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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

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2: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

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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

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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: 32.13 Reductions of Partial Differential Equations

§32.13 Reductions of Partial Differential Equations

… ►Equation (32.13.3) also has the similarity reduction … ►

§32.13(ii) Sine-Gordon Equation

►The sine-Gordon equation … ►

§32.13(iii) Boussinesq Equation

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7: 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

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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

… ► …

8: 28.20 Definitions and Basic Properties

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§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

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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

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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. …

9: 22.19 Physical Applications

… ►Classical motion in one dimension is described by Newton’s equation … ►

§22.19(iii) Nonlinear ODEs and PDEs

►Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. These include the time dependent, and time independent, nonlinear Schrödinger equations (NLSE) (Drazin and Johnson (1993, Chapter 2), Ablowitz and Clarkson (1991, pp. 42, 99)), the Korteweg–de Vries (KdV) equation (Kruskal (1974), Li and Olver (2000)), the sine-Gordon equation, and others; see Drazin and Johnson (1993, Chapter 2) for an overview. … …

10: Bibliography

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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).

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Notes:

English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47

External Links:

ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

Encodings:

BibTeX

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

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The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.31(ii), §28.32(ii).

Encodings:

BibTeX

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