Olver associated Legendre function

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21: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). …

22: 1 Algebraic and Analytic Methods

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23: 14.17 Integrals

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§14.17(iii) Orthogonality Properties

… ►Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013). ►

§14.17(iv) Definite Integrals of Products

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§14.17(v) Laplace Transforms

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§14.17(vi) Mellin Transforms

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24: 10 Bessel Functions

Chapter 10 Bessel Functions

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25: 14.16 Zeros

… ►For all cases concerning ${𝖯}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ we assume that $\nu \ge -\frac{1}{2}$ without loss of generality (see (14.9.5) and (14.9.11)). … ►The zeros of ${𝖰}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ interlace those of ${𝖯}_{\nu }^{\mu }\left(x\right)$. … ►For uniform asymptotic approximations for the zeros of ${𝖯}_n^{-m}\left(x\right)$ in the interval when $n\to \mathrm{\infty }$ with $m$ $(\ge 0)$ fixed, see Olver (1997b, p. 469). ►

§14.16(iii) Interval

… ► ${𝑸}_{\nu }^{\mu }\left(x\right)$ has no zeros in the interval $(1,\mathrm{\infty })$ when $\nu >-1$, and at most one zero in the interval $(1,\mathrm{\infty })$ when .

26: 15 Hypergeometric Function

Chapter 15 Hypergeometric Function

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27: 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

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28: 15.9 Relations to Other Functions

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15.9.16 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{2b};z)=\frac{\sqrt{\pi }}{\mathrm{\Gamma }\left(b\right)}{z}^{-b+(1/2)}{(1-z)}^{(b-a-(1/2))/2}{P}_{a-b-(1/2)}^{-b+(1/2)}\left(\frac{2-z}{2\sqrt{1-z}}\right),$$ $b\ne 0,-1,-2,\mathrm{\dots }$, and .

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E16

Encodings:

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See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.17 $$𝐅(\genfrac{}{}{0pt}{}{a,a+\frac{1}{2}}{c};z)=2^{c-1}{z}^{(1-c)/2}{(1-z)}^{-a+((c-1)/2)}{P}_{2a-c}^{1-c}\left(\frac{1}{\sqrt{1-z}}\right),$$ and .

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.18 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{a+b+\frac{1}{2}};z)=2^{a+b-(1/2)}{(-z)}^{(-a-b+(1/2))/2}{P}_{a-b-(1/2)}^{-a-b+(1/2)}\left(\sqrt{1-z}\right),$$ .

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.19 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{a-b+1};z)=z^{(b-a)/2}{(1-z)}^{-b}{P}_{-b}^{b-a}\left(\frac{1+z}{1-z}\right),$$ and .

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E19

Encodings:

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See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.21 $$𝐅(\genfrac{}{}{0pt}{}{a,1-a}{c};z)={\left(\frac{-z}{1-z}\right)}^{(1-c)/2}{P}_{-a}^{1-c}\left(1-2z\right),$$ .

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.9(iv)

Permalink:

http://dlmf.nist.gov/15.9.E21

Encodings:

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See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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29: 19 Elliptic Integrals

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30: 14.32 Methods of Computation

§14.32 Methods of Computation

►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). … ►

For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).