Spheroidal Wave Functions - 30.14 Wave Equation in Oblate Spheroidal Coordinates

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
30.14#Ex4 0 < ξ 0 𝜉 {\displaystyle{\displaystyle 0<\xi}}
0 < \xi		 

0 < xi
 
0 < \[Xi]
 Skipped - no semantic math Skipped - no semantic math - -
30.14#Ex5 - 1 < η 1 𝜂 {\displaystyle{\displaystyle-1<\eta}}
-1 < \eta		 

- 1 < eta
 
- 1 < \[Eta]
 Skipped - no semantic math Skipped - no semantic math - -
30.14#Ex6 0 ≤ ϕ 0 italic-ϕ {\displaystyle{\displaystyle 0\leq\phi}}
0 \leq \phi		 

0 <= phi
 
0 <= \[Phi]
 Skipped - no semantic math Skipped - no semantic math - -
30.14.E3 h ξ 2 = c 2 ⁢ ( ξ 2 + η 2 ) 1 + ξ 2 superscript subscript ℎ 𝜉 2 superscript 𝑐 2 superscript 𝜉 2 superscript 𝜂 2 1 superscript 𝜉 2 {\displaystyle{\displaystyle h_{\xi}^{2}=\frac{c^{2}(\xi^{2}+\eta^{2})}{1+\xi^% {2}}}}
h_{\xi}^{2} = \frac{c^{2}(\xi^{2}+\eta^{2})}{1+\xi^{2}}		 

(h[xi])^(2) = ((c)^(2)*((xi)^(2)+ (eta)^(2)))/(1 + (xi)^(2))
 
(Subscript[h, \[Xi]])^(2) == Divide[(c)^(2)*(\[Xi]^(2)+ \[Eta]^(2)),1 + \[Xi]^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
30.14.E4 h η 2 = c 2 ⁢ ( ξ 2 + η 2 ) 1 - η 2 superscript subscript ℎ 𝜂 2 superscript 𝑐 2 superscript 𝜉 2 superscript 𝜂 2 1 superscript 𝜂 2 {\displaystyle{\displaystyle h_{\eta}^{2}=\frac{c^{2}(\xi^{2}+\eta^{2})}{1-% \eta^{2}}}}
h_{\eta}^{2} = \frac{c^{2}(\xi^{2}+\eta^{2})}{1-\eta^{2}}		 

(h[eta])^(2) = ((c)^(2)*((xi)^(2)+ (eta)^(2)))/(1 - (eta)^(2))
 
(Subscript[h, \[Eta]])^(2) == Divide[(c)^(2)*(\[Xi]^(2)+ \[Eta]^(2)),1 - \[Eta]^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
30.14.E5 h ϕ 2 = c 2 ⁢ ( ξ 2 + 1 ) ⁢ ( 1 - η 2 ) superscript subscript ℎ italic-ϕ 2 superscript 𝑐 2 superscript 𝜉 2 1 1 superscript 𝜂 2 {\displaystyle{\displaystyle h_{\phi}^{2}=c^{2}(\xi^{2}+1)(1-\eta^{2})}}
h_{\phi}^{2} = c^{2}(\xi^{2}+1)(1-\eta^{2})		 

(h[phi])^(2) = (c)^(2)*((xi)^(2)+ 1)*(1 - (eta)^(2))
 
(Subscript[h, \[Phi]])^(2) == (c)^(2)*(\[Xi]^(2)+ 1)*(1 - \[Eta]^(2))
 Skipped - no semantic math Skipped - no semantic math - -
30.14.E7 d d ξ ⁡ ( ( 1 + ξ 2 ) ⁢ d w 1 d ξ ) - ( λ + γ 2 ⁢ ( 1 + ξ 2 ) - μ 2 1 + ξ 2 ) ⁢ w 1 = 0 derivative 𝜉 1 superscript 𝜉 2 derivative subscript 𝑤 1 𝜉 𝜆 superscript 𝛾 2 1 superscript 𝜉 2 superscript 𝜇 2 1 superscript 𝜉 2 subscript 𝑤 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}}
\deriv{}{\xi}\left((1+\xi^{2})\deriv{w_{1}}{\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
Failed [260 / 300]
Result: -.6665112581-1.154431362*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, w[1] = 1/2*3^(1/2)+1/2*I}


Result: 1.154431362-.6665112581*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, w[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.6666666666666664, -2.309401076758503]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.309401076758503, 0.6666666666666662]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[w, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
30.14.E8 w 1 ⁢ ( ξ ) = a 1 ⁢ S n m ⁢ ( 1 ) ⁡ ( i ⁢ ξ , γ ) + b 1 ⁢ S n m ⁢ ( 2 ) ⁡ ( i ⁢ ξ , γ ) subscript 𝑤 1 𝜉 subscript 𝑎 1 radial-spheroidal-wave-S 𝑚 1 𝑛 𝑖 𝜉 𝛾 subscript 𝑏 1 radial-spheroidal-wave-S 𝑚 2 𝑛 𝑖 𝜉 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)}}
w_{1}(\xi) = a_{1}\radsphwaveS{m}{1}{n}@{i\xi}{\gamma}+b_{1}\radsphwaveS{m}{2}{n}@{i\xi}{\gamma}		 

Error

Subscript[w, 1][\[Xi]] == Subscript[a, 1]*SpheroidalS1[n, m, I*\[Xi], \[Gamma]]+ Subscript[b, 1]*SpheroidalS2[n, m, I*\[Xi], \[Gamma]]
Missing Macro Error Failure - Skipped - Because timed out
30.14.E9 S n m ⁢ ( 1 ) ⁡ ( i ⁢ ξ 0 , γ ) = 0 radial-spheroidal-wave-S 𝑚 1 𝑛 imaginary-unit subscript 𝜉 0 𝛾 0 {\displaystyle{\displaystyle S^{m(1)}_{n}\left(\mathrm{i}\xi_{0},\gamma\right)% =0}}
\radsphwaveS{m}{1}{n}@{\iunit\xi_{0}}{\gamma} = 0		 

Error

SpheroidalS1[n, m, I*Subscript[\[Xi], 0], \[Gamma]] == 0
Missing Macro Error Failure -
Failed [300 / 300]
Result: Complex[-0.30414296182717676, -0.005578569442222112]
Test Values: {Rule[j, 4], Rule[m, 1], Rule[n, 1], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ξ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.05539464168732451, -0.030004541724887247]
Test Values: {Rule[j, 4], Rule[m, 1], Rule[n, 2], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ξ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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