Uniform Asymptotic Expansions of a Class of Integrals in Terms of Modified Bessel Functions, with Application to Confluent Hypergeometric Functions

摘要

The integral F lambda (z, alpha) = the integral between 0 and infinity t sup (lambda-1) e sup (-zt -alpha/t) ft (dt) is considered for large values of z; alpha and lambda are uniformity parameters in (0, inf). The asymptotic expansion is given in terms of the modified Bessel function K lambda (2 sqrt alpha z). The asymptotic nature of the expansion is discussed and error bounds are constructed for the remainders in the expansions. An example for confluent hypergeometric or Whittaker functions is given. In this example the integrals are transformed to standard forms and the mappings are investigated.