30 Spheroidal Wave Functions Properties 30.1 Special Notation 30.3 Eigenvalues

§30.2 Differential Equations

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Permalink:: http://dlmf.nist.gov/30.2
See also:: Annotations for Ch.30

§30.2(i) Spheroidal Differential Equation
§30.2(ii) Other Forms
§30.2(iii) Special Cases

§30.2(i) Spheroidal Differential Equation

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Keywords:: differential equation, singularities, spheroidal differential equation, spheroidal wave functions
Notes:: See Meixner and Schäfke (1954, §3.1).
Permalink:: http://dlmf.nist.gov/30.2.i
See also:: Annotations for §30.2 and Ch.30

30.2.1 $\frac{\mathrm{d}}{\mathrm{d}z}\left((1-z^{2})\frac{\mathrm{d}w}{\mathrm{d}z}% \right)+\left(\lambda+\gamma^{2}(1-z^{2})-\frac{\mu^{2}}{1-z^{2}}\right)w=0.$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : complex variable, : solution to DE, $\lambda$ : real parameter, $\gamma^{2}$ : real parameter and $\mu$ : real parameter
Referenced by:: §30.11(i), §30.12, §30.12, §30.13(iv), §30.14(iv), §30.16(ii), §30.2(ii), §30.2(iii), §30.3(i), §30.4(i), §30.5, §30.6, §30.6
Permalink:: http://dlmf.nist.gov/30.2.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §30.2(i), §30.2 and Ch.30

This equation has regular singularities at $z=\pm 1$ with exponents $\pm\frac{1}{2}\mu$ and an irregular singularity of rank 1 at $z=\infty$ (if $\gamma\neq 0$ ). The equation contains three real parameters $\lambda$ , $\gamma^{2}$ , and $\mu$ . In applications involving prolate spheroidal coordinates $\gamma^{2}$ is positive, in applications involving oblate spheroidal coordinates $\gamma^{2}$ is negative; see §§30.13, 30.14.

§30.2(ii) Other Forms

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Keywords:: Liouville normal form, spheroidal differential equation
Notes:: See Arscott (1964b, §8.1) and Meixner and Schäfke (1954, §3.14).
Permalink:: http://dlmf.nist.gov/30.2.ii
See also:: Annotations for §30.2 and Ch.30

The Liouville normal form of equation (30.2.1) is

30.2.2 $\frac{{\mathrm{d}}^{2}g}{{\mathrm{d}t}^{2}}+\left(\lambda+\frac{1}{4}+\gamma^{% 2}{\sin}^{2}t-\frac{\mu^{2}-\frac{1}{4}}{{\sin}^{2}t}\right)g=0,$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\sin\NVar{z}$ : sine function, $\lambda$ : real parameter, $\gamma^{2}$ : real parameter and $\mu$ : real parameter
Referenced by:: §30.2(iii)
Permalink:: http://dlmf.nist.gov/30.2.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §30.2(ii), §30.2 and Ch.30

30.2.3

$z=\cos t,$

$w(z)=(1-z^{2})^{-\frac{1}{4}}g(t).$

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Symbols:: $\cos\NVar{z}$ : cosine function, : complex variable and : solution to DE
Permalink:: http://dlmf.nist.gov/30.2.E3
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §30.2(ii), §30.2 and Ch.30

With $\zeta=\gamma z$ Equation (30.2.1) changes to

30.2.4 $(\zeta^{2}-\gamma^{2})\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+2\zeta% \frac{\mathrm{d}w}{\mathrm{d}\zeta}+\left(\zeta^{2}-\lambda-\gamma^{2}-\frac{% \gamma^{2}\mu^{2}}{\zeta^{2}-\gamma^{2}}\right)w=0.$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : solution to DE, $\lambda$ : real parameter, $\gamma^{2}$ : real parameter, $\mu$ : real parameter and $\zeta$ : parameter
Referenced by:: §30.2(iii)
Permalink:: http://dlmf.nist.gov/30.2.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §30.2(ii), §30.2 and Ch.30

§30.2(iii) Special Cases

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Keywords:: special cases, spheroidal differential equation
Permalink:: http://dlmf.nist.gov/30.2.iii
See also:: Annotations for §30.2 and Ch.30

If $\gamma=0$ , Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). If $\mu^{2}=\frac{1}{4}$ , Equation (30.2.2) reduces to the Mathieu equation; see (28.2.1). If $\gamma=0$ , Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).

30.1 Special Notation 30.3 Eigenvalues

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§30.2 Differential Equations

Contents

§30.2(i) Spheroidal Differential Equation

§30.2(ii) Other Forms

§30.2(iii) Special Cases