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Manuscript Title: f1: a code to compute Appell's F1 hypergeometric function.
Authors: F.D. Colavecchia
Program title: f1
Catalogue identifier: ADSJ
Journal reference: Comput. Phys. Commun. 157(2004)32
Programming language: Fortran 90.
Computer: PC, SGI Origin2*.
Operating system: Linux, IRIX.
RAM: 1K words
Word size: 32
Keywords: Numerical methods, Special functions, Hypergeometric functions, Appell functions, Gauss function, General purpose, Function.
Classification: 4.7.

Nature of problem:
Computing the Appell F1 function is relevant in atomic collisons and elementary particle physics. It is usually the result of multidimensional integrals involving Coulomb continuum states.

Solution method:
The F1 function has a convergent-series definition for |x|< 1 and |y| < 1, and several analytic continuations for other regions of the variable space. The code tests the values of the variables and selects one of the precedent cases. In the convergence region the program uses the series definition near the origin of coordinates, and a numerical integration of the third-order differential parametric equation for the F1 function Also detects several special cases according to the values of the

The code is restricted to real values of the variables {x,y}. Also, there are some parameter domains that are not covered. These usually imply differences between integer parameters that lead to negative integer arguments of Gamma functions.

Running time:
Depends basically on the variables. The computation of table 4 (64 functions) requires approximately 0.33 seconds in a Athlon 900 MHz processor.