C331 Conical Functions of the First Kind

Function subprograms RFCONC and DFCONC calculate the (real valued) conical function of the first kind

$P$_-{12}+iτ^m(x)

for real $x>-1,\; \tau \ge 0$ , and m=0,1, where $P$_ν^m(x) is the Legendre (or spherical) function of the first kind and $i=-1$ .

On CDC and Cray computers, the double-precision version DFCONC is not available.

Structure:

FUNCTION subprograms
User Entry Names: RFCONC, DFCONC
Obsolete User Entry Names: FCONC $\equiv$ RFCONC
Files Referenced: Unit 6
External References:

CGAMMA,	WGAMMA(C305),	CLGAMA,	WLGAMA(C306),
BESJO,	DBESJ0,	BESJ1,	DBESJ1(C312),
BESIO,	DBESI0,	BESI1,	DBESI1(C313),
RELIKC,	DELIKC,	RELIEC,	DELIEC(C347),
3lMTLMTR(N002), ABEND(Z035)

Usage:

For $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION),

    tFCONC(X,TAU,M)}

X

(type according to t) Variable x.

TAU

(type according to t) The imaginary part of the index, $\tau$ .

M

( INTEGER) Order m. ($M=0$ or $M=1)$ .

Method:

Either (i) series expansions based on the Gaussian hypergeometric function and evaluated by direct summation or from rational approximations, or (ii) asymptotic expressions in terms of Bessel functions. For $\tau =0$ the conical functions (for m = 0,1) can be expressed in terms of complete elliptic integrals. For details see Ref. 1.

Restrictions:

$X\; \ge -1$ , $TAU\; \ge 0$ , $M\; =\; 0$ or 1.

Accuracy:

RFCONC (except on CDC and Cray computers) has full single-precision accuracy. For most values of the argument X, DFCONC (and RFCONC on CDC and Cray computers), an accuracy of not less than 10 significant digits is usually obtained. If x and $\tau$ are not too large the accuracy increases to about 12-13 significant digits.

Error handling:

Error C331.1: or or $M\; \ne 0$ and $M\; \ne 1$ .
Error C331.2: Problems of convergence for a hypergeometric function.
In both cases, the function value is set equal to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

Notes:

This program is an (only formally) modified version of the CPC Program Library Package FCONIC (see Ref. 1).

References:

1. K.S. Kölbig, A program for computing the conical functions of the first kind $P$_1/2+iτ^m(x) for m=0 and m=1, Computer Phys. Comm. 23 (1981) 51--61.
2. M.I. Zhurina and L.N. Karmazina, Tables and formulae for the spherical functions $P$_1/2+iτ^m(z) , (Pergamon Press, Oxford 1966).

C332