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11: 14.24 Analytic Continuation

… ►Let $s$ be an arbitrary integer, and $P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)$ and ${𝑸}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)$ denote the branches obtained from the principal branches by making $\frac{1}{2}s$ circuits, in the positive sense, of the ellipse having $\pm 1$ as foci and passing through $z$. … ►Next, let $P_{\nu ,s}^{-\mu }\left(z\right)$ and ${𝑸}_{\nu ,s}^{\mu }\left(z\right)$ denote the branches obtained from the principal branches by encircling the branch point $1$ (but not the branch point $-1$) $s$ times in the positive sense. …the limiting value being taken in (14.24.4) when $\mu \in \mathbb{Z}$. ►For fixed $z$, other than $\pm 1$ or $\mathrm{\infty }$, each branch of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ is an entire function of each parameter $\nu$ and $\mu$. ►The behavior of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ as $z\to -1$ from the left on the upper or lower side of the cut from $-\mathrm{\infty }$ to $1$ can be deduced from (14.8.7)–(14.8.11), combined with (14.24.1) and (14.24.2) with $s=\pm 1$.

12: 14.25 Integral Representations

… ►The principal values of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ (§14.21(i)) are given by ►

14.25.1 $$P_{\nu }^{-\mu }\left(z\right)=\frac{{\left(z^2-1\right)}^{\mu /2}}{2^{\nu }\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\nu +1}}{{(z+\cosh t)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\mu >\mathrm{\Re }\nu >-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

►

14.25.2 $${𝑸}_{\nu }^{\mu }\left(z\right)=\frac{{\pi }^{1/2}{\left(z^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(z+{(z^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

… ►For corresponding contour integrals, with less restrictions on $\mu$ and $\nu$, see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).

13: 14.14 Continued Fractions

… ►

14.14.1 $$\frac{1}{2}{\left(x^2-1\right)}^{1/2}\frac{P_{\nu }^{\mu }\left(x\right)}{P_{\nu }^{\mu -1}\left(x\right)}=\frac{x_0}{y_0+}\frac{x_1}{y_1+}\frac{x_2}{y_2+}\mathrm{\cdots },$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

… ►

$x_k=\frac{1}{4}(\nu -\mu -k+1)(\nu +\mu +k)\left(x^2-1\right),$

►

$y_k=(\mu +k)x,$

… ►

14.14.3 $$(\nu -\mu )\frac{Q_{\nu }^{\mu }\left(x\right)}{Q_{\nu -1}^{\mu }\left(x\right)}=\frac{x_0}{y_0-}\frac{x_1}{y_1-}\frac{x_2}{y_2-}\mathrm{\cdots },$$ $\nu \ne \mu$,

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

… ►

$x_k=(\nu +\mu +k)(\nu -\mu +k),$

…

14: 14.3 Definitions and Hypergeometric Representations

… ► exists for all values of $\mu$ and $\nu$. is undefined when $\mu +\nu =-1,-2,-3,\mathrm{\dots }$. ►When $\mu =m=0,1,2,\mathrm{\dots }$, (14.3.1) reduces to … ►When $\mu =m=1,2,3,\mathrm{\dots }$, (14.3.6) reduces to … ►As standard solutions of (14.2.2) we take the pair and , where …

15: 13.14 Definitions and Basic Properties

… ►except that $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$. … ►The principal branches correspond to the principal branches of the functions $z^{\frac{1}{2}+\mu }$ and $U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$ on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i). … ►Except when $z=0$, each branch of the functions $M_{\kappa ,\mu }\left(z\right)/\mathrm{\Gamma }\left(2\mu +1\right)$ and $W_{\kappa ,\mu }\left(z\right)$ is entire in $\kappa$ and $\mu$. … ►When $2\mu$ is an integer we may use the results of §13.2(v) with the substitutions $b=2\mu +1$, $a=\mu -\kappa +\frac{1}{2}$, and $W={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}+\mu }w$, where $W$ is the solution of (13.14.1) corresponding to the solution $w$ of (13.2.1). … ►When $2\mu$ is not an integer …

16: 13.24 Series

… ►For expansions of arbitrary functions in series of $M_{\kappa ,\mu }\left(z\right)$ functions see Schäfke (1961b). … ►

13.24.1 $$M_{\kappa ,\mu }\left(z\right)=\mathrm{\Gamma }\left(\kappa +\mu \right)2^{2\kappa +2\mu }{z}^{\frac{1}{2}-\kappa }\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2\kappa +2\mu \right)}_s{\left(2\kappa \right)}_s}{{\left(1+2\mu \right)}_{s}s!}\left(\kappa +\mu +s\right)I_{\kappa +\mu +s}\left(\frac{1}{2}z\right),$$ $2\mu ,\kappa +\mu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.11.2), §13.24(ii)

Permalink:

http://dlmf.nist.gov/13.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

… ►

13.24.2 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=2^{2\mu }{z}^{\mu +\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{p}_s^{(\mu )}(z){\left(2\sqrt{\kappa z}\right)}^{-2\mu -s}{J}_{2\mu +s}\left(2\sqrt{\kappa z}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $s$: nonnegative integer, $z$: complex variable and $p_s$: coefficents

Permalink:

http://dlmf.nist.gov/13.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

►where $p_0^{(\mu )}(z)=1$, $p_1^{(\mu )}(z)=\frac{1}{6}{z}^2$, and higher polynomials $p_s^{(\mu )}(z)$ are defined by ►

13.24.3 $$\exp \left(-\frac{1}{2}z\left(\coth t-\frac{1}{t}\right)\right){\left(\frac{t}{\sinh t}\right)}^{1-2\mu }=\sum _{s=0}^{\mathrm{\infty }}{p}_s^{(\mu )}(z){\left(-\frac{t}{z}\right)}^s.$$

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Symbols:

$\exp z$: exponential function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $s$: nonnegative integer, $z$: complex variable and $p_s$: coefficents

Permalink:

http://dlmf.nist.gov/13.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

…

17: 14.29 Generalizations

… ►

14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

… ►As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when ${\mu }_1={\mu }_2=\mu$. …

18: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►

§13.21(i) Large $\kappa$, Fixed $\mu$

… ►When $\kappa \to \mathrm{\infty }$ through positive real values with $\mu$ ($\ge 0$) fixed … ►Other types of approximations when $\kappa \to \mathrm{\infty }$ through positive real values with $\mu$ ($\ge 0$) fixed are as follows. … ►

§13.21(ii) Large $\kappa$, $0\le \mu \le (1-\delta )\kappa$

… ►For the functions $J_{2\mu }$, $Y_{2\mu }$, $H_{2\mu }^{(1)}$, and $H_{2\mu }^{(2)}$ see §10.2(ii), and for the $\mathrm{env}$ functions associated with $J_{2\mu }$ and $Y_{2\mu }$ see §2.8(iv). …

19: 13.17 Continued Fractions

… ►If $\kappa ,\mu \in ℂ$ such that $\mu \pm (\kappa -\frac{1}{2})\ne -1,-2,-3,\mathrm{\dots }$, then … ►

$u_{2n+1}=-{\displaystyle \frac{\frac{1}{2}+\mu +\kappa +n}{(2\mu +2n+1)(2\mu +2n+2)}},$

►

$u_{2n}={\displaystyle \frac{\frac{1}{2}+\mu -\kappa +n}{(2\mu +2n)(2\mu +2n+1)}}.$

… ►If $\kappa ,\mu \in ℂ$ such that $\mu +\frac{1}{2}\pm (\kappa +1)\ne -1,-2,-3,\mathrm{\dots }$, then … ►

$v_{2n+1}=\frac{1}{2}+\mu -\kappa +n,$

…

20: 13.16 Integral Representations

… ►In this subsection see §§10.2(ii), 10.25(ii) for the functions $J_{2\mu }$, $I_{2\mu }$, and $K_{2\mu }$, and §§15.1, 15.2(i) for ${}_2{}^{}𝐅_1^{}$. … ►If $\frac{1}{2}+\mu -\kappa \ne 0,-1,-2,\mathrm{\dots }$, then … ►If $\frac{1}{2}\pm \mu -\kappa \ne 0,-1,-2,\mathrm{\dots }$, then …where the contour of integration separates the poles of $\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)$ from those of $\mathrm{\Gamma }\left(-\kappa -t\right)$. …where the contour of integration passes all the poles of $\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)$ on the right-hand side.