu-ce 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(z,q\right)=\mathrm{ce}_{\nu% }\left(-z,q\right)}} MathieuCE(nu, q, z)= MathieuCE(nu, q, - z) MathieuC[\[Nu], q, z]= MathieuC[\[Nu], q, - z] Successful Successful - -
28.12.E14 ce ν ⁡ ( - z , q ) = ce - ν ⁡ ( z , q ) Mathieu-ce 𝜈 𝑧 𝑞 Mathieu-ce 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(-z,q\right)=\mathrm{ce}_{-% \nu}\left(z,q\right)}} MathieuCE(nu, q, - z)= MathieuCE(- nu, q, z) MathieuC[\[Nu], q, - z]= MathieuC[- \[Nu], q, z] Failure Failure Error Error
28.12.E15 se ν ⁡ ( z , q ) = - se ν ⁡ ( - z , q ) Mathieu-se 𝜈 𝑧 𝑞 Mathieu-se 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{\nu}\left(z,q\right)=-\mathrm{se}_{% \nu}\left(-z,q\right)}} MathieuSE(nu, q, z)= - MathieuSE(nu, q, - z) MathieuS[\[Nu], q, z]= - MathieuS[\[Nu], q, - z] Successful Successful - -
28.12.E15 - se ν ⁡ ( - z , q ) = - se - ν ⁡ ( z , q ) Mathieu-se 𝜈 𝑧 𝑞 Mathieu-se 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle-\mathrm{se}_{\nu}\left(-z,q\right)=-\mathrm{se}_{% -\nu}\left(z,q\right)}} - MathieuSE(nu, q, - z)= - MathieuSE(- nu, q, z) - MathieuS[\[Nu], q, - z]= - MathieuS[- \[Nu], q, z] Failure Failure Error Error
28.14.E1 me ν ⁡ ( z , q ) = ∑ m = - ∞ ∞ c 2 ⁢ m ν ⁢ ( q ) ⁢ e i ⁢ ( ν + 2 ⁢ m ) ⁢ z Mathieu-me 𝜈 𝑧 𝑞 superscript subscript 𝑚 subscript superscript 𝑐 𝜈 2 𝑚 𝑞 superscript 𝑒 imaginary-unit 𝜈 2 𝑚 𝑧 {\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z,q\right)=\sum_{m=-\infty}% ^{\infty}c^{\nu}_{2m}(q)e^{\mathrm{i}(\nu+2m)z}}} Error Sqrt[2]*MathieuC[\[Nu], q, z]Sum[c(Subscript[c, 2*m])^(\[Nu])*(q)* Exp[I*(\[Nu]+ 2*m)* z], {m, - Infinity, Infinity}] Error Failure - Error
28.14.E2 ce ν ⁡ ( z , q ) = ∑ m = - ∞ ∞ c 2 ⁢ m ν ⁢ ( q ) ⁢ cos ⁡ ( ν + 2 ⁢ m ) ⁢ z Mathieu-ce 𝜈 𝑧 𝑞 superscript subscript 𝑚 subscript superscript 𝑐 𝜈 2 𝑚 𝑞 𝜈 2 𝑚 𝑧 {\displaystyle{\displaystyle\mathrm{ce}_{\nu}\left(z,q\right)=\sum_{m=-\infty}% ^{\infty}c^{\nu}_{2m}(q)\cos(\nu+2m)z}} MathieuCE(nu, q, z)sum(c(c[2*m])^(nu)*(q)* cos((nu + 2*m)* z), m = - infinity..infinity) MathieuC[\[Nu], q, z]Sum[c(Subscript[c, 2*m])^(\[Nu])*(q)* Cos[(\[Nu]+ 2*m)* z], {m, - Infinity, Infinity}] Failure Failure Skip Error
28.14.E3 se ν ⁡ ( z , q ) = ∑ m = - ∞ ∞ c 2 ⁢ m ν ⁢ ( q ) ⁢ sin ⁡ ( ν + 2 ⁢ m ) ⁢ z Mathieu-se 𝜈 𝑧 𝑞 superscript subscript 𝑚 subscript superscript 𝑐 𝜈 2 𝑚 𝑞 𝜈 2 𝑚 𝑧 {\displaystyle{\displaystyle\mathrm{se}_{\nu}\left(z,q\right)=\sum_{m=-\infty}% ^{\infty}c^{\nu}_{2m}(q)\sin(\nu+2m)z}} MathieuSE(nu, q, z)sum(c(c[2*m])^(nu)*(q)* sin((nu + 2*m)* z), m = - infinity..infinity) MathieuS[\[Nu], q, z]Sum[c(Subscript[c, 2*m])^(\[Nu])*(q)* Sin[(\[Nu]+ 2*m)* z], {m, - Infinity, Infinity}] Failure Failure Skip Error
28.19.E4 e i ⁢ ν ⁢ z = ∑ n = - ∞ ∞ c - 2 ⁢ n ν + 2 ⁢ n ⁢ ( q ) ⁢ me ν + 2 ⁢ n ⁡ ( z , q ) superscript 𝑒 imaginary-unit 𝜈 𝑧 superscript subscript 𝑛 subscript superscript 𝑐 𝜈 2 𝑛 2 𝑛 𝑞 Mathieu-me 𝜈 2 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle e^{\mathrm{i}\nu z}=\sum_{n=-\infty}^{\infty}c^{% \nu+2n}_{-2n}(q)\mathrm{me}_{\nu+2n}\left(z,q\right)}} Error Exp[I*\[Nu]*z]Sum[c(Subscript[c, - 2*n])^(\[Nu]+ 2*n)*(q)* Sqrt[2]*MathieuC[\[Nu]+ 2*n, q, z], {n, - Infinity, Infinity}] Error Failure - Error
28.22.E5 g e , 2 ⁢ m ⁢ ( h ) = ( - 1 ) m ⁢ 2 π ⁢ ce 2 ⁢ m ⁡ ( 1 2 ⁢ π , h 2 ) A 0 2 ⁢ m ⁢ ( h 2 ) subscript 𝑔 𝑒 2 𝑚 ℎ superscript 1 𝑚 2 𝜋 Mathieu-ce 2 𝑚 1 2 𝜋 superscript ℎ 2 superscript subscript 𝐴 0 2 𝑚 superscript ℎ 2 {\displaystyle{\displaystyle g_{\mathit{e},2m}(h)=(-1)^{m}\sqrt{\dfrac{2}{\pi}% }\dfrac{\mathrm{ce}_{2m}\left(\frac{1}{2}\pi,h^{2}\right)}{A_{0}^{2m}(h^{2})}}} g[exp(1), 2*m]*(h)=(- 1)^(m)*(MathieuCE(2*m, (h)^(2), (1)/(2)*Pi))/(A(A[0])^(2*m)*((h)^(2))) Subscript[g, E , 2*m]*(h)=(- 1)^(m)*Divide[MathieuC[2*m, (h)^(2), Divide[1,2]*Pi],A(Subscript[A, 0])^(2*m)*((h)^(2))] Failure Failure Error Error
28.22.E6 g e , 2 ⁢ m + 1 ⁢ ( h ) = ( - 1 ) m + 1 ⁢ 2 π ⁢ ce 2 ⁢ m + 1 ′ ⁡ ( 1 2 ⁢ π , h 2 ) h ⁢ A 1 2 ⁢ m + 1 ⁢ ( h 2 ) subscript 𝑔 𝑒 2 𝑚 1 ℎ superscript 1 𝑚 1 2 𝜋 diffop Mathieu-ce 2 𝑚 1 1 1 2 𝜋 superscript ℎ 2 ℎ superscript subscript 𝐴 1 2 𝑚 1 superscript ℎ 2 {\displaystyle{\displaystyle g_{\mathit{e},2m+1}(h)=(-1)^{m+1}\sqrt{\frac{2}{% \pi}}\dfrac{\mathrm{ce}_{2m+1}'\left(\frac{1}{2}\pi,h^{2}\right)}{hA_{1}^{2m+1% }(h^{2})}}} g[exp(1), 2*m + 1]*(h)=(- 1)^(m + 1)*(subs( temp=(1)/(2)*Pi, diff( MathieuCE(2*m + 1, (h)^(2), temp), temp$(1) ) ))/(h*A(A[1])^(2*m + 1)*((h)^(2))) Subscript[g, E , 2*m + 1]*(h)=(- 1)^(m + 1)*Divide[D[MathieuC[2*m + 1, (h)^(2), temp], {temp, 1}]/.temp-> Divide[1,2]*Pi,h*A(Subscript[A, 1])^(2*m + 1)*((h)^(2))] Failure Failure Error Error
28.22.E7 g o , 2 ⁢ m + 1 ⁢ ( h ) = ( - 1 ) m ⁢ 2 π ⁢ se 2 ⁢ m + 1 ⁡ ( 1 2 ⁢ π , h 2 ) h ⁢ B 1 2 ⁢ m + 1 ⁢ ( h 2 ) subscript 𝑔 𝑜 2 𝑚 1 ℎ superscript 1 𝑚 2 𝜋 Mathieu-se 2 𝑚 1 1 2 𝜋 superscript ℎ 2 ℎ superscript subscript 𝐵 1 2 𝑚 1 superscript ℎ 2 {\displaystyle{\displaystyle g_{\mathit{o},2m+1}(h)=(-1)^{m}\sqrt{\dfrac{2}{% \pi}}\dfrac{\mathrm{se}_{2m+1}\left(\frac{1}{2}\pi,h^{2}\right)}{hB_{1}^{2m+1}% (h^{2})}}} g[o , 2*m + 1]*(h)=(- 1)^(m)*(MathieuSE(2*m + 1, (h)^(2), (1)/(2)*Pi))/(h*B(B[1])^(2*m + 1)*((h)^(2))) Subscript[g, o , 2*m + 1]*(h)=(- 1)^(m)*Divide[MathieuS[2*m + 1, (h)^(2), Divide[1,2]*Pi],h*B(Subscript[B, 1])^(2*m + 1)*((h)^(2))] Failure Failure Error Error
28.22.E8 g o , 2 ⁢ m + 2 ⁢ ( h ) = ( - 1 ) m + 1 ⁢ 2 π ⁢ se 2 ⁢ m + 2 ′ ⁡ ( 1 2 ⁢ π , h 2 ) h 2 ⁢ B 2 2 ⁢ m + 2 ⁢ ( h 2 ) subscript 𝑔 𝑜 2 𝑚 2 ℎ superscript 1 𝑚 1 2 𝜋 diffop Mathieu-se 2 𝑚 2 1 1 2 𝜋 superscript ℎ 2 superscript ℎ 2 superscript subscript 𝐵 2 2 𝑚 2 superscript ℎ 2 {\displaystyle{\displaystyle g_{\mathit{o},2m+2}(h)=(-1)^{m+1}\sqrt{\dfrac{2}{% \pi}}\dfrac{\mathrm{se}_{2m+2}'\left(\frac{1}{2}\pi,h^{2}\right)}{h^{2}B_{2}^{% 2m+2}(h^{2})}}} g[o , 2*m + 2]*(h)=(- 1)^(m + 1)*(subs( temp=(1)/(2)*Pi, diff( MathieuSE(2*m + 2, (h)^(2), temp), temp$(1) ) ))/((h)^(2)* B(B[2])^(2*m + 2)*((h)^(2))) Subscript[g, o , 2*m + 2]*(h)=(- 1)^(m + 1)*Divide[D[MathieuS[2*m + 2, (h)^(2), temp], {temp, 1}]/.temp-> Divide[1,2]*Pi,(h)^(2)* B(Subscript[B, 2])^(2*m + 2)*((h)^(2))] Failure Failure Error Error
28.25.E3 ( m + 1 ) ⁢ D m + 1 + + ( ( m + 1 2 ) 2 + ( m + 1 4 ) ⁢ 8 ⁢ i ⁢ h + 2 ⁢ h 2 - a ) ⁢ D m + + ( m - 1 2 ) ⁢ ( 8 ⁢ i ⁢ h ⁢ m ) ⁢ D m - 1 + = 0 𝑚 1 subscript superscript 𝐷 𝑚 1 superscript 𝑚 1 2 2 𝑚 1 4 8 imaginary-unit ℎ 2 superscript ℎ 2 𝑎 subscript superscript 𝐷 𝑚 𝑚 1 2 8 imaginary-unit ℎ 𝑚 superscript subscript 𝐷 𝑚 1 0 {\displaystyle{\displaystyle(m+1)D^{+}_{m+1}+{\left((m+\tfrac{1}{2})^{2}+(m+% \tfrac{1}{4})8\mathrm{i}h+2h^{2}-a\right)D^{+}_{m}}+(m-\tfrac{1}{2})\left(8% \mathrm{i}hm\right)D_{m-1}^{+}=0}} (m + 1)* (D[m + 1])^(+)+((m +(1)/(2))^(2)+(m +(1)/(4))*8*I*h + 2*(h)^(2)- a)* (D[m])^(+)+(m -(1)/(2))*(8*I*h*m)* (D[m - 1])^(+)= 0 (m + 1)* (Subscript[D, m + 1])^(+)+((m +Divide[1,2])^(2)+(m +Divide[1,4])*8*I*h + 2*(h)^(2)- a)* (Subscript[D, m])^(+)+(m -Divide[1,2])*(8*I*h*m)* (Subscript[D, m - 1])^(+)= 0 Error Failure - Error
28.25.E3 ( m + 1 ) ⁢ D m + 1 - + ( ( m + 1 2 ) 2 - ( m + 1 4 ) ⁢ 8 ⁢ i ⁢ h + 2 ⁢ h 2 - a ) ⁢ D m - - ( m - 1 2 ) ⁢ ( 8 ⁢ i ⁢ h ⁢ m ) ⁢ D m - 1 - = 0 𝑚 1 subscript superscript 𝐷 𝑚 1 superscript 𝑚 1 2 2 𝑚 1 4 8 imaginary-unit ℎ 2 superscript ℎ 2 𝑎 subscript superscript 𝐷 𝑚 𝑚 1 2 8 imaginary-unit ℎ 𝑚 superscript subscript 𝐷 𝑚 1 0 {\displaystyle{\displaystyle(m+1)D^{-}_{m+1}+{\left((m+\tfrac{1}{2})^{2}-(m+% \tfrac{1}{4})8\mathrm{i}h+2h^{2}-a\right)D^{-}_{m}}-(m-\tfrac{1}{2})\left(8% \mathrm{i}hm\right)D_{m-1}^{-}=0}} (m + 1)* (D[m + 1])^(-)+((m +(1)/(2))^(2)-(m +(1)/(4))*8*I*h + 2*(h)^(2)- a)* (D[m])^(-)-(m -(1)/(2))*(8*I*h*m)* (D[m - 1])^(-)= 0 (m + 1)* (Subscript[D, m + 1])^(-)+((m +Divide[1,2])^(2)-(m +Divide[1,4])*8*I*h + 2*(h)^(2)- a)* (Subscript[D, m])^(-)-(m -Divide[1,2])*(8*I*h*m)* (Subscript[D, m - 1])^(-)= 0 Error Failure - Error
28.26.E3 ϕ = 2 ⁢ h ⁢ sinh ⁡ z - ( m + 1 2 ) ⁢ arctan ⁡ ( sinh ⁡ z ) italic-ϕ 2 ℎ 𝑧 𝑚 1 2 𝑧 {\displaystyle{\displaystyle\phi=2h\sinh z-\left(m+\tfrac{1}{2}\right)% \operatorname{arctan}\left(\sinh z\right)}} phi = 2*h*sinh(z)-(m +(1)/(2))* arctan(sinh(z)) \[Phi]= 2*h*Sinh[z]-(m +Divide[1,2])* ArcTan[Sinh[z]] Failure Failure
Fail
8.881737466-4.791146621*I <- {h = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
10.37211930-4.302072204*I <- {h = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
11.86250114-3.812997788*I <- {h = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-3.289185494+5.912553089*I <- {h = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 1}
... skip entries to safe data
 Error
28.28.E1 w = cosh ⁡ z ⁢ cos ⁡ t ⁢ cos ⁡ α + sinh ⁡ z ⁢ sin ⁡ t ⁢ sin ⁡ α 𝑤 𝑧 𝑡 𝛼 𝑧 𝑡 𝛼 {\displaystyle{\displaystyle w=\cosh z\cos t\cos\alpha+\sinh z\sin t\sin\alpha}} w = cosh(z)*cos(t)*cos(alpha)+ sinh(z)*sin(t)*sin(alpha) w = Cosh[z]*Cos[t]*Cos[\[Alpha]]+ Sinh[z]*Sin[t]*Sin[\[Alpha]] Failure Failure
Fail
1.558413509-1.537727303*I <- {alpha = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.934763107+4.464613324*I <- {alpha = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.290314455+18.77347892*I <- {alpha = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
9.261119835-14.27924254*I <- {alpha = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.28.E10 0 < ph ⁡ ( h ⁢ ( cosh ⁡ z + 1 ) ) 0 phase ℎ 𝑧 1 {\displaystyle{\displaystyle 0<\operatorname{ph}\left(h(\cosh z+1)\right)}} 0 < argument(h*(cosh(z)+ 1)) 0 < Arg[h*(Cosh[z]+ 1)] Failure Failure
Fail
0. < -.1740759484 <- {h = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
0. < -.1740759484 <- {h = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
0. < -1.744872276 <- {h = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
0. < -1.744872276 <- {h = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.28.E10 0 < ph ⁡ ( h ⁢ ( cosh ⁡ z - 1 ) ) 0 phase ℎ 𝑧 1 {\displaystyle{\displaystyle 0<\operatorname{ph}\left(h(\cosh z-1)\right)}} 0 < argument(h*(cosh(z)- 1)) 0 < Arg[h*(Cosh[z]- 1)] Failure Failure
Fail
0. < -1.118030274 <- {h = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
0. < -1.118030274 <- {h = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
0. < -2.688826601 <- {h = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
0. < -2.688826601 <- {h = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.28.E10 ph ⁡ ( h ⁢ ( cosh ⁡ z + 1 ) ) < π phase ℎ 𝑧 1 𝜋 {\displaystyle{\displaystyle\operatorname{ph}\left(h(\cosh z+1)\right)<\pi}} argument(h*(cosh(z)+ 1))< Pi Arg[h*(Cosh[z]+ 1)]< Pi Failure Failure Skip Error
28.28.E10 ph ⁡ ( h ⁢ ( cosh ⁡ z - 1 ) ) < π phase ℎ 𝑧 1 𝜋 {\displaystyle{\displaystyle\operatorname{ph}\left(h(\cosh z-1)\right)<\pi}} argument(h*(cosh(z)- 1))< Pi Arg[h*(Cosh[z]- 1)]< Pi Failure Failure Skip Error
28.28#Ex4 R ⁢ ( z , t ) = ( 1 2 ⁢ ( cosh ⁡ ( 2 ⁢ z ) + cos ⁡ ( 2 ⁢ t ) ) ) 1 / 2 𝑅 𝑧 𝑡 superscript 1 2 2 𝑧 2 𝑡 1 2 {\displaystyle{\displaystyle R(z,t)=\left(\tfrac{1}{2}(\cosh\left(2z\right)+% \cos\left(2t\right))\right)^{\ifrac{1}{2}}}} R*(z , t)=((1)/(2)*(cosh(2*z)+ cos(2*t)))^((1)/(2)) R*(z , t)=(Divide[1,2]*(Cosh[2*z]+ Cos[2*t]))^(Divide[1,2]) Failure Failure Error Error
28.28#Ex5 R ⁢ ( z , 0 ) = cosh ⁡ z 𝑅 𝑧 0 𝑧 {\displaystyle{\displaystyle R(z,0)=\cosh z}} R*(z , 0)= cosh(z) R*(z , 0)= Cosh[z] Failure Failure Error Error
28.28#Ex6 e 2 ⁢ i ⁢ ϕ = cosh ⁡ ( z + i ⁢ t ) cosh ⁡ ( z - i ⁢ t ) superscript 𝑒 2 imaginary-unit italic-ϕ 𝑧 imaginary-unit 𝑡 𝑧 imaginary-unit 𝑡 {\displaystyle{\displaystyle e^{2\mathrm{i}\phi}=\dfrac{\cosh\left(z+\mathrm{i% }t\right)}{\cosh\left(z-\mathrm{i}t\right)}}} exp(2*I*phi)=(cosh(z + I*t))/(cosh(z - I*t)) Exp[2*I*\[Phi]]=Divide[Cosh[z + I*t],Cosh[z - I*t]] Failure Failure
Fail
.5583951073e-1+.1820881038e-1*I <- {phi = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.5598730513e-1-.1787687390e-1*I <- {phi = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
8.866720637+.1820881038e-1*I <- {phi = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
8.019859366+2.615210924*I <- {phi = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.28.E28 α ν , m ( 1 ) = 1 2 ⁢ π ⁢ ∫ 0 2 ⁢ π sin ⁡ t ⁢ me ν ⁡ ( t , h 2 ) ⁢ me - ν - 2 ⁢ m - 1 ⁡ ( t , h 2 ) ⁢ d t subscript superscript 𝛼 1 𝜈 𝑚 1 2 𝜋 superscript subscript 0 2 𝜋 𝑡 Mathieu-me 𝜈 𝑡 superscript ℎ 2 Mathieu-me 𝜈 2 𝑚 1 𝑡 superscript ℎ 2 𝑡 {\displaystyle{\displaystyle\alpha^{(1)}_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi% }\sin t\mathrm{me}_{\nu}\left(t,h^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{% 2}\right)\mathrm{d}t}} Error (Subscript[\[Alpha], \[Nu], m])^(1)=Divide[1,2*Pi]*Integrate[Sin[t]*Sqrt[2]*MathieuC[\[Nu], (h)^(2), t]*Sqrt[2]*MathieuC[- \[Nu]- 2*m - 1, (h)^(2), t], {t, 0, 2*Pi}] Error Failure - Error
28.29.E3 ∫ 0 π Q ⁢ ( z ) ⁢ d z = 0 superscript subscript 0 𝜋 𝑄 𝑧 𝑧 0 {\displaystyle{\displaystyle\int_{0}^{\pi}Q(z)\mathrm{d}z=0}} int(Q*(z), z = 0..Pi)= 0 Integrate[Q*(z), {z, 0, Pi}]= 0 Failure Failure Skip Error
28.29.E6 - 1 < ℜ ⁡ ν 1 𝜈 {\displaystyle{\displaystyle-1<\Re\nu}} - 1 < Re(nu) - 1 < Re[\[Nu]] Failure Failure
Fail
-1. < -1.414213562 <- {nu = -2^(1/2)-I*2^(1/2)}
-1. < -1.414213562 <- {nu = -2^(1/2)+I*2^(1/2)}
 Error
28.29.E6 ℜ ⁡ ν ≤ 1 𝜈 1 {\displaystyle{\displaystyle\Re\nu<=1}} Re(nu)< = 1 Re[\[Nu]]< = 1 Failure Failure
Fail
1.414213562 <= 1. <- {nu = 2^(1/2)+I*2^(1/2)}
1.414213562 <= 1. <- {nu = 2^(1/2)-I*2^(1/2)}
 Error
28.29.E7 w ⁢ ( z + π ) = e π ⁢ i ⁢ ν ⁢ w ⁢ ( z ) 𝑤 𝑧 𝜋 superscript 𝑒 𝜋 imaginary-unit 𝜈 𝑤 𝑧 {\displaystyle{\displaystyle w(z+\pi)=e^{\pi\mathrm{i}\nu}w(z)}} w*(z + Pi)= exp(Pi*I*nu)*w*(z) w*(z + Pi)= Exp[Pi*I*\[Nu]]*w*(z) Failure Failure
Fail
4.397533327+8.455409696*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
8.455409696+4.488232549*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
4.488232549+.4303561791*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
.4303561791+4.397533327*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.29.E13 w ⁢ ( z + π ) + w ⁢ ( z - π ) = 2 ⁢ cos ⁡ ( π ⁢ ν ) ⁢ w ⁢ ( z ) 𝑤 𝑧 𝜋 𝑤 𝑧 𝜋 2 𝜋 𝜈 𝑤 𝑧 {\displaystyle{\displaystyle w(z+\pi)+w(z-\pi)=2\cos\left(\pi\nu\right)w(z)}} w*(z + Pi)+ w*(z - Pi)= 2*cos(Pi*nu)*w*(z) w*(z + Pi)+ w*(z - Pi)= 2*Cos[Pi*\[Nu]]*w*(z) Failure Failure
Fail
327.7574484+98.56031892*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
98.56031892-327.7574484*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-327.7574484-98.56031892*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-98.56031892+327.7574484*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.30.E2 1 2 ⁢ π ⁢ ∫ 0 2 ⁢ π w m ⁢ ( x ) ⁢ w n ⁢ ( x ) ⁢ d x = δ m , n 1 2 𝜋 superscript subscript 0 2 𝜋 subscript 𝑤 𝑚 𝑥 subscript 𝑤 𝑛 𝑥 𝑥 Kronecker 𝑚 𝑛 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}w_{m}(x)w_{n}(x)% \mathrm{d}x=\delta_{m,n}}} (1)/(2*Pi)*int(w[m]*(x)* w[n]*(x), x = 0..2*Pi)= KroneckerDelta[m, n] Divide[1,2*Pi]*Integrate[Subscript[w, m]*(x)* Subscript[w, n]*(x), {x, 0, 2*Pi}]= KroneckerDelta[m, n] Failure Failure Skip Error
28.31#Ex4 W ⁢ ( z ) = w ⁢ ( z ) ⁢ exp ⁡ ( - 1 4 ⁢ ξ ⁢ cos ⁡ ( 2 ⁢ z ) ) 𝑊 𝑧 𝑤 𝑧 1 4 𝜉 2 𝑧 {\displaystyle{\displaystyle W(z)=w(z)\exp\left(-\tfrac{1}{4}\xi\cos\left(2z% \right)\right)}} W*(z)= w*(z)* exp(-(1)/(4)*xi*cos(2*z)) W*(z)= w*(z)* Exp[-Divide[1,4]*\[Xi]*Cos[2*z]] Failure Failure
Fail
-16.39510758+26.39608289*I <- {W = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
66.37576538-162.5747100*I <- {W = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
16.39510758-26.39608289*I <- {W = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-66.37576538+162.5747100*I <- {W = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.31.E4 w e , s ⁢ ( z ) = ∑ ℓ = 0 ∞ A 2 ⁢ ℓ + s ⁢ cos ⁡ ( 2 ⁢ ℓ + s ) ⁢ z subscript 𝑤 𝑒 𝑠 𝑧 superscript subscript ℓ 0 subscript 𝐴 2 ℓ 𝑠 2 ℓ 𝑠 𝑧 {\displaystyle{\displaystyle w_{\mathit{e},s}(z)=\sum_{\ell=0}^{\infty}A_{2% \ell+s}\cos(2\ell+s)z}} w[exp(1), s]*(z)= sum(A[2*ell + s]*cos((2*ell + s)* z), ell = 0..infinity) Subscript[w, E , s]*(z)= Sum[Subscript[A, 2*\[ScriptL]+ s]*Cos[(2*\[ScriptL]+ s)* z], {\[ScriptL], 0, Infinity}] Error Failure - Error
28.31.E5 w o , s ⁢ ( z ) = ∑ ℓ = 0 ∞ B 2 ⁢ ℓ + s ⁢ sin ⁡ ( 2 ⁢ ℓ + s ) ⁢ z subscript 𝑤 𝑜 𝑠 𝑧 superscript subscript ℓ 0 subscript 𝐵 2 ℓ 𝑠 2 ℓ 𝑠 𝑧 {\displaystyle{\displaystyle w_{\mathit{o},s}(z)=\sum_{\ell=0}^{\infty}B_{2% \ell+s}\sin(2\ell+s)z}} w[o , s]*(z)= sum(B[2*ell + s]*sin((2*ell + s)* z), ell = 0..infinity) Subscript[w, o , s]*(z)= Sum[Subscript[B, 2*\[ScriptL]+ s]*Sin[(2*\[ScriptL]+ s)* z], {\[ScriptL], 0, Infinity}] Error Failure - Error
28.31.E12 1 π ⁢ ∫ 0 2 ⁢ π ( C p m ⁢ ( x , ξ ) ) 2 ⁢ d x = 1 π ⁢ ∫ 0 2 ⁢ π ( S p m ⁢ ( x , ξ ) ) 2 ⁢ d x 1 𝜋 superscript subscript 0 2 𝜋 superscript superscript subscript 𝐶 𝑝 𝑚 𝑥 𝜉 2 𝑥 1 𝜋 superscript subscript 0 2 𝜋 superscript superscript subscript 𝑆 𝑝 𝑚 𝑥 𝜉 2 𝑥 {\displaystyle{\displaystyle\dfrac{1}{\pi}\int_{0}^{2\pi}\left(C_{p}^{m}(x,\xi% )\right)^{2}\mathrm{d}x=\dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi)% \right)^{2}\mathrm{d}x}} int((C(C[p])^(m)*(x , xi))^(2), x = 0..2*Pi)=int((S(S[p])^(m)*(x , xi))^(2), x = 0..2*Pi) Integrate[(C(Subscript[C, p])^(m)*(x , \[Xi]))^(2), {x, 0, 2*Pi}]=Integrate[(S(Subscript[S, p])^(m)*(x , \[Xi]))^(2), {x, 0, 2*Pi}] Failure Failure Skip Error
28.31.E12 1 π ⁢ ∫ 0 2 ⁢ π ( S p m ⁢ ( x , ξ ) ) 2 ⁢ d x = 1 1 𝜋 superscript subscript 0 2 𝜋 superscript superscript subscript 𝑆 𝑝 𝑚 𝑥 𝜉 2 𝑥 1 {\displaystyle{\displaystyle\dfrac{1}{\pi}\int_{0}^{2\pi}\left(S_{p}^{m}(x,\xi% )\right)^{2}\mathrm{d}x=1}} int((S(S[p])^(m)*(x , xi))^(2), x = 0..2*Pi)= 1 Integrate[(S(Subscript[S, p])^(m)*(x , \[Xi]))^(2), {x, 0, 2*Pi}]= 1 Failure Failure Skip Error
28.31.E21 ∫ 0 2 ⁢ π ℎ𝑐 p m 1 ⁢ ( x , ξ ) ⁢ ℎ𝑐 p m 2 ⁢ ( x , ξ ) ⁢ d x = ∫ 0 2 ⁢ π ℎ𝑠 p m 1 ⁢ ( x , ξ ) ⁢ ℎ𝑠 p m 2 ⁢ ( x , ξ ) ⁢ d x superscript subscript 0 2 𝜋 superscript subscript ℎ𝑐 𝑝 subscript 𝑚 1 𝑥 𝜉 superscript subscript ℎ𝑐 𝑝 subscript 𝑚 2 𝑥 𝜉 𝑥 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑠 𝑝 subscript 𝑚 1 𝑥 𝜉 superscript subscript ℎ𝑠 𝑝 subscript 𝑚 2 𝑥 𝜉 𝑥 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathit{hc}_{p}^{m_{1}}(x,\xi)% \mathit{hc}_{p}^{m_{2}}(x,\xi)\mathrm{d}x=\int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1% }}(x,\xi)\mathit{hs}_{p}^{m_{2}}(x,\xi)\mathrm{d}x}} int(h*s(s[p])^(m[1])*(x , xi)* h*s(s[p])^(m[2])*(x , xi), x = 0..2*Pi) Integrate[h*s(Subscript[s, p])^(Subscript[m, 1])*(x , \[Xi])* h*s(Subscript[s, p])^(Subscript[m, 2])*(x , \[Xi]), {x, 0, 2*Pi}] Failure Failure Skip Error
28.31.E21 ∫ 0 2 ⁢ π ℎ𝑠 p m 1 ⁢ ( x , ξ ) ⁢ ℎ𝑠 p m 2 ⁢ ( x , ξ ) ⁢ d x = 0 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑠 𝑝 subscript 𝑚 1 𝑥 𝜉 superscript subscript ℎ𝑠 𝑝 subscript 𝑚 2 𝑥 𝜉 𝑥 0 {\displaystyle{\displaystyle\int_{0}^{2\pi}\mathit{hs}_{p}^{m_{1}}(x,\xi)% \mathit{hs}_{p}^{m_{2}}(x,\xi)\mathrm{d}x=0}} int(h*s(s[p])^(m[1])*(x , xi)* h*s(s[p])^(m[2])*(x , xi), x = 0..2*Pi)= 0 Integrate[h*s(Subscript[s, p])^(Subscript[m, 1])*(x , \[Xi])* h*s(Subscript[s, p])^(Subscript[m, 2])*(x , \[Xi]), {x, 0, 2*Pi}]= 0 Failure Failure Skip Error
28.31.E22 ∫ u 0 u ∞ ∫ 0 2 ⁢ π ℎ𝑐 p 1 m 1 ⁢ ( u , ξ ) ⁢ ℎ𝑐 p 1 m 1 ⁢ ( v , ξ ) ⁢ ℎ𝑐 p 2 m 2 ⁢ ( u , ξ ) ⁢ ℎ𝑐 p 2 m 2 ⁢ ( v , ξ ) ⁢ ( cos ⁡ ( 2 ⁢ u ) - cos ⁡ ( 2 ⁢ v ) ) ⁢ d v ⁢ d u = 0 superscript subscript subscript 𝑢 0 subscript 𝑢 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑐 subscript 𝑝 1 subscript 𝑚 1 𝑢 𝜉 superscript subscript ℎ𝑐 subscript 𝑝 1 subscript 𝑚 1 𝑣 𝜉 superscript subscript ℎ𝑐 subscript 𝑝 2 subscript 𝑚 2 𝑢 𝜉 superscript subscript ℎ𝑐 subscript 𝑝 2 subscript 𝑚 2 𝑣 𝜉 2 𝑢 2 𝑣 𝑣 𝑢 0 {\displaystyle{\displaystyle\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc% }_{p_{1}}^{m_{1}}(u,\xi)\mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hc}_{p_{2}}^% {m_{2}}(u,\xi)\mathit{hc}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-% \cos\left(2v\right)\right)\mathrm{d}v\mathrm{d}u=0}} int(int(h*c(c[p[1]])^(m[1])*(u , xi)* h*c(c[p[1]])^(m[1])*(v , xi)* h*c(c[p[2]])^(m[2])*(u , xi)* h*c(c[p[2]])^(m[2])*(v , xi)*(cos(2*u)- cos(2*v)), v = 0..2*Pi), u = u[0]..u[infinity])= 0 Integrate[Integrate[h*c(Subscript[c, Subscript[p, 1]])^(Subscript[m, 1])*(u , \[Xi])* h*c(Subscript[c, Subscript[p, 1]])^(Subscript[m, 1])*(v , \[Xi])* h*c(Subscript[c, Subscript[p, 2]])^(Subscript[m, 2])*(u , \[Xi])* h*c(Subscript[c, Subscript[p, 2]])^(Subscript[m, 2])*(v , \[Xi])*(Cos[2*u]- Cos[2*v]), {v, 0, 2*Pi}], {u, Subscript[u, 0], Subscript[u, Infinity]}]= 0 Failure Failure Skip Error
28.31.E23 ∫ u 0 u ∞ ∫ 0 2 ⁢ π ℎ𝑠 p 1 m 1 ⁢ ( u , ξ ) ⁢ ℎ𝑠 p 1 m 1 ⁢ ( v , ξ ) ⁢ ℎ𝑠 p 2 m 2 ⁢ ( u , ξ ) ⁢ ℎ𝑠 p 2 m 2 ⁢ ( v , ξ ) ⁢ ( cos ⁡ ( 2 ⁢ u ) - cos ⁡ ( 2 ⁢ v ) ) ⁢ d v ⁢ d u = 0 superscript subscript subscript 𝑢 0 subscript 𝑢 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑠 subscript 𝑝 1 subscript 𝑚 1 𝑢 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 1 subscript 𝑚 1 𝑣 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 2 subscript 𝑚 2 𝑢 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 2 subscript 𝑚 2 𝑣 𝜉 2 𝑢 2 𝑣 𝑣 𝑢 0 {\displaystyle{\displaystyle\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hs% }_{p_{1}}^{m_{1}}(u,\xi)\mathit{hs}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^% {m_{2}}(u,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-% \cos\left(2v\right)\right)\mathrm{d}v\mathrm{d}u=0}} int(int(h*s(s[p[1]])^(m[1])*(u , xi)* h*s(s[p[1]])^(m[1])*(v , xi)* h*s(s[p[2]])^(m[2])*(u , xi)* h*s(s[p[2]])^(m[2])*(v , xi)*(cos(2*u)- cos(2*v)), v = 0..2*Pi), u = u[0]..u[infinity])= 0 Integrate[Integrate[h*s(Subscript[s, Subscript[p, 1]])^(Subscript[m, 1])*(u , \[Xi])* h*s(Subscript[s, Subscript[p, 1]])^(Subscript[m, 1])*(v , \[Xi])* h*s(Subscript[s, Subscript[p, 2]])^(Subscript[m, 2])*(u , \[Xi])* h*s(Subscript[s, Subscript[p, 2]])^(Subscript[m, 2])*(v , \[Xi])*(Cos[2*u]- Cos[2*v]), {v, 0, 2*Pi}], {u, Subscript[u, 0], Subscript[u, Infinity]}]= 0 Failure Failure Skip Error
28.31.E24 ∫ u 0 u ∞ ∫ 0 2 ⁢ π ℎ𝑐 p 1 m 1 ⁢ ( u , ξ ) ⁢ ℎ𝑐 p 1 m 1 ⁢ ( v , ξ ) ⁢ ℎ𝑠 p 2 m 2 ⁢ ( u , ξ ) ⁢ ℎ𝑠 p 2 m 2 ⁢ ( v , ξ ) ⁢ ( cos ⁡ ( 2 ⁢ u ) - cos ⁡ ( 2 ⁢ v ) ) ⁢ d v ⁢ d u = 0 superscript subscript subscript 𝑢 0 subscript 𝑢 superscript subscript 0 2 𝜋 superscript subscript ℎ𝑐 subscript 𝑝 1 subscript 𝑚 1 𝑢 𝜉 superscript subscript ℎ𝑐 subscript 𝑝 1 subscript 𝑚 1 𝑣 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 2 subscript 𝑚 2 𝑢 𝜉 superscript subscript ℎ𝑠 subscript 𝑝 2 subscript 𝑚 2 𝑣 𝜉 2 𝑢 2 𝑣 𝑣 𝑢 0 {\displaystyle{\displaystyle\int_{u_{0}}^{u_{\infty}}\int_{0}^{2\pi}\mathit{hc% }_{p_{1}}^{m_{1}}(u,\xi)\mathit{hc}_{p_{1}}^{m_{1}}(v,\xi)\mathit{hs}_{p_{2}}^% {m_{2}}(u,\xi)\mathit{hs}_{p_{2}}^{m_{2}}(v,\xi)\*\left(\cos\left(2u\right)-% \cos\left(2v\right)\right)\mathrm{d}v\mathrm{d}u=0}} int(int(h*c(c[p[1]])^(m[1])*(u , xi)* h*c(c[p[1]])^(m[1])*(v , xi)* h*s(s[p[2]])^(m[2])*(u , xi)* h*s(s[p[2]])^(m[2])*(v , xi)*(cos(2*u)- cos(2*v)), v = 0..2*Pi), u = u[0]..u[infinity])= 0 Integrate[Integrate[h*c(Subscript[c, Subscript[p, 1]])^(Subscript[m, 1])*(u , \[Xi])* h*c(Subscript[c, Subscript[p, 1]])^(Subscript[m, 1])*(v , \[Xi])* h*s(Subscript[s, Subscript[p, 2]])^(Subscript[m, 2])*(u , \[Xi])* h*s(Subscript[s, Subscript[p, 2]])^(Subscript[m, 2])*(v , \[Xi])*(Cos[2*u]- Cos[2*v]), {v, 0, 2*Pi}], {u, Subscript[u, 0], Subscript[u, Infinity]}]= 0 Failure Failure Skip Error
28.32#Ex1 x = c ⁢ cosh ⁡ ξ ⁢ cos ⁡ η 𝑥 𝑐 𝜉 𝜂 {\displaystyle{\displaystyle x=c\cosh\xi\cos\eta}} x = c*cosh(xi)*cos(eta) x = c*Cosh[\[Xi]]*Cos[\[Eta]] Failure Failure
Fail
-4.329890955-5.329890955*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), x = 1}
-3.329890955-5.329890955*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), x = 2}
-2.329890955-5.329890955*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), x = 3}
4.167193694+6.839909304*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Error
28.32#Ex2 y = c ⁢ sinh ⁡ ξ ⁢ sin ⁡ η 𝑦 𝑐 𝜉 𝜂 {\displaystyle{\displaystyle y=c\sinh\xi\sin\eta}} y = c*sinh(xi)*sin(eta) y = c*Sinh[\[Xi]]*Sin[\[Eta]] Failure Failure
Fail
7.675321591-6.675321591*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), y = 1}
8.675321591-6.675321591*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), y = 2}
9.675321591-6.675321591*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), y = 3}
-7.254122865+4.581407255*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)-I*2^(1/2), y = 1}
... skip entries to safe data
 Error
28.32.E2 ∂ 2 ⁡ V ∂ ⁡ x 2 + ∂ 2 ⁡ V ∂ ⁡ y 2 + k 2 ⁢ V = 0 partial-derivative 𝑉 𝑥 2 partial-derivative 𝑉 𝑦 2 superscript 𝑘 2 𝑉 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}V}{{\partial x}^{2}}+\frac{{% \partial}^{2}V}{{\partial y}^{2}}+k^{2}V=0}} diff(V, [x$(2)])+ diff(V, [y$(2)])+ (k)^(2)* V = 0 D[V, {x, 2}]+ D[V, {y, 2}]+ (k)^(2)* V = 0 Failure Failure
Fail
1.414213562+1.414213562*I <- {V = 2^(1/2)+I*2^(1/2), k = 1}
5.656854248+5.656854248*I <- {V = 2^(1/2)+I*2^(1/2), k = 2}
12.72792206+12.72792206*I <- {V = 2^(1/2)+I*2^(1/2), k = 3}
1.414213562-1.414213562*I <- {V = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Error
28.32.E3 ∂ 2 ⁡ V ∂ ⁡ ξ 2 + ∂ 2 ⁡ V ∂ ⁡ η 2 + 1 2 ⁢ c 2 ⁢ k 2 ⁢ ( cosh ⁡ ( 2 ⁢ ξ ) - cos ⁡ ( 2 ⁢ η ) ) ⁢ V = 0 partial-derivative 𝑉 𝜉 2 partial-derivative 𝑉 𝜂 2 1 2 superscript 𝑐 2 superscript 𝑘 2 2 𝜉 2 𝜂 𝑉 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}V}{{\partial\xi}^{2}}+\frac{{% \partial}^{2}V}{{\partial\eta}^{2}}+\frac{1}{2}c^{2}k^{2}(\cosh\left(2\xi% \right)-\cos\left(2\eta\right))V=0}} diff(V, [xi$(2)])+ diff(V, [eta$(2)])+(1)/(2)*(c)^(2)* (k)^(2)*(cosh(2*xi)- cos(2*eta))* V = 0 D[V, {\[Xi], 2}]+ D[V, {\[Eta], 2}]+Divide[1,2]*(c)^(2)* (k)^(2)*(Cosh[2*\[Xi]]- Cos[2*\[Eta]])* V = 0 Failure Failure
Fail
-14.69086244-14.69086244*I <- {V = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), k = 1}
-58.76344974-58.76344974*I <- {V = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), k = 2}
-132.2177619-132.2177619*I <- {V = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), k = 3}
-14.69086244-14.69086244*I <- {V = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), xi = -2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Error
28.32.E4 ∂ 2 ⁡ K ∂ ⁡ z 2 - ∂ 2 ⁡ K ∂ ⁡ ζ 2 = 2 ⁢ q ⁢ ( cos ⁡ ( 2 ⁢ z ) - cos ⁡ ( 2 ⁢ ζ ) ) ⁢ K partial-derivative 𝐾 𝑧 2 partial-derivative 𝐾 𝜁 2 2 𝑞 2 𝑧 2 𝜁 𝐾 {\displaystyle{\displaystyle\frac{{\partial}^{2}K}{{\partial z}^{2}}-\frac{{% \partial}^{2}K}{{\partial\zeta}^{2}}=2q\left(\cos\left(2z\right)-\cos\left(2% \zeta\right)\right)K}} diff(K, [z$(2)])- diff(K, [zeta$(2)])= 2*q*(cos(2*z)- cos(2*zeta))* K D[K, {z, 2}]- D[K, {\[zeta], 2}]= 2*q*(Cos[2*z]- Cos[2*\[zeta]])* K Failure Failure
Fail
-41.55203380-0.*I <- {K = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
-41.55203380-0.*I <- {K = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
41.55203380-0.*I <- {K = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
41.55203380-0.*I <- {K = 2^(1/2)+I*2^(1/2), q = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
... skip entries to safe data
 Error
28.32#Ex3 x 1 = 1 2 ⁢ c ⁢ ( cosh ⁡ ( 2 ⁢ α ) + cos ⁡ ( 2 ⁢ β ) - cosh ⁡ ( 2 ⁢ γ ) ) subscript 𝑥 1 1 2 𝑐 2 𝛼 2 𝛽 2 𝛾 {\displaystyle{\displaystyle x_{1}=\tfrac{1}{2}c\left(\cosh\left(2\alpha\right% )+\cos\left(2\beta\right)-\cosh\left(2\gamma\right)\right)}} x[1]=(1)/(2)*c*(cosh(2*alpha)+ cos(2*beta)- cosh(2*gamma)) Subscript[x, 1]=Divide[1,2]*c*(Cosh[2*\[Alpha]]+ Cos[2*\[Beta]]- Cosh[2*\[Gamma]]) Failure Failure
Fail
14.06853191+14.06853191*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[1] = 2^(1/2)+I*2^(1/2)}
14.06853191+11.24010479*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[1] = 2^(1/2)-I*2^(1/2)}
11.24010479+11.24010479*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[1] = -2^(1/2)-I*2^(1/2)}
11.24010479+14.06853191*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[1] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.32#Ex4 x 2 = 2 ⁢ c ⁢ cosh ⁡ α ⁢ cos ⁡ β ⁢ sinh ⁡ γ subscript 𝑥 2 2 𝑐 𝛼 𝛽 𝛾 {\displaystyle{\displaystyle x_{2}=2c\cosh\alpha\cos\beta\sinh\gamma}} x[2]= 2*c*cosh(alpha)*cos(beta)*sinh(gamma) Subscript[x, 2]= 2*c*Cosh[\[Alpha]]*Cos[\[Beta]]*Sinh[\[Gamma]] Failure Failure
Fail
-5.086190384-5.086190384*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[2] = 2^(1/2)+I*2^(1/2)}
-5.086190384-7.914617508*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[2] = 2^(1/2)-I*2^(1/2)}
-7.914617508-7.914617508*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[2] = -2^(1/2)-I*2^(1/2)}
-7.914617508-5.086190384*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[2] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.32#Ex5 x 3 = 2 ⁢ c ⁢ sinh ⁡ α ⁢ sin ⁡ β ⁢ cosh ⁡ γ subscript 𝑥 3 2 𝑐 𝛼 𝛽 𝛾 {\displaystyle{\displaystyle x_{3}=2c\sinh\alpha\sin\beta\cosh\gamma}} x[3]= 2*c*sinh(alpha)*sin(beta)*cosh(gamma) Subscript[x, 3]= 2*c*Sinh[\[Alpha]]*Sin[\[Beta]]*Cosh[\[Gamma]] Failure Failure
Fail
17.05136734-14.22294022*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[3] = 2^(1/2)+I*2^(1/2)}
17.05136734-17.05136734*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[3] = 2^(1/2)-I*2^(1/2)}
14.22294022-17.05136734*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[3] = -2^(1/2)-I*2^(1/2)}
14.22294022-14.22294022*I <- {alpha = 2^(1/2)+I*2^(1/2), beta = 2^(1/2)+I*2^(1/2), c = 2^(1/2)+I*2^(1/2), x[3] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
28.33.E1 ∂ 2 ⁡ W ∂ ⁡ x 2 + ∂ 2 ⁡ W ∂ ⁡ y 2 - ρ τ ⁢ ∂ 2 ⁡ W ∂ ⁡ t 2 = 0 partial-derivative 𝑊 𝑥 2 partial-derivative 𝑊 𝑦 2 𝜌 𝜏 partial-derivative 𝑊 𝑡 2 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}W}{{\partial x}^{2}}+\frac{{% \partial}^{2}W}{{\partial y}^{2}}-\frac{\rho}{\tau}\frac{{\partial}^{2}W}{{% \partial t}^{2}}=0}} diff(W, [x$(2)])+ diff(W, [y$(2)])-(rho)/(tau)*diff(W, [t$(2)])= 0 D[W, {x, 2}]+ D[W, {y, 2}]-Divide[\[Rho],\[Tau]]*D[W, {t, 2}]= 0 Successful Successful - -
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