C320 Jacobian Elliptic Functions sn, cn, dn for Complex Argument

Function subprograms CELFUN and WELFUN calculate, for complex argument z and real modulus k, the Jacobian elliptic functions $sn(z,k)$ , $cn(z,k)$ and $dn(z,k)$ . The function $sn(z,k)$ is the inverse of the elliptic integral of the first kind and is defined implicitly by

$z\; =\; \int$₀^{sn( z, k )}{dw(1-w²)(1-k²w²)}(k²≤1).

The functions $cn(z,k)$ and $dn(z,k)$ are defined by

For k = 0 and $k2=\; 1$ these functions are elementary:

Note that the Jacobian elliptic functions are doubly-periodic in the z-plane. For details see the relevant treatises or handbooks.

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

Structure:

SUBROUTINE subprograms
User Entry Names: CELFUN, WELFUN
External References: MTLMTR (N002), ABEND (Z035)

Usage:

For $t=C$ (type COMPLEX), $t=W$ (type COMPLEX*16),

    CALL tELFUN(Z,AK2,SN,CN,DN)

Z

(type according to t) The argument z.

AK2

( REAL for $t=C$ , DOUBLE PRECISION for $t=W$ ) The value of $k2$ (the square of the modulus).

SN

(type according to t) On exit, $SN=sn(Z,k)$ .

CN

(type according to t) On exit, $CN=cn(Z,k)$ .

DN

(type according to t) On exit, $DN=dn(Z,k)$ .

Method:

The Jacobian elliptic functions with complex argument z=x+iy are computed from their representations in terms of Jacobian elliptic functions with real arguments x or y (Ref. 1, formula 125.01). See also the Short Write-up for ELFUN (C318).

Accuracy:

CELFUN (except on CDC and Cray computers) has full single-precision accuracy. For most values of the arguments, WELFUN (and CELFUN on CDC and Cray computers) has an accuracy of approximately two significant digits less than the machine precision.

Restrictions:

$|Re\hspace{0.17em}z|\; \le 3K(k)$ , $|Im\hspace{0.17em}z|\; \le 3K(k\text{'})$

where $k\text{'}=1-k2$ is the complementary modulus, and $K(x)$ is the complete elliptic integral of the first kind. (See entries RELIKC and DELIKC in RELI1C (C347)).

Error handling:

Error C320.1: $|AK2|\; >\; 1$ . The function value is set equal to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

References:

1. P.F. Byrd and M.D. Friedman, Handbook of elliptic integrals for engineers and scientists, 2nd ed., Springer-Verlag Berlin (1971).

C321