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31: 3.6 Linear Difference Equations

§3.6 Linear Difference Equations

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§3.6(ii) Homogeneous Equations

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§3.6(iv) Inhomogeneous Equations

… ►The difference equation … …

32: 10.73 Physical Applications

… ► … ► … ►This equation governs problems in acoustic and electromagnetic wave propagation. … … ►

§10.73(ii) Spherical Bessel Functions

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33: 32.5 Integral Equations

§32.5 Integral Equations

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34: 28.29 Definitions and Basic Properties

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§28.29(i) Hill’s Equation

►A generalization of Mathieu’s equation (28.2.1) is Hill’s equation … ► … ► … ► …

35: 28.30 Expansions in Series of Eigenfunctions

§28.30 Expansions in Series of Eigenfunctions

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§28.30(i) Real Variable

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28.30.1 $$w_m^{\prime \prime }+({\widehat{\lambda }}_m+Q(x))w_m=0,$$

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Symbols:

$m$: integer, $x$: real variable, $w(z)$: Mathieu’s equation solution and ${\widehat{\lambda }}_m$: characteristic values

Permalink:

http://dlmf.nist.gov/28.30.E1

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Annotations for §28.30(i), §28.30 and Ch.28

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28.30.3 $$f(x)=\sum _{m=0}^{\mathrm{\infty }}{f}_m{w}_m(x),$$

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Symbols:

$m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E3

Encodings:

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See also:

Annotations for §28.30(i), §28.30 and Ch.28

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28.30.4 $$f_m=\frac{1}{2\pi }{\int }_0^{2\pi }f(x)w_m(x)dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E4

Encodings:

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See also:

Annotations for §28.30(i), §28.30 and Ch.28

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36: 32.17 Methods of Computation

… ►The Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003). …

37: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

38: 10.72 Mathematical Applications

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§10.72(i) Differential Equations with Turning Points

… ►The canonical form of differential equation for these problems is given by … ►

§10.72(ii) Differential Equations with Poles

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§10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point

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39: 29.9 Stability

§29.9 Stability

►The Lamé equation (29.2.1) with specified values of $k,h,\nu$ is called stable if all of its solutions are bounded on $ℝ$; otherwise the equation is called unstable. …

40: 32.1 Special Notation

… ►The functions treated in this chapter are the solutions of the Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$.