C348 Elliptic Integral for Complex Argument

Function subprograms CELINT and WELINT calculate, for complex argument z=x+iy and real complementary modulus $k\text{'}$ a general elliptic integral of the second kind:

$F(z,k\text{'},a,b)\; =\; \int$₀^z{a+bζ²(1+ζ²)(1+ζ²)(1+k'²ζ²)} dζ(k'²≥0, Re(z) ≥0),

which contains the elliptic integrals of the first and second kind as special cases:

$F$1(z,k')	$=$	$\int$0z{dζ(1+ζ2)(1+k'2ζ2)}	$=$	$F(z,k\text{'},1,1),$
$F$2(z,k')	$=$	$\int$0z{dζ(1+ζ)2}{1+k'2ζ21+ζ2}	$=$	$F(z,k\text{'},1,k\text{'}2).$

The double-precision version WELINT is available only on computers which support a COMPLEX*16 Fortran data type.

Structure:

FUNCTION subprograms
User Entry Names: CELINT, WELINT
Files Referenced: Unit 6
External References: MTLMTR (N002), ABEND (Z035)

Usage:

In any arithmetic expression, with $AKP=k\text{'}$ , CELINT(Z,AKP,A,B) or WELINT(Z,AKP,A,B) has the value $F(Z,k\text{'},A,B)$ ,

where CELINT is of type COMPLEX, WELINT is of type COMPLEX*16, Z is of the same type as the function name, and AKP, A, B are of type REAL for CELINT and of type DOUBLE PRECISION for WELINT.

Method:

The evaluation of $F$ is based on the Gauss transformation. For details, in particular for the conformal mapping provided by $F$ , see Ref. 1.

Accuracy:

CELINT (except on CDC and Cray computers) has full single-precision accuracy. For most values of the arguments, WELINT (and CELINT on CDC and Cray computers) has an accuracy of approximately one significant digit less than the machine precision.

Error handling:

Error C348.1: Re . The function value is set equal to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

Notes:

For other forms of the elliptic integrals see the write-up for RELI1 (C346).

The subprogram is based on the Algol60 procedure elco2 given in Ref. 1.

References:

1. R. Bulirsch, Numerical calculation of elliptic integrals and elliptic functions, Numer. Math. 7 (1965) 78--90.

C349