The expansion of the confluent hypergeometric function on the positive real axis

TL;DR

This paper investigates the asymptotic expansion of the Kummer function ${}_1F_1(a; b; z)$ along the positive real axis, providing a correct algebraic contribution form and validating it with numerical results.

Contribution

It offers a precise asymptotic expansion of the Kummer function on the positive real axis, clarifying the subdominant algebraic term for non-integer parameters.

Findings

Derived the correct form of the subdominant algebraic contribution.

Validated the asymptotic expansion with numerical results.

Enhanced understanding of the Kummer function's behavior at infinity.

Abstract

The asymptotic expansion of the Kummer function ${}_1F_1(a; b; z)$ is examined as $z\to+\infty$ on the Stokes line $\arg\,z=0$. The correct form of the subdominant algebraic contribution is obtained for non-integer $a$. Numerical results demonstrating the accuracy of the expansion are given.