Uniform, Exponentially Improved, Asymptotic Expansions for the Confluent Hypergeometric Function and Other Integral Transforms

A new generalized asymptotic expansion is constructed for the confluent hypergeometric function $U(a,a - b + 1,z)$ in which the parameters a and b are real or complex constants, and z is a large complex variable. The expansion is expressed in terms of generalized exponential integrals (or, equivalently, incomplete Gamma functions). It has a larger region of validity and greater accuracy than the conventional expansions of Poincaré type; moreover, it provides insight into the manner in which the Poincaré expansions change smoothly, albeit rapidly, from one to the other in the vicinity of the so-called Stokes lines. The expansion is accompanied by strict error bounds in the most important part of its region of validity. The method used is quite general and can be applied to other functions that are representable as transforms of Laplace or Stieltjes type.