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1: 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

…

2: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

… ►

15.10.1 $$z(1-z)\frac{d^{2}w}{{dz}^2}+\left(c-(a+b+1)z\right)\frac{dw}{dz}-abw=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)

Permalink:

http://dlmf.nist.gov/15.10.E1

Encodings:

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Annotations for §15.10(i), §15.10 and Ch.15

►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: …

3: 31.2 Differential Equations

§31.2 Differential Equations

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§31.2(i) Heun’s Equation

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31.2.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{ϵ}{z-a}\right)\frac{dw}{dz}+\frac{\alpha \beta z-q}{z(z-1)(z-a)}w=0,$$ $\alpha +\beta +1=\gamma +\delta +ϵ$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8

Permalink:

http://dlmf.nist.gov/31.2.E1

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Annotations for §31.2(i), §31.2 and Ch.31

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§31.2(v) Heun’s Equation Automorphisms

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Composite Transformations

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4: 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).

5: 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

… ► …

6: 28.2 Definitions and Basic Properties

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§28.2(i) Mathieu’s Equation

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28.2.1 $$w^{\prime \prime }+(a-2q\cos \left(2z\right))w=0.$$

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Symbols:

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

A&S Ref:

20.3.1 20.1.1 (in slightly different form)

Referenced by:

§1.18(vii), Figure 28.17.1, Figure 28.17.1, §28.17, §28.17, §28.2(ii), §28.2(ii), §28.2(iii), §28.2(iv), §28.20(i), §28.29(i), §28.32(i), §28.32(i), 1st item, 2nd item, §28.33(iii), item (a), item (c), item (c), §28.5(i), §28.8(iv), §28.8(iv), §28.8(iv), §30.2(iii)

Permalink:

http://dlmf.nist.gov/28.2.E1

Encodings:

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Annotations for §28.2(i), §28.2 and Ch.28

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

§28.2(iv) Floquet Solutions

… ► …

7: 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

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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:

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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. …

8: 31.10 Integral Equations and Representations

… ►and the kernel $𝒦(z,t)$ is a solution of the partial differential equation … ►

Kernel Functions

… ►and the kernel $𝒦(z;s,t)$ is a solution of the partial differential equation … ►

Kernel Functions

… ►leads to the kernel equation …

9: 10.61 Definitions and Basic Properties

… ►When $\nu =0$ suffices on $\mathrm{ber}$, $\mathrm{bei}$, $\ker$, and $\mathrm{kei}$ are usually suppressed. ►Most properties of ${\mathrm{ber}}_{\nu }x$, ${\mathrm{bei}}_{\nu }x$, ${\ker }_{\nu }x$, and ${\mathrm{kei}}_{\nu }x$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. ►

§10.61(ii) Differential Equations

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$${\ker }_{-\nu }x=\cos \left(\nu \pi \right){\ker }_{\nu }x-\sin \left(\nu \pi \right){\mathrm{kei}}_{\nu }x,$$

… ►

$${\ker }_{-n}x={(-1)}^n{\ker }_{n}x,{\mathrm{kei}}_{-n}x={(-1)}^n{\mathrm{kei}}_{n}x.$$

…

10: 10.73 Physical Applications

… ► … ► … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►The analysis of the current distribution in circular conductors leads to the Kelvin functions $\mathrm{ber}x$, $\mathrm{bei}x$, $\ker x$, and $\mathrm{kei}x$. …