f1: a code to compute Appell's F_{1} hypergeometric function
Abstract
In this work we present the FORTRAN code to compute the hypergeometric function F_{1}( α, β_{1}, β_{2}, γ, x, y) of Appell. The program can compute the F_{1} function for real values of the variables { x, y}, and complex values of the parameters { α, β_{1}, β_{2}, γ}. The code uses different strategies to calculate the function according to the ideas outlined in [F.D. Colavecchia et al., Comput. Phys. Comm. 138 (1) (2001) 29]. Program summaryTitle of the program: f1 Catalogue identifier: ADSJ Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSJ Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: none Computers: PC compatibles, SGI Origin2∗ Operating system under which the program has been tested: Linux, IRIX Programming language used: Fortran 90 Memory required to execute with typical data: 4 kbytes No. of bits in a word: 32 No. of bytes in distributed program, including test data, etc.: 52 325 Distribution format: tar gzip file External subprograms used: Numerical Recipes hypgeo [W.H. Press et al., Numerical Recipes in Fortran 77, Cambridge Univ. Press, 1996] or chyp routine of R.C. Forrey [J. Comput. Phys. 137 (1997) 79], rkf45 [L.F. Shampine and H.H. Watts, Rep. SAND760585, 1976]. Keywords: Numerical methods, special functions, hypergeometric functions, Appell functions, Gauss function Nature of the physical problem: Computing the Appell F_{1} function is relevant in atomic collisions and elementary particle physics. It is usually the result of multidimensional integrals involving Coulomb continuum states. Method of solution: The F_{1} function has a convergentseries 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 thirdorder differential parametric equation for the F_{1} function. Also detects several special cases according to the values of the parameters. Restrictions on the complexity of the problem: 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. Typical running time: Depends basically on the variables. The computation of Table 4 of [F.D. Colavecchia et al., Comput. Phys. Comm. 138 (1) (2001) 29] (64 functions) requires approximately 0.33 s in a Athlon 900 MHz processor.
 Publication:

Computer Physics Communications
 Pub Date:
 February 2004
 DOI:
 10.1016/S00104655(03)004909
 Bibcode:
 2004CoPhC.157...32C