§14.20 Conical (or Mehler) Functions

Throughout §14.20 we assume that $\nu =-\frac{1}{2}+\mathrm{i}\tau$, with $\mu \ge 0$ and $\tau \ge 0$. (14.2.2) takes the form

14.20.1		$$\left(1-x^2\right)\frac{d^{2}w}{{dx}^2}-2x\frac{dw}{dx}-\left({\tau }^2+\frac{1}{4}+\frac{{\mu }^2}{1-x^2}\right)w=0.$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $x$: real variable, $\tau$: real variable and $\mu$: general order Referenced by: §14.20(i), §14.20(i), 2nd item Permalink: http://dlmf.nist.gov/14.20.E1 Encodings: TeX, pMML, png See also: Annotations for §14.20(i), §14.20 and Ch.14

Solutions are known as conical or Mehler functions. For and $\tau >0$, a numerically satisfactory pair of real conical functions is ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ and ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right)$.

Another real-valued solution ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ of (14.20.1) was introduced in Dunster (1991). This is defined by

14.20.2		$${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\mathrm{\Re }\left({\mathrm{e}}^{\mu \pi \mathrm{i}}{\mathsf{Q}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)\right)-\frac{1}{2}\pi \sin \left(\mu \pi \right){\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right).$$
ⓘ Defines: ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function Symbols: : Ferrers function of the first kind, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $\tau$: real variable and $\mu$: general order Referenced by: §14.20(i), §14.23 Permalink: http://dlmf.nist.gov/14.20.E2 Encodings: TeX, pMML, png See also: Annotations for §14.20(i), §14.20 and Ch.14

Equivalently,

14.20.3		$${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\begin{array}{l}\frac{\pi {\mathrm{e}}^{-\tau \pi }\sin \left(\mu \pi \right)\sinh \left(\tau \pi \right)}{2({\cosh }^2\left(\tau \pi \right)-{\sin }^2\left(\mu \pi \right))}{\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)\\ +\frac{\pi ({\mathrm{e}}^{-\tau \pi }{\cos }^2\left(\mu \pi \right)+\sinh \left(\tau \pi \right))}{2({\cosh }^2\left(\tau \pi \right)-{\sin }^2\left(\mu \pi \right))}{\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right).\end{array}$$
ⓘ Symbols: : Ferrers function of the first kind, ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $\tau$: real variable and $\mu$: general order Referenced by: §14.20(i), §14.20(iii), §14.20(ix) Permalink: http://dlmf.nist.gov/14.20.E3 Encodings: TeX, pMML, png See also: Annotations for §14.20(i), §14.20 and Ch.14

${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ exists except when $\mu =\frac{1}{2},\frac{3}{2},\mathrm{\dots }$ and $\tau =0$; compare §14.3(i). It is an important companion solution to ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ when $\tau$ is large; compare §§14.20(vii), 14.20(viii), and 10.25(iii).

14.20.4		$$𝒲\left\{{\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right),{\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right)\right\}=\frac{2}{{|\mathrm{\Gamma }\left(\mu +\frac{1}{2}+\mathrm{i}\tau \right)|}^2(1-x^2)}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $𝒲$: Wronskian, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order Referenced by: §14.20(i) Permalink: http://dlmf.nist.gov/14.20.E4 Encodings: TeX, pMML, png See also: Annotations for §14.20(i), §14.20 and Ch.14

14.20.5		$$𝒲\left\{{\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right),{\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)\right\}=\frac{\pi ({\mathrm{e}}^{-\tau \pi }{\cos }^2\left(\mu \pi \right)+\sinh \left(\tau \pi \right))}{{|\mathrm{\Gamma }\left(\mu +\frac{1}{2}+\mathrm{i}\tau \right)|}^2({\cosh }^2\left(\tau \pi \right)-{\sin }^2\left(\mu \pi \right))(1-x^2)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $\tau$: real variable and $\mu$: general order Referenced by: §14.20(i) Permalink: http://dlmf.nist.gov/14.20.E5 Encodings: TeX, pMML, png See also: Annotations for §14.20(i), §14.20 and Ch.14

provided that ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ exists.

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):

14.20.6		$$P_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{\mathrm{i}{\mathrm{e}}^{-\mu \pi \mathrm{i}}}{\sinh \left(\tau \pi \right){\left|\mathrm{\Gamma }\left(\mu +\frac{1}{2}+\mathrm{i}\tau \right)\right|}^2}\left(Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)-Q_{-\frac{1}{2}-\mathrm{i}\tau }^{\mu }\left(x\right)\right),$$
$\tau \ne 0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order Referenced by: §14.20(i) Permalink: http://dlmf.nist.gov/14.20.E6 Encodings: TeX, pMML, png See also: Annotations for §14.20(i), §14.20 and Ch.14

The behavior of ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(\pm x\right)$ as $x\to 1-$ is given in §14.8(i). For $\mu >0$ and $x\to 1-$,

14.20.7		$${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)\sim \frac{1}{2}\mathrm{\Gamma }\left(\mu \right){\left(\frac{2}{1-x}\right)}^{\mu /2},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $\sim$: asymptotic equality, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order Referenced by: §14.20(iii) Permalink: http://dlmf.nist.gov/14.20.E7 Encodings: TeX, pMML, png See also: Annotations for §14.20(iii), §14.20 and Ch.14

14.20.8		$${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)\sim \frac{\pi \mathrm{\Gamma }\left(\mu \right)({\mathrm{e}}^{-\tau \pi }{\cos }^2\left(\mu \pi \right)+\sinh \left(\tau \pi \right))}{2({\cosh }^2\left(\tau \pi \right)-{\sin }^2\left(\mu \pi \right)){\left|\mathrm{\Gamma }\left(\mu +\frac{1}{2}+\mathrm{i}\tau \right)\right|}^2}{\left(\frac{2}{1-x}\right)}^{\mu /2}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $\tau$: real variable and $\mu$: general order Referenced by: §14.20(iii) Permalink: http://dlmf.nist.gov/14.20.E8 Encodings: TeX, pMML, png See also: Annotations for §14.20(iii), §14.20 and Ch.14

When ,

14.20.9		$${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }\left(\cos \theta \right)=\frac{2}{\pi }{\int }_0^{\theta }\frac{\cosh \left(\tau \varphi \right)}{\sqrt{2(\cos \varphi -\cos \theta )}}d\varphi .$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\tau$: real variable and : variable A&S Ref: 8.12.3 Referenced by: §14.20(iv), §14.20(v) Permalink: http://dlmf.nist.gov/14.20.E9 Encodings: TeX, pMML, png See also: Annotations for §14.20(iv), §14.20 and Ch.14

14.20.10		$${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }\left(\cos \theta \right)=1+\frac{4{\tau }^2+1^2}{2^2}{\sin }^2\left(\frac{1}{2}\theta \right)+\frac{\left(4{\tau }^2+1^2\right)\left(4{\tau }^2+3^2\right)}{2^2\cdot 4^2}{\sin }^4\left(\frac{1}{2}\theta \right)+\mathrm{\cdots },$$
$0\le \theta \le \pi$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and $\tau$: real variable A&S Ref: 8.12.1 Referenced by: §14.20(v) Permalink: http://dlmf.nist.gov/14.20.E10 Encodings: TeX, pMML, png See also: Annotations for §14.20(v), §14.20 and Ch.14

From (14.20.9) or (14.20.10) it is evident that ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }\left(\cos \theta \right)$ is positive for real $\theta$.

14.20.11		$$f(\tau )=\frac{\tau }{\pi }\sinh \left(\tau \pi \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\mathrm{i}\tau \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\mathrm{i}\tau \right){\int }_1^{\mathrm{\infty }}{P}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)g(x)dx,$$
ⓘ 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, $dx$: differential of $x$, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $x$: real variable, $\tau$: real variable and $\mu$: general order Permalink: http://dlmf.nist.gov/14.20.E11 Encodings: TeX, pMML, png See also: Annotations for §14.20(vi), §14.20 and Ch.14

where

14.20.12		$$g(x)={\int }_0^{\mathrm{\infty }}{P}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)f(\tau )d\tau .$$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $x$: real variable, $\tau$: real variable and $\mu$: general order Permalink: http://dlmf.nist.gov/14.20.E12 Encodings: TeX, pMML, png See also: Annotations for §14.20(vi), §14.20 and Ch.14

Special cases:

14.20.13		$$P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)=\frac{\cosh \left(\tau \pi \right)}{\pi }{\int }_1^{\mathrm{\infty }}\frac{P_{-\frac{1}{2}+\mathrm{i}\tau }\left(t\right)}{x+t}dt,$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable and $\tau$: real variable Permalink: http://dlmf.nist.gov/14.20.E13 Encodings: TeX, pMML, png See also: Annotations for §14.20(vi), §14.20 and Ch.14

14.20.14		$$\pi {\int }_0^{\mathrm{\infty }}\frac{\tau \tanh \left(\tau \pi \right)}{\cosh \left(\tau \pi \right)}{P}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)P_{-\frac{1}{2}+\mathrm{i}\tau }\left(y\right)d\tau =\frac{1}{y+x}.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\int$: integral, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable and $\tau$: real variable Permalink: http://dlmf.nist.gov/14.20.E14 Encodings: TeX, pMML, png See also: Annotations for §14.20(vi), §14.20 and Ch.14

For $\tau \to \mathrm{\infty }$ and fixed $\mu$,

14.20.15		${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(\cos \theta \right)$	$={\displaystyle \frac{1}{{\tau }^{\mu }}}{\left({\displaystyle \frac{\theta }{\sin \theta }}\right)}^{1/2}{I}_{\mu }\left(\tau \theta \right)\left(1+O\left(1/\tau \right)\right),$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, : Ferrers function of the first kind, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $\tau$: real variable, $\mu$: general order and : variable Referenced by: §14.20(vii) Permalink: http://dlmf.nist.gov/14.20.E15 Encodings: TeX, pMML, png See also: Annotations for §14.20(vii), §14.20 and Ch.14
14.20.16		${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(\cos \theta \right)$	$={\displaystyle \frac{1}{{\tau }^{\mu }}}{\left({\displaystyle \frac{\theta }{\sin \theta }}\right)}^{1/2}{K}_{\mu }\left(\tau \theta \right)\left(1+O\left(1/\tau \right)\right),$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $\tau$: real variable, $\mu$: general order and : variable Referenced by: §14.20(vii) Permalink: http://dlmf.nist.gov/14.20.E16 Encodings: TeX, pMML, png See also: Annotations for §14.20(vii), §14.20 and Ch.14

uniformly for $\theta \in (0,\pi -\delta ]$, where $I$ and $K$ are the modified Bessel functions (§10.25(ii)) and $\delta$ is an arbitrary constant such that . For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). See also Žurina and Karmazina (1966).

In this subsection and §14.20(ix), $A$ and $\delta$ denote arbitrary constants such that $A>0$ and .

As $\tau \to \mathrm{\infty }$,

14.20.17		$${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\sigma (\mu ,\tau ){\left(\frac{{\alpha }^2+\eta }{1+{\alpha }^2-x^2}\right)}^{1/4}{I}_{\mu }\left(\tau {\eta }^{1/2}\right)\left(1+O\left(1/\tau \right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, : Ferrers function of the first kind, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\tau$: real variable, $\mu$: general order, $\alpha$, $\sigma (\mu ,\tau )$ and $\eta$ Permalink: http://dlmf.nist.gov/14.20.E17 Encodings: TeX, pMML, png See also: Annotations for §14.20(viii), §14.20 and Ch.14

14.20.18		$${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\sigma (\mu ,\tau ){\left(\frac{{\alpha }^2+\eta }{1+{\alpha }^2-x^2}\right)}^{1/4}{K}_{\mu }\left(\tau {\eta }^{1/2}\right)\left(1+O\left(1/\tau \right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable, $\tau$: real variable, $\mu$: general order, $\alpha$, $\sigma (\mu ,\tau )$ and $\eta$ Referenced by: §14.20(vii) Permalink: http://dlmf.nist.gov/14.20.E18 Encodings: TeX, pMML, png See also: Annotations for §14.20(viii), §14.20 and Ch.14

uniformly for $x\in [-1+\delta ,1)$ and $\mu \in [0,A\tau ]$. Here

14.20.19		$$\alpha =\mu /\tau ,$$
ⓘ Symbols: $\tau$: real variable, $\mu$: general order and $\alpha$ Permalink: http://dlmf.nist.gov/14.20.E19 Encodings: TeX, pMML, png See also: Annotations for §14.20(viii), §14.20 and Ch.14

14.20.20		$$\sigma (\mu ,\tau )=\frac{\exp \left(\mu -\tau \arctan \alpha \right)}{{\left({\mu }^2+{\tau }^2\right)}^{\mu /2}}.$$
ⓘ Symbols: $\exp z$: exponential function, $\arctan z$: arctangent function, $\tau$: real variable, $\mu$: general order, $\alpha$ and $\sigma (\mu ,\tau )$ Permalink: http://dlmf.nist.gov/14.20.E20 Encodings: TeX, pMML, png See also: Annotations for §14.20(viii), §14.20 and Ch.14

The variable $\eta$ is defined implicitly by

14.20.21		$$\begin{array}{l}{\left({\alpha }^2+\eta \right)}^{1/2}+\frac{1}{2}\alpha \ln \eta -\alpha \ln \left({\left({\alpha }^2+\eta \right)}^{1/2}+\alpha \right)\\ =\arccos \left(\frac{x}{{\left(1+{\alpha }^2\right)}^{1/2}}\right)+\frac{\alpha }{2}\ln \left(\frac{1+{\alpha }^2+\left({\alpha }^2-1\right)x^2-2\alpha x{\left(1+{\alpha }^2-x^2\right)}^{1/2}}{\left(1+{\alpha }^2\right)\left(1-x^2\right)}\right),\end{array}$$
ⓘ Symbols: $\arccos z$: arccosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $\eta$ Permalink: http://dlmf.nist.gov/14.20.E21 Encodings: TeX, pMML, png See also: Annotations for §14.20(viii), §14.20 and Ch.14

where the inverse trigonometric functions take their principal values. The interval is mapped one-to-one to the interval , with the points $x=-1$ and $x=1$ corresponding to $\eta =\mathrm{\infty }$ and $\eta =0$, respectively.

As $\mu \to \mathrm{\infty }$,

14.20.22		$${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{\beta \exp \left(\mu \beta \arctan \beta \right)}{\mathrm{\Gamma }\left(\mu +1\right){\left(1+{\beta }^2\right)}^{\mu /2}}\frac{{\mathrm{e}}^{-\mu \rho }}{{\left(1+{\beta }^2-x^2{\beta }^2\right)}^{1/4}}\left(1+O\left(\frac{1}{\mu }\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\arctan z$: arctangent function, $x$: real variable, $\tau$: real variable, $\mu$: general order, $\beta$ and $\rho$ Referenced by: §14.20(ix) Permalink: http://dlmf.nist.gov/14.20.E22 Encodings: TeX, pMML, png See also: Annotations for §14.20(ix), §14.20 and Ch.14

uniformly for $x\in (-1,1)$ and $\tau \in [0,A\mu ]$. Here

14.20.23		$$\beta =\tau /\mu ,$$
ⓘ Symbols: $\tau$: real variable, $\mu$: general order and $\beta$ Permalink: http://dlmf.nist.gov/14.20.E23 Encodings: TeX, pMML, png See also: Annotations for §14.20(ix), §14.20 and Ch.14

and the variable $\rho$ is defined by

14.20.24		$$\rho =\begin{array}{l}\frac{1}{2}\ln \left(\frac{\left(1-{\beta }^2\right)x^2+1+{\beta }^2+2x{\left(1+{\beta }^2-{\beta }^2{x}^2\right)}^{1/2}}{1-x^2}\right)+\beta \arctan \left(\frac{\beta x}{\sqrt{1+{\beta }^2-{\beta }^2{x}^2}}\right)\\ -\frac{1}{2}\ln \left(1+{\beta }^2\right),\end{array}$$
ⓘ Symbols: $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\beta$ and $\rho$ Permalink: http://dlmf.nist.gov/14.20.E24 Encodings: TeX, pMML, png See also: Annotations for §14.20(ix), §14.20 and Ch.14

with the inverse tangent taking its principal value. The interval is mapped one-to-one to the interval , with the points $x=-1$, $x=0$, and $x=1$ corresponding to $\rho =-\mathrm{\infty }$, $\rho =0$, and $\rho =\mathrm{\infty }$, respectively.

With the same conditions, the corresponding approximation for ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right)$ is obtainable by replacing ${\mathrm{e}}^{-\mu \rho }$ by ${\mathrm{e}}^{\mu \rho }$ on the right-hand side of (14.20.22). Approximations for ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ and ${\widehat{\mathsf{Q}}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ can then be achieved via (14.9.7) and (14.20.3).

For extensions to complex arguments (including the range ), asymptotic expansions, and explicit error bounds, see Dunster (1991). For the case of purely imaginary order and argument see Dunster (2013).

For zeros of ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ see Hobson (1931, §237).

For integrals with respect to $\tau$ involving ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$, see Prudnikov et al. (1990, pp. 218–228).