Numerical Representation of the Incomplete Gamma Function of Complex Argument

Details Numerical Representation of the Incomplete Gamma Function of Complex Argument Numerical Representation of the Incomplete Gamma Function of Complex Argument

2003-06-11

arxiv.org/abs/math/0306184v2

Numerical Algorithms 36 (3) (2004) 247-264

Mathematics

Numerical Analysis

REVTeX4, 48 pages, 16 PostScript figures. Corrected typos in Eqs. (46), (47) and on bottom p. 43. Added Ref [32]

Scientific paper

10.1023/B:NUMA.0000040063.91709.

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 z-plane, which stores a static matrix of higher derivatives. This is the obvious extension to a procedure often in use for real-valued z.

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