§30.2 Differential Equations

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

§30.2(i) Spheroidal Differential Equation

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

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $w$ : 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
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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

Notes:: See Arscott (1964b, §8.1) and Meixner and Schäfke (1954, §3.14).
Keywords:: spheroidal differential equation
Permalink:: http://dlmf.nist.gov/30.2.ii

The Liouville normal form of equation (30.2.1) is

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : 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
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30.2.3

$z=\mathop{\cos\/}\nolimits t,$

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $z$ : complex variable and $w$ : solution to DE
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With $\zeta=\gamma z$ Equation (30.2.1) changes to

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $w$ : 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
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§30.2(iii) Special Cases

Keywords:: spheroidal differential equation
Permalink:: http://dlmf.nist.gov/30.2.iii

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).

§30.2 Differential Equations

Contents

§30.2(i) Spheroidal Differential Equation

§30.2(ii) Other Forms

§30.2(iii) Special Cases