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

Author(s): H.-H. Umstätter Library: MATHLIB
Submitter: K.S. Kölbig Submitted: 30.01.1980
Language: Fortran Revised: 07.06.1992

Function subprograms CELFUN and WELFUN calculate, for complex argument z and real modulus k, the Jacobian elliptic functions tex2html_wrap_inline128 , tex2html_wrap_inline130 and tex2html_wrap_inline132 . The function tex2html_wrap_inline134 is the inverse of the elliptic integral of the first kind and is defined implicitly by

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The functions tex2html_wrap_inline138 and tex2html_wrap_inline140 are defined by

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For k = 0 and tex2html_wrap_inline146 these functions are elementary:

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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, ABEND

Usage:

For tex2html_wrap_inline154 (type COMPLEX), tex2html_wrap_inline156 (type COMPLEX*16), 

    CALL tELFUN(Z,AK2,SN,CN,DN)
Z
(type according to t) The argument z.
AK2
(REAL for tex2html_wrap_inline160 , DOUBLE PRECISION for tex2html_wrap_inline162 ) The value of tex2html_wrap_inline164 (the square of the modulus).
SN
(type according to t) On exit, tex2html_wrap_inline166 .
CN
(type according to t) On exit, tex2html_wrap_inline168 .
DN
(type according to t) On exit, tex2html_wrap_inline170 .

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:

tex2html_wrap_inline178 , tex2html_wrap_inline180 where tex2html_wrap_inline182 is the complementary modulus, and tex2html_wrap_inline184 is the complete elliptic integral of the first kind. (See entries RELIKC and DELIKC in RELI1C (C347)).

Error handling:

Error C320.1: tex2html_wrap_inline186 . 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).
tex2html_wrap_inline188

Michel Goossens Tue Jun 4 21:40:26 METDST 1996