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ZHYPG2 - calculation of the hypergeometric functions 2F1(a,2,c,z)
with complex parameters and complex argument


Author: O.Chuluunbaatar
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Language: Fortran

The function ZHYPG2 computes the hypergeometric functions fig1 with complex parameters figa, figc and an argument figz [1]. For calculation of the hypergeometric function fig1 it used the beforehand calculated values of the hypergeometric functions and their derivatives by means of the subroutine HYPGEO [2] at points

fig2

The considered procedure was constructed on the basis of the algorithms published in [3]. The described algorithm allows one to save calculation time of the multidimensional integrals, whose kernel contains hypergeometric functions fig1, approximately 10-60 times depending on accuracy of the calculation (10-4-10-14) in comparison with a direct use of the subroutine HYPGEO at each value figz. This type of calculations occurs in investigation of the single and double ionizations by electron impact of two-nuclear molecules (see papers [4,5]).

References:

  1. Abramowitz M. and Stegun I. Handbook of mathematical functions, National Bureau of Standards Applied Mathematics series. 55. 1964.
  2. Press W.H., Teukolsky S.A., Vetterling W.T. and Flannery B.P. Numerical recipes: The art of scientific computing. Cambridge University Press, Cambridge, 1986.
  3. Chuluunbaatar O. Bulletin of Tver State University: Ser. Applied Mathematics. 2008, ¹ = 26(86), pp. 47-64.
  4. Chuluunbaatar O., Joulakian B.B., Tsookhuu Kh. and Vinitsky S.I. J. Phys. B, 2004, v. 37, pp. 2607-2616.
  5. Chuluunbaatar O., Joulakian B.B., Puzynin I.V., Tsookhuu Kh. and Vinitsky S.I. J. Phys. B, 2008, v. 41, pp.015204-1-6.

Structure:

Type: - FUNCTION
Name - ZHYPG2
Internal subroutines - ZSUM, ZGAMMA, DRHYP, CGAMA, HYPGEO, HYPDRV, HYPSER, ODEINT, BSSTEP, MMID, PZEXTR

Usage:

ZFUNC = ZHYPG2(ZA,ZC,ZZ,NMAX,EPS)

INPUT: ZA, ZC, ZZ, NMAX, EPS, where:

ZA - double complex number, contains value of parameter fig
ZC - double complex number, contains value of parameter fig.
ZZ - double complex number, contains value of argument fig.
NMAX - integer number, the maximum number of summation of truncated Gaussian series.
EPS - double precision number, given accuracy.

Sources with example and description are submitted. Example:

fig


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