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Manuscript Title: A simple procedure for numerical approximation of the Fm(z) functions with complex argument.
Authors: L. Jakab
Program title: FZRI
Catalogue identifier: ACFO_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 31(1984)89
Programming language: Fortran.
Computer: FELIX C-256.
Operating system: SIRIS 2/3.
RAM: 8K words
Word size: 32
Keywords: General purpose, Auxiliary functions, Travelling gaussian type orbitals, Electronic transitions, Scattering.
Classification: 4.7.

Nature of problem:
Molecular integrals over a basis of complex Gaussian type orbitals. The integrals can then be evaluated analytically in terms of the complex variable auxiliary function Fm(z).

Solution method:
The Re Fm(z), real and Im Fm(z), imaginary parts of the Fm(z) functions are calculated by a 2N-point Gauss-Legendre quadrature formula.

During the calculation of the Fm(z) functions up to the order m, the desired accuracy is achieved on a limited domain of the complex plane only. The program should not be used outside this domain or for the computation of functions of order higher than m. The bounds on Re(z) and Im(z) are:
(a) Im(z) <=8 -24<= Re(z) <=16
(b) Im(z) =0 -30<= Re(z) <=20
Since, Fm(z+) = Fm+(z) computations are performed on the upper half of the complex plane only.

Unusual features:
The relative error of the computations is equal to or less than 10**-9 for the domains represented on Fig.1. The program needs minor alterations if higher accuracy on a larger domain is required. For the domains represented on Fig.2. the relative error is 10**-12 or less.

Running time:
The execution time depends strongly on the number of quadrature points used. The calculation of 266 pairs (Re Fm and Im Fm) of functions with N=16 required 7.12 s on the FELIX C-256 computer. The compilation time was 14 s.