# §14.20 Conical (or Mehler) Functions

## §14.20(i) Definitions and Wronskians

Throughout §14.20 we assume that , with and . (14.2.2) takes the form

Solutions are known as conical or Mehler functions. For and , a numerically satisfactory pair of real conical functions is and .

Another real-valued solution of (14.20.1) was introduced in Dunster (1991). This is defined by

Equivalently,

exists except when and ; compare §14.3(i). It is an important companion solution to when is large; compare §§14.20(vii), 14.20(viii), and 10.25(iii).

provided that exists.

Lastly, for the range , is a real-valued solution of (14.20.1); in terms of (which are complex-valued in general):

## §14.20(ii) Graphics

 Figure 14.20.1: . Symbols: : Ferrers function of the first kind, : real variable and : real variable Permalink: http://dlmf.nist.gov/14.20.F1 Encodings: pdf, png Figure 14.20.2: . Symbols: : conical function, : real variable and : real variable Permalink: http://dlmf.nist.gov/14.20.F2 Encodings: pdf, png
 Figure 14.20.3: . Symbols: : Ferrers function of the first kind, : real variable and : real variable Permalink: http://dlmf.nist.gov/14.20.F3 Encodings: pdf, png Figure 14.20.4: , . (This function does not exist when .) Symbols: : conical function, : real variable and : real variable Permalink: http://dlmf.nist.gov/14.20.F4 Encodings: pdf, png
 Figure 14.20.5: . Symbols: : Ferrers function of the first kind, : real variable and : real variable Permalink: http://dlmf.nist.gov/14.20.F5 Encodings: pdf, png Figure 14.20.6: . Symbols: : conical function, : real variable and : real variable Permalink: http://dlmf.nist.gov/14.20.F6 Encodings: pdf, png
 Figure 14.20.7: . Symbols: : Ferrers function of the first kind, : real variable and : real variable Permalink: http://dlmf.nist.gov/14.20.F7 Encodings: pdf, png Figure 14.20.8: . Symbols: : conical function, : real variable and : real variable Permalink: http://dlmf.nist.gov/14.20.F8 Encodings: pdf, png

## §14.20(v) Trigonometric Expansion

14.20.10.

From (14.20.9) or (14.20.10) it is evident that is positive for real .

## §14.20(vii) Asymptotic Approximations: Large , Fixed

For and fixed ,

uniformly for , where and are the modified Bessel functions (§10.25(ii)) and 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).

## §14.20(viii) Asymptotic Approximations: Large ,

In this subsection and §14.20(ix), and denote arbitrary constants such that and .

The variable 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 and corresponding to and , respectively.

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

## §14.20(ix) Asymptotic Approximations: Large ,

with the inverse tangent taking its principal value. The interval is mapped one-to-one to the interval , with the points , , and corresponding to , , and , respectively.

With the same conditions, the corresponding approximation for is obtainable by replacing by on the right-hand side of (14.20.22). Approximations for and 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).

## §14.20(x) Zeros and Integrals

For zeros of see Hobson (1931, §237).

For integrals with respect to involving , see Prudnikov et al. (1990, pp. 218–228).