Spheroidal Wave Functions - 30.12 Generalized and Coulomb Spheroidal Functions
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 30.12.E1 | \deriv{}{z}\left((1-z^{2})\deriv{w}{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
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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 | Failed [300 / 300] Result: 1.965860183+1.904969718*I
Test Values: {alpha = 3/2, 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, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}
Result: 2.453469468+.8348874183e-1*I
Test Values: {alpha = 3/2, 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, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[3.299038105676658, 0.7500000000000002]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], 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]]]}
Result: Complex[1.2990381056766578, -2.7141016151377553]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], 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[2, 3]], Pi]]]}
... skip entries to safe data |
| 30.12.E2 | \deriv{}{z}\left((1-z^{2})\deriv{w}{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
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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 | Failed [300 / 300] Result: 4.416822075-5.340220804*I
Test Values: {alpha = 3/2, 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, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}
Result: 7.649621884+3.083488740*I
Test Values: {alpha = 3/2, 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, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[5.749999999999999, -6.495190528383291]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], 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]]]}
Result: Complex[3.749999999999999, -9.959292143521045]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], 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[2, 3]], Pi]]]}
... skip entries to safe data |