# §14.20 Conical (or Mehler) Functions

## §14.20(i) Definitions and Wronskians

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

 14.20.1 $\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{% \mathrm{d}w}{\mathrm{d}x}-\left(\tau^{2}+\frac{1}{4}+\frac{\mu^{2}}{1-x^{2}}% \right)w=0.$

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

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

 14.20.2 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\Re\left(e^{\mu% \pi i}\mathsf{Q}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\right)-\tfrac{1}{2}% \pi\sin\left(\mu\pi\right)\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right).$ ⓘ Defines: $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$: conical function Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $\Re$: real part, $\sin\NVar{z}$: 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 14

Equivalently,

 14.20.3 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)=\frac{% \pi e^{-\tau\pi}\sin\left(\mu\pi\right)\sinh\left(\tau\pi\right)}{2({\cosh^{2}% }\left(\tau\pi\right)-{\sin^{2}}\left(\mu\pi\right))}\mathsf{P}^{-\mu}_{-\frac% {1}{2}+\mathrm{i}\tau}\left(x\right)+\frac{\pi(e^{-\tau\pi}{\cos^{2}}\left(\mu% \pi\right)+\sinh\left(\tau\pi\right))}{2({\cosh^{2}}\left(\tau\pi\right)-{\sin% ^{2}}\left(\mu\pi\right))}\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left% (-x\right).$

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

 14.20.4 $\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right% ),\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(-x\right)\right\}=\frac% {2}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}(1-x^{2})}.$
 14.20.5 $\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right% ),\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)% \right\}=\frac{\pi(e^{-\tau\pi}{\cos^{2}}\left(\mu\pi\right)+\sinh\left(\tau% \pi\right))}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}({\cosh^{2% }}\left(\tau\pi\right)-{\sin^{2}}\left(\mu\pi\right))(1-x^{2})},$

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

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

 14.20.6 $P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{ie^{-\mu\pi i}}{\sinh\left(% \tau\pi\right)\left|\Gamma\left(\mu+\frac{1}{2}+i\tau\right)\right|^{2}}\*% \left(Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)-Q^{\mu}_{-\frac{1}{2}-i\tau}% \left(x\right)\right),$ $\tau\neq 0$.

## §14.20(iii) Behavior as $x\to 1$

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

 14.20.7 $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\sim\tfrac{1}{2}% \Gamma\left(\mu\right)\left(\frac{2}{1-x}\right)^{\mu/2},$
 14.20.8 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\sim\frac{\pi% \Gamma\left(\mu\right)(e^{-\tau\pi}{\cos^{2}}\left(\mu\pi\right)+\sinh\left(% \tau\pi\right))}{2({\cosh^{2}}\left(\tau\pi\right)-{\sin^{2}}\left(\mu\pi% \right)){\left|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)\right|^{2}}}% \*\left(\frac{2}{1-x}\right)^{\mu/2}.$

## §14.20(iv) Integral Representation

When $0<\theta<\pi$,

 14.20.9 $\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=\frac{2}{\pi}\int_{0}^{% \theta}\frac{\cosh\left(\tau\phi\right)}{\sqrt{2(\cos\phi-\cos\theta)}}\mathrm% {d}\phi.$

## §14.20(v) Trigonometric Expansion

 14.20.10 $\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=1+\frac{4\tau^{2}+1^{2}% }{2^{2}}{\sin^{2}}\left(\tfrac{1}{2}\theta\right)+\frac{\left(4\tau^{2}+1^{2}% \right)\left(4\tau^{2}+3^{2}\right)}{2^{2}\cdot 4^{2}}{\sin^{4}}\left(\tfrac{1% }{2}\theta\right)+\cdots,$ $0\leq\theta\leq\pi$.

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

## §14.20(vi) Generalized Mehler–Fock Transformation

 14.20.11 $f(\tau)=\frac{\tau}{\pi}\sinh\left(\tau\pi\right)\Gamma\left(\tfrac{1}{2}-\mu+% i\tau\right)\*\Gamma\left(\tfrac{1}{2}-\mu-i\tau\right)\int_{1}^{\infty}P^{\mu% }_{-\frac{1}{2}+i\tau}\left(x\right)g(x)\mathrm{d}x,$

where

 14.20.12 $g(x)=\int_{0}^{\infty}P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)f(\tau)\mathrm% {d}\tau.$

Special cases:

 14.20.13 $P_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\cosh\left(\tau\pi\right)}{\pi}\int% _{1}^{\infty}\frac{P_{-\frac{1}{2}+i\tau}\left(t\right)}{x+t}\mathrm{d}t,$
 14.20.14 $\pi\int_{0}^{\infty}\frac{\tau\tanh\left(\tau\pi\right)}{\cosh\left(\tau\pi% \right)}P_{-\frac{1}{2}+i\tau}\left(x\right)P_{-\frac{1}{2}+i\tau}\left(y% \right)\mathrm{d}\tau=\frac{1}{y+x}.$

## §14.20(vii) Asymptotic Approximations: Large $\tau$, Fixed $\mu$

For $\tau\to\infty$ and fixed $\mu$,

 14.20.15 $\displaystyle\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)$ $\displaystyle=\frac{1}{\tau^{\mu}}\left(\frac{\theta}{\sin\theta}\right)^{1/2}% I_{\mu}\left(\tau\theta\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right),$ 14.20.16 $\displaystyle\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)$ $\displaystyle=\frac{1}{\tau^{\mu}}\left(\frac{\theta}{\sin\theta}\right)^{1/2}% K_{\mu}\left(\tau\theta\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right),$

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 $0<\delta<\pi$. For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). See also Žurina and Karmazina (1966).

## §14.20(viii) Asymptotic Approximations: Large $\tau$, $0\leq\mu\leq A\tau$

In this subsection and §14.20(ix), $A$ and $\delta$ denote arbitrary constants such that $A>0$ and $0<\delta<2$.

As $\tau\to\infty$,

 14.20.17 $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\sigma(\mu,\tau)\left(% \frac{\alpha^{2}+\eta}{1+\alpha^{2}-x^{2}}\right)^{1/4}I_{\mu}\left(\tau\eta^{% 1/2}\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right),$
 14.20.18 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\sigma(\mu,\tau% )\left(\frac{\alpha^{2}+\eta}{1+\alpha^{2}-x^{2}}\right)^{1/4}K_{\mu}\left(% \tau\eta^{1/2}\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right),$

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 14
 14.20.20 $\sigma(\mu,\tau)=\frac{\exp\left(\mu-\tau\operatorname{arctan}\alpha\right)}{% \left(\mu^{2}+\tau^{2}\right)^{\mu/2}}.$

The variable $\eta$ is defined implicitly by

 14.20.21 ${\left(\alpha^{2}+\eta\right)^{1/2}+\tfrac{1}{2}\alpha\ln\eta-\alpha\ln\left(% \left(\alpha^{2}+\eta\right)^{1/2}+\alpha\right)}={\operatorname{arccos}\left(% \frac{x}{\left(1+\alpha^{2}\right)^{1/2}}\right)+\frac{\alpha}{2}\ln\left(% \frac{1+\alpha^{2}+\left(\alpha^{2}-1\right)x^{2}-2\alpha x\left(1+\alpha^{2}-% x^{2}\right)^{1/2}}{\left(1+\alpha^{2}\right)\left(1-x^{2}\right)}\right)},$

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

## §14.20(ix) Asymptotic Approximations: Large $\mu$, $0\leq\tau\leq A\mu$

As $\mu\to\infty$,

 14.20.22 $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\beta\exp\left(\mu% \beta\operatorname{arctan}\beta\right)}{\Gamma\left(\mu+1\right)\left(1+\beta^% {2}\right)^{\mu/2}}\frac{e^{-\mu\rho}}{\left(1+\beta^{2}-x^{2}\beta^{2}\right)% ^{1/4}}\left(1+O\left(\frac{1}{\mu}\right)\right),$

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 14

and the variable $\rho$ is defined by

 14.20.24 $\rho=\frac{1}{2}\ln\left(\frac{\left(1-\beta^{2}\right)x^{2}+1+\beta^{2}+2x% \left(1+\beta^{2}-\beta^{2}x^{2}\right)^{1/2}}{1-x^{2}}\right)+\beta% \operatorname{arctan}\left(\frac{\beta x}{\sqrt{1+\beta^{2}-\beta^{2}x^{2}}}% \right)-\frac{1}{2}\ln\left(1+\beta^{2}\right),$

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

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

For extensions to complex arguments (including the range $1), asymptotic expansions, and explicit error bounds, see Dunster (1991). For the case of purely imaginary order and argument see Dunster (2013).

## §14.20(x) Zeros and Integrals

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

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