§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

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

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

Equivalently,

${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ 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 ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ when $\tau$ is large; compare §§14.20(vii), 14.20(viii), and 10.25(iii).

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

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

When ,

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

where

Special cases:

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

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

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

The variable $\eta$ is defined implicitly by

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

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

and the variable $\rho$ is defined by

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 ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right)$ is obtainable by replacing ${\mathrm{e}}^{-\mu \rho }$ by ${\mathrm{e}}^{\mu \rho }$ on the right-hand side of (14.20.22). Approximations for ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ and ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ 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 ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ see Hobson (1931, §237).

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