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1: 30.1 Special Notation

… ►The main functions treated in this chapter are the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ and the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$, and $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. ►

Other Notations

►Flammer (1957) and Abramowitz and Stegun (1964) use ${\lambda }_{mn}(\gamma )$ for ${\lambda }_n^m\left({\gamma }^2\right)+{\gamma }^2$, $R_{mn}^{(j)}(\gamma ,z)$ for $S_n^{m(j)}(z,\gamma )$, and …

2: 30.11 Radial Spheroidal Wave Functions

§30.11 Radial Spheroidal Wave Functions

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§30.11(i) Definitions

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Connection Formulas

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§30.11(ii) Graphics

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§30.11(iv) Wronskian

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3: 30.2 Differential Equations

§30.2 Differential Equations

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§30.2(i) Spheroidal Differential Equation

… ► … ►The Liouville normal form of equation (30.2.1) is … ►

§30.2(iii) Special Cases

…

4: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

… ►This is the hypergeometric differential equation. … ► … ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …

5: 31.2 Differential Equations

§31.2 Differential Equations

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§31.2(i) Heun’s Equation

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§31.2(ii) Normal Form of Heun’s Equation

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§31.2(v) Heun’s Equation Automorphisms

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Composite Transformations

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6: 29.2 Differential Equations

§29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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§29.2(ii) Other Forms

… ►Equation (29.2.10) is a special case of Heun’s equation (31.2.1).

7: 32.2 Differential Equations

§32.2 Differential Equations

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§32.2(i) Introduction

►The six Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are as follows: … ►

§32.2(ii) Renormalizations

… ► …

8: 28.2 Definitions and Basic Properties

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§28.2(i) Mathieu’s Equation

… ►This is the characteristic equation of Mathieu’s equation (28.2.1). … ►

§28.2(iv) Floquet Solutions

… ► … ►

Distribution

…

9: 28.20 Definitions and Basic Properties

… ►

§28.20(i) Modified Mathieu’s Equation

►When $z$ is replaced by $\pm \mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation: ►

28.20.1 $$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.1.2 (in slightly different form) 20.8.6

Referenced by:

§28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)

Permalink:

http://dlmf.nist.gov/28.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

… ►

28.20.2 $$({\zeta }^2-1)w^{\prime \prime }+\zeta w^{\prime }+\left(4q{\zeta }^2-2q-a\right)w=0,$$ $\zeta =\cosh z$.

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Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

Referenced by:

§28.20(iii)

Permalink:

http://dlmf.nist.gov/28.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

… ►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. …

10: 18.42 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …