CWPL

This function evaluates the Weierstrass’ ℘ function in the lemniscatic case for complex argument with unit period parallelogram.

Function Return Value

CWPL — Complex function value. (Output)

Required Arguments

Z — Complex argument for which the function value is desired. (Input)

FORTRAN 90 Interface

Generic: CWPL (Z)

Specific: The specific interface names are C_CWPL and Z_CWPL.

FORTRAN 77 Interface

Complex:      CWPL (Z)

Double complex:    The double complex name is ZWPL.

Description

The Weierstrass’ ℘ function, ℘(z) = ℘(z ∣ ω, ωʹ), is an elliptic function of order two with periods 2 ω and 2 ωʹ and a double pole at z = 0. CWPL(Z) computes ℘(z ∣ ω, ωʹ) with 2  ω = 1 and 2 ωʹ = i.

The input argument is first reduced to the fundamental parallelogram of all z satisfying –1/2 ≤ ℜz ≤ 1/2 and –1/2  ≤ ℑz ≤ 1/2. Then, a rational approximation is used.

All arguments are valid with the exception of the lattice points z = m + ni, which are the poles of CWPL. If the argument is a lattice point, then b = AMACH(2), the largest floating‑point number, is returned. If the argument has modulus greater than 10ɛ−1, then NaN (not a number) is returned. Here, ɛ = AMACH(4) is the machine precision.

Function CWPL is based on code by Eckhardt (1980). Also, see Eckhardt (1977).

Example

In this example, ℘(0.25 + 0.25i) is computed and printed.

 

USE CWPL_INT

USE UMACH_INT

 

IMPLICIT NONE

! Declare variables

INTEGER NOUT

COMPLEX VALUE, Z

! Compute

Z = (0.25, 0.25)

VALUE = CWPL(Z)

! Print the results

CALL UMACH (2, NOUT)

WRITE (NOUT,99999) Z, VALUE

99999 FORMAT (' CWPL(', F6.3, ',', F6.3, ') = (', &

F6.3, ',', F6.3, ')')

END

Output

 

CWPL( 0.250, 0.250) = ( 0.000,-6.875)