*(English)*Zbl 1097.33503

The present paper (Part II) is a sequel to the paper [Part I, cf. ibid. 457, No. 2016, 2835–2853 (2001; Zbl 1039.33500)] corresponding to the preceding review in which was described a method of hyperasymptotic evaluation using (absolutely convergent) Hadamard expansions to the confluent hypergeometric functions in the case of real variables. The author here discusses the fact of that the Hadamard expansions there developed, fail to be satisfactory when the argument $z$ is complex, by using firstly the modified Bessel function ${I}_{\nu}\left(z\right)$ as an illustrative example. It is found that the incomplete gamma function $P(a,z)$ (when $a>0$ and $z$ is large and positive) no longer exhibits (as in Part I) a certain cut-off property that now in the Hadamard expansion for ${I}_{\nu}\left(z\right)$ makes to change from approximately unity when $a\lesssim z$ to a rapid decay when $a\gtrsim z$ arising as a consequence, a loss of accuracy in the expansions for computing ${I}_{\nu}\left(z\right)$. The same difficulty appears in the computation of the confluent hypergeometric functions ${}_{1}{F}_{1}(a;b;z)$ and $U(a;b;z)$ corresponding to the Hadamard expansions developed in Part I. To overcome this loss of accuracy, the Hadamard expansions are then modified in order to produce expansions in which the incomplete gamma functions have an argument proportional to $\left|z\right|$, so that the simple cut-off property can be again applied.

The resulting representations for both types of hypergeometric functions (valid when ${R}_{e}b>{R}_{e}a>0$), consists of an infinite number of Hadamard expansions which are associated with a decreasing sequence of subdominant exponential levels (when $\left|z\right|$ is large) dependent of $argz$.

Some numerical examples to illustrate the use of the Hadamard expansions of complex variable $z$, are finally given.

##### MSC:

33C15 | Confluent hypergeometric functions, Whittaker functions, ${}_{1}{F}_{1}$ |

33F05 | Numerical approximation and evaluation of special functions |

65D20 | Computation of special functions, construction of tables |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |