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31: 31.14 General Fuchsian Equation

… ►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …

32: 14.3 Definitions and Hypergeometric Representations

… ►The following are real-valued solutions of (14.2.2) when $\mu$, $\nu \in ℝ$ and . … ►

Ferrers Function of the Second Kind

… ►The following are solutions of (14.2.2) when $\mu$, $\nu \in ℝ$ and . … ►

Associated Legendre Function of the Second Kind

… ►As standard solutions of (14.2.2) we take the pair and , where …

33: 10.45 Functions of Imaginary Order

… ►and ${\tilde{I}}_{\nu }\left(x\right)$, ${\tilde{K}}_{\nu }\left(x\right)$ are real and linearly independent solutions of (10.45.1): … ►In consequence of (10.45.5)–(10.45.7), ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either ${\tilde{I}}_{\nu }\left(x\right)$ and $(1/\pi )\sinh \left(\pi \nu \right){\tilde{K}}_{\nu }\left(x\right)$, or ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu \ne 0$ or $\nu =0$. …

34: 1.13 Differential Equations

… ►For an extensive collection of solutions of differential equations of the first, second, and higher orders see Kamke (1977). …

35: 14.20 Conical (or Mehler) Functions

… ►Lastly, for the range , $P_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ is a real-valued solution of (14.20.1); in terms of $Q_{-\frac{1}{2}\pm \mathrm{i}\tau }^{\mu }\left(x\right)$ (which are complex-valued in general): …

36: 10.27 Connection Formulas

… ►Other solutions of (10.25.1) are $I_{-\nu }\left(z\right)$ and $K_{-\nu }\left(z\right)$. …

37: 14.21 Definitions and Basic Properties

… ►Standard solutions: the associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $P_{\nu }^{-\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{-\nu -1}^{\mu }\left(z\right)$. …When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). … ►

§14.21(ii) Numerically Satisfactory Solutions

►When $\mathrm{\Re }\nu \ge -\frac{1}{2}$ and $\mathrm{\Re }\mu \ge 0$, a numerically satisfactory pair of solutions of (14.21.1) in the half-plane $|\mathrm{ph}z|\le \frac{1}{2}\pi$ is given by $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$. …

38: 29.10 Lamé Functions with Imaginary Periods

… ►

29.10.3 $$\frac{d^{2}w}{{d{z}^{\prime }}^2}+(h^{\prime }-\nu (\nu +1)k_{}^{\prime }{}_{}{}^2{\mathrm{sn}}^2(z^{\prime },k^{\prime }))w=0.$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

… ►are solutions of (29.2.1). The first and the fourth functions have period $2\mathrm{i}{K}^{\prime }$; the second and the third have period $4\mathrm{i}{K}^{\prime }$. …

39: 32.6 Hamiltonian Structure

… ►are solutions of (32.6.3) and (32.6.4). ►

§32.6(iii) Second Painlevé Equation

… ►are solutions of (32.6.10) and (32.6.11). … ►are solutions of (32.6.17) and (32.6.18). … ►are solutions of (32.6.25) and (32.6.26). …

40: 2.8 Differential Equations with a Parameter

… ►

2.8.26 $$W_{n,2}(u,\xi )={\xi }^{1/2}{K}_{\nu }\left(u{\xi }^{1/2}\right)\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}-\xi K_{\nu +1}\left(u{\xi }^{1/2}\right)\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+{\xi }^{1/2}{K}_{\nu }\left(u{\xi }^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Referenced by:

§2.8(iv)

Permalink:

http://dlmf.nist.gov/2.8.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

… ►

2.8.30 $$W_{n,4}(u,\xi )={|\xi |}^{1/2}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)-|\xi |Y_{\nu +1}\left(u{|\xi |}^{1/2}\right)\left(\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+O\left(\frac{1}{u^{2n-2}}\right)\right).$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Referenced by:

§2.8(iv), §2.8(iv)

Permalink:

http://dlmf.nist.gov/2.8.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

… ►

2.8.36 $$W_{n,4}(u,\xi )={|\xi |}^{1/2}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}-|\xi |Y_{\nu +1}\left(u{|\xi |}^{1/2}\right)\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+1}}+{|\xi |}^{1/2}\mathrm{env}{Y}_{\nu }\left(u{|\xi |}^{1/2}\right)O\left(\frac{1}{u^{2n-1}}\right),$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $W_{n,j}(u,\xi )$: solution, $\nu$: real nonnegative constant, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Permalink:

http://dlmf.nist.gov/2.8.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

…