Results of Spheroidal Wave Functions

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
30.1#Ex3 S m ⁒ n ( 1 ) ⁒ ( Ξ³ , 0 ) = ( - 1 ) m ⁒ 𝖯 n m ⁑ ( 0 ) subscript superscript 𝑆 1 π‘š 𝑛 𝛾 0 superscript 1 π‘š Ferrers-Legendre-P-first-kind π‘š 𝑛 0 {\displaystyle{\displaystyle S^{(1)}_{mn}(\gamma,0)=(-1)^{m}\mathsf{P}^{m}_{n}% \left(0\right)}} (S[m*n])^(1)*(gamma , 0)=(- 1)^(m)* LegendreP(n, m, 0) (Subscript[S, m*n])^(1)*(\[Gamma], 0)=(- 1)^(m)* LegendreP[n, m, 0] Failure Failure Error Error
30.2.E1 d d z ⁑ ( ( 1 - z 2 ) ⁒ d w d z ) + ( Ξ» + Ξ³ 2 ⁒ ( 1 - z 2 ) - ΞΌ 2 1 - z 2 ) ⁒ w = 0 derivative 𝑧 1 superscript 𝑧 2 derivative 𝑀 𝑧 πœ† superscript 𝛾 2 1 superscript 𝑧 2 superscript πœ‡ 2 1 superscript 𝑧 2 𝑀 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left((1-z^{2})\frac% {\mathrm{d}w}{\mathrm{d}z}\right)+\left(\lambda+\gamma^{2}(1-z^{2})-\frac{\mu^% {2}}{1-z^{2}}\right)w=0}} diff(((1 - (z)^(2))*diff(w, z))+(lambda + (gamma)^(2)*(1 - (z)^(2))-((mu)^(2))/(1 - (z)^(2)))* w, z)= 0 D[((1 - (z)^(2))*D[w, z])+(\[Lambda]+ (\[Gamma])^(2)*(1 - (z)^(2))-Divide[(\[Mu])^(2),1 - (z)^(2)])* w, z]= 0 Failure Failure
Fail
-1.660899653-1.779610239*I <- {mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-3.551236539+1.660899654*I <- {mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.660899653+1.779610239*I <- {mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
3.551236539-1.660899654*I <- {mu = 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[30.33910034602076, 0.8858131487889274] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ³, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[33.660899653979236, -0.8858131487889274] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ³, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[30.33910034602076, 0.8858131487889274] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ³, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[33.660899653979236, -0.8858131487889274] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ξ³, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ΞΌ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
30.2.E2 d 2 g d t 2 + ( Ξ» + 1 4 + Ξ³ 2 ⁒ sin 2 ⁑ t - ΞΌ 2 - 1 4 sin 2 ⁑ t ) ⁒ g = 0 derivative 𝑔 𝑑 2 πœ† 1 4 superscript 𝛾 2 2 𝑑 superscript πœ‡ 2 1 4 2 𝑑 𝑔 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}g}{{\mathrm{d}t}^{2}}+\left(% \lambda+\frac{1}{4}+\gamma^{2}{\sin^{2}}t-\frac{\mu^{2}-\frac{1}{4}}{{\sin^{2}% }t}\right)g=0}} diff(g, [t$(2)])+(lambda +(1)/(4)+ (gamma)^(2)* (sin(t))^(2)-((mu)^(2)-(1)/(4))/((sin(t))^(2)))* g = 0 D[g, {t, 2}]+(\[Lambda]+Divide[1,4]+ (\[Gamma])^(2)* (Sin[t])^(2)-Divide[(\[Mu])^(2)-Divide[1,4],(Sin[t])^(2)])* g = 0 Failure Failure
Fail
2.795111666+5.673160258*I <- {g = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2)}
4.636944094+5.150079117*I <- {g = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), t = 2^(1/2)-I*2^(1/2)}
2.795111666+5.673160258*I <- {g = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), t = -2^(1/2)-I*2^(1/2)}
4.636944094+5.150079117*I <- {g = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), t = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Skip
30.2#Ex1 z = cos ⁑ t 𝑧 𝑑 {\displaystyle{\displaystyle z=\cos t}} z = cos(t) z = Cos[t] Failure Failure
Fail
1.074539570+3.325606671*I <- {t = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.074539570+.497179547*I <- {t = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.753887554+.497179547*I <- {t = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-1.753887554+3.325606671*I <- {t = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[1.0745395706783705, 3.3256066725373055] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.0745395706783705, 0.4971795477911152] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.7538875540678198, 0.4971795477911152] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.7538875540678198, 3.3256066725373055] <- {Rule[t, 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
30.2.E4 ( ΞΆ 2 - Ξ³ 2 ) ⁒ d 2 w d ΞΆ 2 + 2 ⁒ ΞΆ ⁒ d w d ΞΆ + ( ΞΆ 2 - Ξ» - Ξ³ 2 - Ξ³ 2 ⁒ ΞΌ 2 ΞΆ 2 - Ξ³ 2 ) ⁒ w = 0 superscript 𝜁 2 superscript 𝛾 2 derivative 𝑀 𝜁 2 2 𝜁 derivative 𝑀 𝜁 superscript 𝜁 2 πœ† superscript 𝛾 2 superscript 𝛾 2 superscript πœ‡ 2 superscript 𝜁 2 superscript 𝛾 2 𝑀 0 {\displaystyle{\displaystyle(\zeta^{2}-\gamma^{2})\frac{{\mathrm{d}}^{2}w}{{% \mathrm{d}\zeta}^{2}}+2\zeta\frac{\mathrm{d}w}{\mathrm{d}\zeta}+\left(\zeta^{2% }-\lambda-\gamma^{2}-\frac{\gamma^{2}\mu^{2}}{\zeta^{2}-\gamma^{2}}\right)w=0}} ((zeta)^(2)- (gamma)^(2))* diff(w, [zeta$(2)])+ 2*zeta*diff(w, zeta)+((zeta)^(2)- lambda - (gamma)^(2)-((gamma)^(2)* (mu)^(2))/((zeta)^(2)- (gamma)^(2)))* w = 0 ((\[zeta])^(2)- (\[Gamma])^(2))* D[w, {\[zeta], 2}]+ 2*\[zeta]*D[w, \[zeta]]+((\[zeta])^(2)- \[Lambda]- (\[Gamma])^(2)-Divide[(\[Gamma])^(2)* (\[Mu])^(2),(\[zeta])^(2)- (\[Gamma])^(2)])* w = 0 Failure Failure
Fail
-6.634953849+.7567079812*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
5.614631034-9.621124116*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
-6.634953849+.7567079812*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
5.614631034-9.621124116*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
30.11#Ex5 S n m ⁒ ( 3 ) ⁑ ( z , Ξ³ ) = S n m ⁒ ( 1 ) ⁑ ( z , Ξ³ ) + i ⁒ S n m ⁒ ( 2 ) ⁑ ( z , Ξ³ ) radial-spheroidal-wave-S π‘š 3 𝑛 𝑧 𝛾 radial-spheroidal-wave-S π‘š 1 𝑛 𝑧 𝛾 imaginary-unit radial-spheroidal-wave-S π‘š 2 𝑛 𝑧 𝛾 {\displaystyle{\displaystyle S^{m(3)}_{n}\left(z,\gamma\right)=S^{m(1)}_{n}% \left(z,\gamma\right)+\mathrm{i}S^{m(2)}_{n}\left(z,\gamma\right)}} Error SpheroidalS3[n, m, z, \[Gamma]]= SpheroidalS1[n, m, z, \[Gamma]]+ I*SpheroidalS2[n, m, z, \[Gamma]] Error Failure - Skip
30.11#Ex6 S n m ⁒ ( 4 ) ⁑ ( z , Ξ³ ) = S n m ⁒ ( 1 ) ⁑ ( z , Ξ³ ) - i ⁒ S n m ⁒ ( 2 ) ⁑ ( z , Ξ³ ) radial-spheroidal-wave-S π‘š 4 𝑛 𝑧 𝛾 radial-spheroidal-wave-S π‘š 1 𝑛 𝑧 𝛾 imaginary-unit radial-spheroidal-wave-S π‘š 2 𝑛 𝑧 𝛾 {\displaystyle{\displaystyle S^{m(4)}_{n}\left(z,\gamma\right)=S^{m(1)}_{n}% \left(z,\gamma\right)-\mathrm{i}S^{m(2)}_{n}\left(z,\gamma\right)}} Error SpheroidalS4[n, m, z, \[Gamma]]= SpheroidalS1[n, m, z, \[Gamma]]- I*SpheroidalS2[n, m, z, \[Gamma]] Error Failure - Skip
30.11.E7 𝒲 ⁑ { S n m ⁒ ( 1 ) ⁑ ( z , Ξ³ ) , S n m ⁒ ( 2 ) ⁑ ( z , Ξ³ ) } = 1 Ξ³ ⁒ ( z 2 - 1 ) Wronskian radial-spheroidal-wave-S π‘š 1 𝑛 𝑧 𝛾 radial-spheroidal-wave-S π‘š 2 𝑛 𝑧 𝛾 1 𝛾 superscript 𝑧 2 1 {\displaystyle{\displaystyle\mathscr{W}\left\{S^{m(1)}_{n}\left(z,\gamma\right% ),S^{m(2)}_{n}\left(z,\gamma\right)\right\}=\frac{1}{\gamma(z^{2}-1)}}} Error Wronskian[{SpheroidalS1[n, m, z, \[Gamma]], SpheroidalS2[n, m, z, \[Gamma]]}, z]=Divide[1,\[Gamma]*((z)^(2)- 1)] Error Failure - Skip
30.12.E1 d d z ⁑ ( ( 1 - z 2 ) ⁒ d w d z ) + ( Ξ» + Ξ± ⁒ z + Ξ³ 2 ⁒ ( 1 - z 2 ) - ΞΌ 2 1 - z 2 ) ⁒ w = 0 derivative 𝑧 1 superscript 𝑧 2 derivative 𝑀 𝑧 πœ† 𝛼 𝑧 superscript 𝛾 2 1 superscript 𝑧 2 superscript πœ‡ 2 1 superscript 𝑧 2 𝑀 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left((1-z^{2})\frac% {\mathrm{d}w}{\mathrm{d}z}\right)+{\left(\lambda+\alpha z+\gamma^{2}(1-z^{2})-% \frac{\mu^{2}}{1-z^{2}}\right)w}=0}} diff(((1 - (z)^(2))*diff(w, z))+(lambda + alpha*z + (gamma)^(2)*(1 - (z)^(2))-((mu)^(2))/(1 - (z)^(2)))* w, z)= 0 D[((1 - (z)^(2))*D[w, z])+(\[Lambda]+ \[Alpha]*z + (\[Gamma])^(2)*(1 - (z)^(2))-Divide[(\[Mu])^(2),1 - (z)^(2)])* w, z]= 0 Failure Failure
Fail
-1.660899654+2.220389759*I <- {alpha = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-3.551236539+5.660899652*I <- {alpha = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.660899655+5.779610238*I <- {alpha = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
3.551236538+2.339100344*I <- {alpha = 2^(1/2)+I*2^(1/2), mu = 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
 Skip
30.12.E2 d d z ⁑ ( ( 1 - z 2 ) ⁒ d w d z ) + ( Ξ» + Ξ³ 2 ⁒ ( 1 - z 2 ) - Ξ± ⁒ ( Ξ± + 1 ) z 2 - ΞΌ 2 1 - z 2 ) ⁒ w = 0 derivative 𝑧 1 superscript 𝑧 2 derivative 𝑀 𝑧 πœ† superscript 𝛾 2 1 superscript 𝑧 2 𝛼 𝛼 1 superscript 𝑧 2 superscript πœ‡ 2 1 superscript 𝑧 2 𝑀 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left((1-z^{2})\frac% {\mathrm{d}w}{\mathrm{d}z}\right)+\left(\lambda+\gamma^{2}(1-z^{2})-\frac{% \alpha(\alpha+1)}{z^{2}}-\frac{\mu^{2}}{1-z^{2}}\right)w=0}} diff(((1 - (z)^(2))*diff(w, z))+(lambda + (gamma)^(2)*(1 - (z)^(2))-(alpha*(alpha + 1))/((z)^(2))-((mu)^(2))/(1 - (z)^(2)))* w, z)= 0 D[((1 - (z)^(2))*D[w, z])+(\[Lambda]+ (\[Gamma])^(2)*(1 - (z)^(2))-Divide[\[Alpha]*(\[Alpha]+ 1),(z)^(2)]-Divide[(\[Mu])^(2),1 - (z)^(2)])* w, z]= 0 Failure Failure
Fail
1.046207128-2.486717021*I <- {alpha = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-4.258343320-1.046207128*I <- {alpha = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.046207128+2.486717021*I <- {alpha = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
4.258343320+1.046207128*I <- {alpha = 2^(1/2)+I*2^(1/2), mu = 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
 Skip
30.13#Ex1 x = c ⁒ ( ΞΎ 2 - 1 ) ⁒ ( 1 - Ξ· 2 ) ⁒ cos ⁑ Ο• π‘₯ 𝑐 superscript πœ‰ 2 1 1 superscript πœ‚ 2 italic-Ο• {\displaystyle{\displaystyle x=c\sqrt{(\xi^{2}-1)(1-\eta^{2})}\cos\phi}} x = c*sqrt(((xi)^(2)- 1)*(1 - (eta)^(2)))*cos(phi) x = c*Sqrt[((\[Xi])^(2)- 1)*(1 - (\[Eta])^(2))]*Cos[\[Phi]] Failure Failure Skip Skip
30.13#Ex2 y = c ⁒ ( ΞΎ 2 - 1 ) ⁒ ( 1 - Ξ· 2 ) ⁒ sin ⁑ Ο• 𝑦 𝑐 superscript πœ‰ 2 1 1 superscript πœ‚ 2 italic-Ο• {\displaystyle{\displaystyle y=c\sqrt{(\xi^{2}-1)(1-\eta^{2})}\sin\phi}} y = c*sqrt(((xi)^(2)- 1)*(1 - (eta)^(2)))*sin(phi) y = c*Sqrt[((\[Xi])^(2)- 1)*(1 - (\[Eta])^(2))]*Sin[\[Phi]] Failure Failure Skip Error
30.13.E3 h ΞΎ 2 = ( βˆ‚ ⁑ x βˆ‚ ⁑ ΞΎ ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ ΞΎ ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ ΞΎ ) 2 superscript subscript β„Ž πœ‰ 2 superscript partial-derivative π‘₯ πœ‰ 2 superscript partial-derivative 𝑦 πœ‰ 2 superscript partial-derivative 𝑧 πœ‰ 2 {\displaystyle{\displaystyle h_{\xi}^{2}=\left(\frac{\partial x}{\partial\xi}% \right)^{2}+\left(\frac{\partial y}{\partial\xi}\right)^{2}+\left(\frac{% \partial z}{\partial\xi}\right)^{2}}} (h[xi])^(2)=(diff(x, xi))^(2)+(diff(y, xi))^(2)+(diff(z, xi))^(2) (Subscript[h, \[Xi]])^(2)=(D[x, \[Xi]])^(2)+(D[y, \[Xi]])^(2)+(D[z, \[Xi]])^(2) Failure Failure
Fail
0.+3.999999998*I <- {h[xi] = 2^(1/2)+I*2^(1/2)}
0.-3.999999998*I <- {h[xi] = 2^(1/2)-I*2^(1/2)}
0.+3.999999998*I <- {h[xi] = -2^(1/2)-I*2^(1/2)}
0.-3.999999998*I <- {h[xi] = -2^(1/2)+I*2^(1/2)}
 Error
30.13.E3 ( βˆ‚ ⁑ x βˆ‚ ⁑ ΞΎ ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ ΞΎ ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ ΞΎ ) 2 = c 2 ⁒ ( ΞΎ 2 - Ξ· 2 ) ΞΎ 2 - 1 superscript partial-derivative π‘₯ πœ‰ 2 superscript partial-derivative 𝑦 πœ‰ 2 superscript partial-derivative 𝑧 πœ‰ 2 superscript 𝑐 2 superscript πœ‰ 2 superscript πœ‚ 2 superscript πœ‰ 2 1 {\displaystyle{\displaystyle\left(\frac{\partial x}{\partial\xi}\right)^{2}+% \left(\frac{\partial y}{\partial\xi}\right)^{2}+\left(\frac{\partial z}{% \partial\xi}\right)^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{\xi^{2}-1}}} (diff(x, xi))^(2)+(diff(y, xi))^(2)+(diff(z, xi))^(2)=((c)^(2)*((xi)^(2)- (eta)^(2)))/((xi)^(2)- 1) (D[x, \[Xi]])^(2)+(D[y, \[Xi]])^(2)+(D[z, \[Xi]])^(2)=Divide[(c)^(2)*((\[Xi])^(2)- (\[Eta])^(2)),(\[Xi])^(2)- 1] Failure Failure
Fail
1.882352941-7.529411761*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)}
1.882352941-7.529411761*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)}
-1.882352941-7.529411761*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)}
-1.882352941-7.529411761*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)}
... skip entries to safe data
 Error
30.13.E4 h Ξ· 2 = ( βˆ‚ ⁑ x βˆ‚ ⁑ Ξ· ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ Ξ· ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ Ξ· ) 2 superscript subscript β„Ž πœ‚ 2 superscript partial-derivative π‘₯ πœ‚ 2 superscript partial-derivative 𝑦 πœ‚ 2 superscript partial-derivative 𝑧 πœ‚ 2 {\displaystyle{\displaystyle h_{\eta}^{2}=\left(\frac{\partial x}{\partial\eta% }\right)^{2}+\left(\frac{\partial y}{\partial\eta}\right)^{2}+\left(\frac{% \partial z}{\partial\eta}\right)^{2}}} (h[eta])^(2)=(diff(x, eta))^(2)+(diff(y, eta))^(2)+(diff(z, eta))^(2) (Subscript[h, \[Eta]])^(2)=(D[x, \[Eta]])^(2)+(D[y, \[Eta]])^(2)+(D[z, \[Eta]])^(2) Failure Failure
Fail
0.+3.999999998*I <- {h[eta] = 2^(1/2)+I*2^(1/2)}
0.-3.999999998*I <- {h[eta] = 2^(1/2)-I*2^(1/2)}
0.+3.999999998*I <- {h[eta] = -2^(1/2)-I*2^(1/2)}
0.-3.999999998*I <- {h[eta] = -2^(1/2)+I*2^(1/2)}
 Error
30.13.E4 ( βˆ‚ ⁑ x βˆ‚ ⁑ Ξ· ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ Ξ· ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ Ξ· ) 2 = c 2 ⁒ ( ΞΎ 2 - Ξ· 2 ) 1 - Ξ· 2 superscript partial-derivative π‘₯ πœ‚ 2 superscript partial-derivative 𝑦 πœ‚ 2 superscript partial-derivative 𝑧 πœ‚ 2 superscript 𝑐 2 superscript πœ‰ 2 superscript πœ‚ 2 1 superscript πœ‚ 2 {\displaystyle{\displaystyle\left(\frac{\partial x}{\partial\eta}\right)^{2}+% \left(\frac{\partial y}{\partial\eta}\right)^{2}+\left(\frac{\partial z}{% \partial\eta}\right)^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{1-\eta^{2}}}} (diff(x, eta))^(2)+(diff(y, eta))^(2)+(diff(z, eta))^(2)=((c)^(2)*((xi)^(2)- (eta)^(2)))/(1 - (eta)^(2)) (D[x, \[Eta]])^(2)+(D[y, \[Eta]])^(2)+(D[z, \[Eta]])^(2)=Divide[(c)^(2)*((\[Xi])^(2)- (\[Eta])^(2)),1 - (\[Eta])^(2)] Failure Failure
Fail
-1.882352941-7.529411761*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)}
-1.882352941-7.529411761*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)}
1.882352941-7.529411761*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)}
1.882352941-7.529411761*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)}
... skip entries to safe data
 Error
30.13.E5 h Ο• 2 = ( βˆ‚ ⁑ x βˆ‚ ⁑ Ο• ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ Ο• ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ Ο• ) 2 superscript subscript β„Ž italic-Ο• 2 superscript partial-derivative π‘₯ italic-Ο• 2 superscript partial-derivative 𝑦 italic-Ο• 2 superscript partial-derivative 𝑧 italic-Ο• 2 {\displaystyle{\displaystyle h_{\phi}^{2}=\left(\frac{\partial x}{\partial\phi% }\right)^{2}+\left(\frac{\partial y}{\partial\phi}\right)^{2}+\left(\frac{% \partial z}{\partial\phi}\right)^{2}}} (h[phi])^(2)=(diff(x, phi))^(2)+(diff(y, phi))^(2)+(diff(z, phi))^(2) (Subscript[h, \[Phi]])^(2)=(D[x, \[Phi]])^(2)+(D[y, \[Phi]])^(2)+(D[z, \[Phi]])^(2) Failure Failure
Fail
0.+3.999999998*I <- {h[phi] = 2^(1/2)+I*2^(1/2)}
0.-3.999999998*I <- {h[phi] = 2^(1/2)-I*2^(1/2)}
0.+3.999999998*I <- {h[phi] = -2^(1/2)-I*2^(1/2)}
0.-3.999999998*I <- {h[phi] = -2^(1/2)+I*2^(1/2)}
 Error
30.13.E5 ( βˆ‚ ⁑ x βˆ‚ ⁑ Ο• ) 2 + ( βˆ‚ ⁑ y βˆ‚ ⁑ Ο• ) 2 + ( βˆ‚ ⁑ z βˆ‚ ⁑ Ο• ) 2 = c 2 ⁒ ( ΞΎ 2 - 1 ) ⁒ ( 1 - Ξ· 2 ) superscript partial-derivative π‘₯ italic-Ο• 2 superscript partial-derivative 𝑦 italic-Ο• 2 superscript partial-derivative 𝑧 italic-Ο• 2 superscript 𝑐 2 superscript πœ‰ 2 1 1 superscript πœ‚ 2 {\displaystyle{\displaystyle\left(\frac{\partial x}{\partial\phi}\right)^{2}+% \left(\frac{\partial y}{\partial\phi}\right)^{2}+\left(\frac{\partial z}{% \partial\phi}\right)^{2}=c^{2}(\xi^{2}-1)(1-\eta^{2})}} (diff(x, phi))^(2)+(diff(y, phi))^(2)+(diff(z, phi))^(2)= (c)^(2)*((xi)^(2)- 1)*(1 - (eta)^(2)) (D[x, \[Phi]])^(2)+(D[y, \[Phi]])^(2)+(D[z, \[Phi]])^(2)= (c)^(2)*((\[Xi])^(2)- 1)*(1 - (\[Eta])^(2)) Failure Failure
Fail
31.99999997-59.99999989*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)}
0.+67.99999989*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)}
31.99999997-59.99999989*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)}
0.+67.99999989*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)}
... skip entries to safe data
 Error
30.13.E9 d d ΞΎ ⁑ ( ( 1 - ΞΎ 2 ) ⁒ d w 1 d ΞΎ ) + ( Ξ» + Ξ³ 2 ⁒ ( 1 - ΞΎ 2 ) - ΞΌ 2 1 - ΞΎ 2 ) ⁒ w 1 = 0 derivative πœ‰ 1 superscript πœ‰ 2 derivative subscript 𝑀 1 πœ‰ πœ† superscript 𝛾 2 1 superscript πœ‰ 2 superscript πœ‡ 2 1 superscript πœ‰ 2 subscript 𝑀 1 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}\xi}\left((1-\xi^{2})% \frac{\mathrm{d}w_{1}}{\mathrm{d}\xi}\right)+\left(\lambda+\gamma^{2}(1-\xi^{2% })-\frac{\mu^{2}}{1-\xi^{2}}\right)w_{1}=0}} diff(((1 - (xi)^(2))*diff(w[1], xi))+(lambda + (gamma)^(2)*(1 - (xi)^(2))-((mu)^(2))/(1 - (xi)^(2)))* w[1], xi)= 0 D[((1 - (\[Xi])^(2))*D[Subscript[w, 1], \[Xi]])+(\[Lambda]+ (\[Gamma])^(2)*(1 - (\[Xi])^(2))-Divide[(\[Mu])^(2),1 - (\[Xi])^(2)])* Subscript[w, 1], \[Xi]]= 0 Failure Failure
Fail
-1.660899653-1.779610239*I <- {mu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), w[1] = 2^(1/2)+I*2^(1/2)}
-1.779610239+1.660899653*I <- {mu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), w[1] = 2^(1/2)-I*2^(1/2)}
1.660899653+1.779610239*I <- {mu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), w[1] = -2^(1/2)-I*2^(1/2)}
1.779610239-1.660899653*I <- {mu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), w[1] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
30.13.E10 d d Ξ· ⁑ ( ( 1 - Ξ· 2 ) ⁒ d w 2 d Ξ· ) + ( Ξ» + Ξ³ 2 ⁒ ( 1 - Ξ· 2 ) - ΞΌ 2 1 - Ξ· 2 ) ⁒ w 2 = 0 derivative πœ‚ 1 superscript πœ‚ 2 derivative subscript 𝑀 2 πœ‚ πœ† superscript 𝛾 2 1 superscript πœ‚ 2 superscript πœ‡ 2 1 superscript πœ‚ 2 subscript 𝑀 2 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}\eta}\left((1-\eta^{2}% )\frac{\mathrm{d}w_{2}}{\mathrm{d}\eta}\right)+\left(\lambda+\gamma^{2}(1-\eta% ^{2})-\frac{\mu^{2}}{1-\eta^{2}}\right)w_{2}=0}} diff(((1 - (eta)^(2))*diff(w[2], eta))+(lambda + (gamma)^(2)*(1 - (eta)^(2))-((mu)^(2))/(1 - (eta)^(2)))* w[2], eta)= 0 D[((1 - (\[Eta])^(2))*D[Subscript[w, 2], \[Eta]])+(\[Lambda]+ (\[Gamma])^(2)*(1 - (\[Eta])^(2))-Divide[(\[Mu])^(2),1 - (\[Eta])^(2)])* Subscript[w, 2], \[Eta]]= 0 Failure Failure
Fail
-1.660899653-1.779610239*I <- {eta = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w[2] = 2^(1/2)+I*2^(1/2)}
-1.779610239+1.660899653*I <- {eta = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w[2] = 2^(1/2)-I*2^(1/2)}
1.660899653+1.779610239*I <- {eta = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w[2] = -2^(1/2)-I*2^(1/2)}
1.779610239-1.660899653*I <- {eta = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2), w[2] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
30.13.E11 d 2 w 3 d Ο• 2 + ΞΌ 2 ⁒ w 3 = 0 derivative subscript 𝑀 3 italic-Ο• 2 superscript πœ‡ 2 subscript 𝑀 3 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w_{3}}{{\mathrm{d}\phi}^{2}}% +\mu^{2}w_{3}=0}} diff(w[3], [phi$(2)])+ (mu)^(2)* w[3]= 0 D[Subscript[w, 3], {\[Phi], 2}]+ (\[Mu])^(2)* Subscript[w, 3]= 0 Failure Failure
Fail
-5.656854245+5.656854245*I <- {mu = 2^(1/2)+I*2^(1/2), w[3] = 2^(1/2)+I*2^(1/2)}
5.656854245+5.656854245*I <- {mu = 2^(1/2)+I*2^(1/2), w[3] = 2^(1/2)-I*2^(1/2)}
5.656854245-5.656854245*I <- {mu = 2^(1/2)+I*2^(1/2), w[3] = -2^(1/2)-I*2^(1/2)}
-5.656854245-5.656854245*I <- {mu = 2^(1/2)+I*2^(1/2), w[3] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
30.13.E12 w 3 ⁒ ( Ο• ) = a 3 ⁒ cos ⁑ ( m ⁒ Ο• ) + b 3 ⁒ sin ⁑ ( m ⁒ Ο• ) subscript 𝑀 3 italic-Ο• subscript π‘Ž 3 π‘š italic-Ο• subscript 𝑏 3 π‘š italic-Ο• {\displaystyle{\displaystyle w_{3}(\phi)=a_{3}\cos\left(m\phi\right)+b_{3}\sin% \left(m\phi\right)}} w[3]*(phi)= a[3]*cos(m*phi)+ b[3]*sin(m*phi) Subscript[w, 3]*(\[Phi])= Subscript[a, 3]*Cos[m*\[Phi]]+ Subscript[b, 3]*Sin[m*\[Phi]] Failure Failure
Fail
-5.799465200+2.753260584*I <- {phi = 2^(1/2)+I*2^(1/2), a[3] = 2^(1/2)+I*2^(1/2), b[3] = 2^(1/2)+I*2^(1/2), w[3] = 2^(1/2)+I*2^(1/2), m = 1}
-7.291659687+26.73735930*I <- {phi = 2^(1/2)+I*2^(1/2), a[3] = 2^(1/2)+I*2^(1/2), b[3] = 2^(1/2)+I*2^(1/2), w[3] = 2^(1/2)+I*2^(1/2), m = 2}
87.76595466+48.56777479*I <- {phi = 2^(1/2)+I*2^(1/2), a[3] = 2^(1/2)+I*2^(1/2), b[3] = 2^(1/2)+I*2^(1/2), w[3] = 2^(1/2)+I*2^(1/2), m = 3}
-1.799465202-1.246739414*I <- {phi = 2^(1/2)+I*2^(1/2), a[3] = 2^(1/2)+I*2^(1/2), b[3] = 2^(1/2)+I*2^(1/2), w[3] = 2^(1/2)-I*2^(1/2), m = 1}
... skip entries to safe data
 Error
30.13.E14 w 1 ⁒ ( ΞΎ ) = a 1 ⁒ S n m ⁒ ( 1 ) ⁑ ( ΞΎ , Ξ³ ) + b 1 ⁒ S n m ⁒ ( 2 ) ⁑ ( ΞΎ , Ξ³ ) subscript 𝑀 1 πœ‰ subscript π‘Ž 1 radial-spheroidal-wave-S π‘š 1 𝑛 πœ‰ 𝛾 subscript 𝑏 1 radial-spheroidal-wave-S π‘š 2 𝑛 πœ‰ 𝛾 {\displaystyle{\displaystyle w_{1}(\xi)=a_{1}S^{m(1)}_{n}\left(\xi,\gamma% \right)+b_{1}S^{m(2)}_{n}\left(\xi,\gamma\right)}} Error Subscript[w, 1]*(\[Xi])= Subscript[a, 1]*SpheroidalS1[n, m, \[Xi], \[Gamma]]+ Subscript[b, 1]*SpheroidalS2[n, m, \[Xi], \[Gamma]] Error Failure - Error
30.13.E15 S n m ⁒ ( 1 ) ⁑ ( ΞΎ 0 , Ξ³ ) = 0 radial-spheroidal-wave-S π‘š 1 𝑛 subscript πœ‰ 0 𝛾 0 {\displaystyle{\displaystyle S^{m(1)}_{n}\left(\xi_{0},\gamma\right)=0}} Error SpheroidalS1[n, m, Subscript[\[Xi], 0], \[Gamma]]= 0 Error Failure - Error
30.14#Ex1 x = c ⁒ ( ΞΎ 2 + 1 ) ⁒ ( 1 - Ξ· 2 ) ⁒ cos ⁑ Ο• π‘₯ 𝑐 superscript πœ‰ 2 1 1 superscript πœ‚ 2 italic-Ο• {\displaystyle{\displaystyle x=c\sqrt{(\xi^{2}+1)(1-\eta^{2})}\cos\phi}} x = c*sqrt(((xi)^(2)+ 1)*(1 - (eta)^(2)))*cos(phi) x = c*Sqrt[((\[Xi])^(2)+ 1)*(1 - (\[Eta])^(2))]*Cos[\[Phi]] Failure Failure
Fail
-12.12586397+9.164618552*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), x = 1}
-11.12586397+9.164618552*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), x = 2}
-10.12586397+9.164618552*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), x = 3}
6.707496333+14.95670498*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
 Error
30.14#Ex2 y = c ⁒ ( ΞΎ 2 + 1 ) ⁒ ( 1 - Ξ· 2 ) ⁒ sin ⁑ Ο• 𝑦 𝑐 superscript πœ‰ 2 1 1 superscript πœ‚ 2 italic-Ο• {\displaystyle{\displaystyle y=c\sqrt{(\xi^{2}+1)(1-\eta^{2})}\sin\phi}} y = c*sqrt(((xi)^(2)+ 1)*(1 - (eta)^(2)))*sin(phi) y = c*Sqrt[((\[Xi])^(2)+ 1)*(1 - (\[Eta])^(2))]*Sin[\[Phi]] Failure Failure
Fail
-9.78594361-14.30505684*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), y = 1}
-8.78594361-14.30505684*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), y = 2}
-7.78594361-14.30505684*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), y = 3}
-15.49391918+6.994416394*I <- {c = 2^(1/2)+I*2^(1/2), eta = 2^(1/2)+I*2^(1/2), phi = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)-I*2^(1/2), y = 1}
... skip entries to safe data
 Error
30.14.E7 d d ΞΎ ⁑ ( ( 1 + ΞΎ 2 ) ⁒ d w 1 d ΞΎ ) - ( Ξ» + Ξ³ 2 ⁒ ( 1 + ΞΎ 2 ) - ΞΌ 2 1 + ΞΎ 2 ) ⁒ w 1 = 0 derivative πœ‰ 1 superscript πœ‰ 2 derivative subscript 𝑀 1 πœ‰ πœ† superscript 𝛾 2 1 superscript πœ‰ 2 superscript πœ‡ 2 1 superscript πœ‰ 2 subscript 𝑀 1 0 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}\xi}\left((1+\xi^{2})% \frac{\mathrm{d}w_{1}}{\mathrm{d}\xi}\right)-\left(\lambda+\gamma^{2}(1+\xi^{2% })-\frac{\mu^{2}}{1+\xi^{2}}\right)w_{1}=0}} diff(((1 + (xi)^(2))*diff(w[1], xi))-(lambda + (gamma)^(2)*(1 + (xi)^(2))-((mu)^(2))/(1 + (xi)^(2)))* w[1], xi)= 0 D[((1 + (\[Xi])^(2))*D[Subscript[w, 1], \[Xi]])-(\[Lambda]+ (\[Gamma])^(2)*(1 + (\[Xi])^(2))-Divide[(\[Mu])^(2),1 + (\[Xi])^(2)])* Subscript[w, 1], \[Xi]]= 0 Failure Failure
Fail
-1.660899654-3.551236539*I <- {mu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), w[1] = 2^(1/2)+I*2^(1/2)}
-3.551236539+1.660899654*I <- {mu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), w[1] = 2^(1/2)-I*2^(1/2)}
1.660899654+3.551236539*I <- {mu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), w[1] = -2^(1/2)-I*2^(1/2)}
3.551236539-1.660899654*I <- {mu = 2^(1/2)+I*2^(1/2), xi = 2^(1/2)+I*2^(1/2), w[1] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
30.14.E8 w 1 ⁒ ( ΞΎ ) = a 1 ⁒ S n m ⁒ ( 1 ) ⁑ ( i ⁒ ΞΎ , Ξ³ ) + b 1 ⁒ S n m ⁒ ( 2 ) ⁑ ( i ⁒ ΞΎ , Ξ³ ) subscript 𝑀 1 πœ‰ subscript π‘Ž 1 radial-spheroidal-wave-S π‘š 1 𝑛 𝑖 πœ‰ 𝛾 subscript 𝑏 1 radial-spheroidal-wave-S π‘š 2 𝑛 𝑖 πœ‰ 𝛾 {\displaystyle{\displaystyle w_{1}(\xi)=a_{1}S^{m(1)}_{n}\left(i\xi,\gamma% \right)+b_{1}S^{m(2)}_{n}\left(i\xi,\gamma\right)}} Error Subscript[w, 1]*(\[Xi])= Subscript[a, 1]*SpheroidalS1[n, m, I*\[Xi], \[Gamma]]+ Subscript[b, 1]*SpheroidalS2[n, m, I*\[Xi], \[Gamma]] Error Failure - Error
30.14.E9 S n m ⁒ ( 1 ) ⁑ ( i ⁒ ΞΎ 0 , Ξ³ ) = 0 radial-spheroidal-wave-S π‘š 1 𝑛 imaginary-unit subscript πœ‰ 0 𝛾 0 {\displaystyle{\displaystyle S^{m(1)}_{n}\left(\mathrm{i}\xi_{0},\gamma\right)% =0}} Error SpheroidalS1[n, m, I*Subscript[\[Xi], 0], \[Gamma]]= 0 Error Failure - Error
30.15.E3 ∫ - Ο„ Ο„ sin ⁑ Οƒ ⁒ ( t - s ) Ο€ ⁒ ( t - s ) ⁒ Ο• n ⁒ ( s ) ⁒ d s = Ξ› n ⁒ Ο• n ⁒ ( t ) superscript subscript 𝜏 𝜏 𝜎 𝑑 𝑠 πœ‹ 𝑑 𝑠 subscript italic-Ο• 𝑛 𝑠 𝑠 subscript Ξ› 𝑛 subscript italic-Ο• 𝑛 𝑑 {\displaystyle{\displaystyle\int_{-\tau}^{\tau}\frac{\sin\sigma(t-s)}{\pi(t-s)% }\phi_{n}(s)\mathrm{d}s=\Lambda_{n}\phi_{n}(t)}} int((sin(sigma*(t - s)))/(Pi*(t - s))*phi[n]*(s), s = - tau..tau)= Lambda[n]*phi[n]*(t) Integrate[Divide[Sin[\[Sigma]*(t - s)],Pi*(t - s)]*Subscript[\[Phi], n]*(s), {s, - \[Tau], \[Tau]}]= Subscript[\[CapitalLambda], n]*Subscript[\[Phi], n]*(t) Failure Failure Skip Error
30.15.E4 ∫ - ∞ ∞ e - i ⁒ t ⁒ Ο‰ ⁒ Ο• n ⁒ ( t ) ⁒ d t = ( - i ) n ⁒ 2 ⁒ Ο€ ⁒ Ο„ Οƒ ⁒ Ξ› n ⁒ Ο• n ⁒ ( Ο„ Οƒ ⁒ Ο‰ ) ⁒ Ο‡ Οƒ ⁒ ( Ο‰ ) superscript subscript superscript 𝑒 imaginary-unit 𝑑 πœ” subscript italic-Ο• 𝑛 𝑑 𝑑 superscript imaginary-unit 𝑛 2 πœ‹ 𝜏 𝜎 subscript Ξ› 𝑛 subscript italic-Ο• 𝑛 𝜏 𝜎 πœ” subscript πœ’ 𝜎 πœ” {\displaystyle{\displaystyle\int_{-\infty}^{\infty}e^{-\mathrm{i}t\omega}\phi_% {n}(t)\mathrm{d}t=(-\mathrm{i})^{n}\sqrt{\frac{2\pi\tau}{\sigma\Lambda_{n}}}% \phi_{n}\left(\frac{\tau}{\sigma}\omega\right)\chi_{\sigma}(\omega)}} int(exp(- I*t*omega)*phi[n]*(t), t = - infinity..infinity)=(- I)^(n)*sqrt((2*Pi*tau)/(sigma*Lambda[n]))*phi[n]*((tau)/(sigma)*omega)* chi[sigma]*(omega) Integrate[Exp[- I*t*\[Omega]]*Subscript[\[Phi], n]*(t), {t, - Infinity, Infinity}]=(- I)^(n)*Sqrt[Divide[2*Pi*\[Tau],\[Sigma]*Subscript[\[CapitalLambda], n]]]*Subscript[\[Phi], n]*(Divide[\[Tau],\[Sigma]]*\[Omega])* Subscript[\[Chi], \[Sigma]]*(\[Omega]) Failure Failure Skip Error
30.15.E5 ∫ - Ο„ Ο„ e - i ⁒ t ⁒ Ο‰ ⁒ Ο• n ⁒ ( t ) ⁒ d t = ( - i ) n ⁒ 2 ⁒ Ο€ ⁒ Ο„ ⁒ Ξ› n Οƒ ⁒ Ο• n ⁒ ( Ο„ Οƒ ⁒ Ο‰ ) superscript subscript 𝜏 𝜏 superscript 𝑒 imaginary-unit 𝑑 πœ” subscript italic-Ο• 𝑛 𝑑 𝑑 superscript imaginary-unit 𝑛 2 πœ‹ 𝜏 subscript Ξ› 𝑛 𝜎 subscript italic-Ο• 𝑛 𝜏 𝜎 πœ” {\displaystyle{\displaystyle\int_{-\tau}^{\tau}e^{-\mathrm{i}t\omega}\phi_{n}(% t)\mathrm{d}t=(-\mathrm{i})^{n}\sqrt{\frac{2\pi\tau\Lambda_{n}}{\sigma}}\phi_{% n}\left(\frac{\tau}{\sigma}\omega\right)}} int(exp(- I*t*omega)*phi[n]*(t), t = - tau..tau)=(- I)^(n)*sqrt((2*Pi*tau*Lambda[n])/(sigma))*phi[n]*((tau)/(sigma)*omega) Integrate[Exp[- I*t*\[Omega]]*Subscript[\[Phi], n]*(t), {t, - \[Tau], \[Tau]}]=(- I)^(n)*Sqrt[Divide[2*Pi*\[Tau]*Subscript[\[CapitalLambda], n],\[Sigma]]]*Subscript[\[Phi], n]*(Divide[\[Tau],\[Sigma]]*\[Omega]) Failure Failure Skip Error
30.15.E7 ∫ - Ο„ Ο„ Ο• k ⁒ ( t ) ⁒ Ο• n ⁒ ( t ) ⁒ d t = Ξ› n ⁒ Ξ΄ k , n superscript subscript 𝜏 𝜏 subscript italic-Ο• π‘˜ 𝑑 subscript italic-Ο• 𝑛 𝑑 𝑑 subscript Ξ› 𝑛 Kronecker π‘˜ 𝑛 {\displaystyle{\displaystyle\int_{-\tau}^{\tau}\phi_{k}(t)\phi_{n}(t)\mathrm{d% }t=\Lambda_{n}\delta_{k,n}}} int(phi[k]*(t)* phi[n]*(t), t = - tau..tau)= Lambda[n]*KroneckerDelta[k, n] Integrate[Subscript[\[Phi], k]*(t)* Subscript[\[Phi], n]*(t), {t, - \[Tau], \[Tau]}]= Subscript[\[CapitalLambda], n]*KroneckerDelta[k, n] Failure Failure Skip Error
30.15.E8 ∫ - ∞ ∞ Ο• k ⁒ ( t ) ⁒ Ο• n ⁒ ( t ) ⁒ d t = Ξ΄ k , n superscript subscript subscript italic-Ο• π‘˜ 𝑑 subscript italic-Ο• 𝑛 𝑑 𝑑 Kronecker π‘˜ 𝑛 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}\phi_{k}(t)\phi_{n}(t)% \mathrm{d}t=\delta_{k,n}}} int(phi[k]*(t)* phi[n]*(t), t = - infinity..infinity)= KroneckerDelta[k, n] Integrate[Subscript[\[Phi], k]*(t)* Subscript[\[Phi], n]*(t), {t, - Infinity, Infinity}]= KroneckerDelta[k, n] Failure Failure Skip Error
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