On the computation of incomplete gamma functions in the complex domain

摘要

Some new continued fractions for incomplete gamma functions γ(a, z) and Γ(a, z), with a and z complex, are derived. For many of these expansions, it is shown that the approximants can be evaluated by a numerically stable backward recurrence algorithm. Truncation error bounds for these approximations are discussed with applicable references cited. Consideration is given to important special cases such as the error and complementary error functions and exponential integrals. In connection with the error function we discuss the ‘anomalous convergence’ of general T-fractions (two-point Padé approximants) first pointed out and studied by Gautschi.