Numerical Representations of the Incomplete Gamma Function of ComplexValued Argument
Abstract
Various approaches to the numerical representation of the Incomplete Gamma Function F_m(z) for complex arguments z and small integer indexes m are compared with respect to numerical fitness (accuracy and speed). We consider power series, Laurent series, Gautschi's approximation to the Faddeeva function, classical numerical methods of treating the standard integral representation, and others not yet covered by the literature. The most suitable scheme is the construction of Taylor expansions around nodes of a regular, fixed grid in the zplane, which stores a static matrix of higher derivatives. This is the obvious extension to a procedure often in use for realvalued z.
 Publication:

Numerical Algorithms
 Pub Date:
 2004
 DOI:
 10.1023/B:NUMA.0000040063.91709.58
 arXiv:
 arXiv:math/0306184
 Bibcode:
 2004NuAlg..36..247M
 Keywords:

 Mathematics  Numerical Analysis;
 33C15;
 33E50;
 33F05
 EPrint:
 REVTeX4, 48 pages, 16 PostScript figures. Corrected typos in Eqs. (46), (47) and on bottom p. 43. Added Ref [32]