14.19 Toroidal (or Ring) Functions14.21 Definitions and Basic Properties

§14.20 Conical (or Mehler) Functions

Contents

§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{{d}^{2}w}{{dx}^{2}}-2x\frac{dw}{dx}-\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<x<1 and \tau>0, a numerically satisfactory pair of real conical functions is \mathop{\mathsf{P}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right) and \mathop{\mathsf{P}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(-x\right).

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

14.20.2 \mathop{\widehat{\mathsf{Q}}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right)=\realpart{\left(e^{{\mu\pi i}}\mathop{\mathsf{Q}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right)\right)}-\tfrac{1}{2}\pi\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)\mathop{\mathsf{P}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right).

Equivalently,

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

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

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

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

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

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

§14.20(ii) Graphics

See accompanying text
Figure 14.20.2: \mathop{\widehat{\mathsf{Q}}^{{0}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right),\tau=0,\tfrac{1}{2},1,2,4. Magnify
See accompanying text
Figure 14.20.4: \mathop{\widehat{\mathsf{Q}}^{{-1/2}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right), \tau=\tfrac{1}{2},1,2,4. (This function does not exist when \tau=0.) Magnify
See accompanying text
Figure 14.20.6: \mathop{\widehat{\mathsf{Q}}^{{-1}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right),\tau=0,\tfrac{1}{2},1,2,4. Magnify
See accompanying text
Figure 14.20.8: \mathop{\widehat{\mathsf{Q}}^{{-2}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right),\tau=0,\tfrac{1}{2},1,2,4. Magnify

§14.20(iv) Integral Representation

§14.20(v) Trigonometric Expansion

14.20.10 \mathop{\mathsf{P}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=1+\frac{4\tau^{2}+1^{2}}{2^{2}}{\mathop{\sin\/}\nolimits^{{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}}{\mathop{\sin\/}\nolimits^{{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 \mathop{\mathsf{P}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right) is positive for real \theta.

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

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

uniformly for \theta\in(0,\pi-\delta], where \mathop{I\/}\nolimits and \mathop{K\/}\nolimits 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.

The variable \eta is defined implicitly by

14.20.21 {\left(\alpha^{2}+\eta\right)^{{1/2}}+\tfrac{1}{2}\alpha\mathop{\ln\/}\nolimits\eta-\alpha\mathop{\ln\/}\nolimits\!\left(\left(\alpha^{2}+\eta\right)^{{1/2}}+\alpha\right)}={\mathop{\mathrm{arccos}\/}\nolimits\!\left(\frac{x}{\left(1+\alpha^{2}\right)^{{1/2}}}\right)+\frac{\alpha}{2}\mathop{\ln\/}\nolimits\!\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<x<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.

For extensions to complex arguments (including the range 1<x<\infty), asymptotic expansions, and explicit error bounds, see Dunster (1991).

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

with the inverse tangent taking its principal value. The interval -1<x<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 \mathop{\mathsf{P}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\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 \mathop{\mathsf{P}^{{\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right) and \mathop{\widehat{\mathsf{Q}}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right) can then be achieved via (14.9.7) and (14.20.3).

For extensions to complex arguments (including the range 1<x<\infty), asymptotic expansions, and explicit error bounds, see Dunster (1991).

§14.20(x) Zeros and Integrals

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

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