Methods for evaluation of the Euler gamma function in the complex field and their application in the computational electromagnetics

Four methods for calculation of the Euler gamma function Γ(z) of complex variable, are presented. The first of them uses the development in a power series for 1 / Γ(z), (|z| <; ∞) and the second and third - that of ln Γ(1 + z) by means of the Riemann zeta function ζ(m) (m - a positive integer, larger than 1) in case |z| <; 1 and |z| <; 2, resp. A scheme is proposed for diminishing the imaginary part of z, employing the Gauss multiplication theorem for Γ(nz), (n - a natural number) and its real one through the same theorem in combination with the recurrence relations for Γ(z) which is the key point in the procedures mentioned. The fourth approach harnesses the asymptotic expansion of function ln Γ(z), (z→ ∞, |arg z| <; π) stated with the help of the Bernoulli´s numbers B_2n. The different techniques are juxtaposed. Comments on the advantages and drawbacks of each of them, are given. The application of the algorithms suggested in the computation of the complex confluent hypergeometric functions, appearing in certain problems of the computational electromagnetics, is debated.