Asymptotic expansions for the generalised trigonometric integral and its zeros

TL;DR

This paper derives asymptotic expansions and error bounds for the generalized trigonometric integral and its zeros, providing insights into their behavior for large complex arguments and real zeros.

Contribution

It introduces new asymptotic expansions for the integral and its zeros, with explicit error bounds and enveloping properties, advancing understanding of their asymptotic behavior.

Findings

Asymptotic expansions for the integral and zeros are established.

Explicit error bounds for the asymptotics are derived.

The expansions are enveloping for real variables.

Abstract

In this paper, we investigate the asymptotic properties of the generalised trigonometric integral $\operatorname{ti}(a, z, \alpha)$ and its associated modulus and phase functions for large complex values of $z$. We derive asymptotic expansions for these functions, accompanied by explicit and computable error bounds. For real values of $a$, the function $\operatorname{ti}(a, z, \alpha)$ possesses infinitely many positive real zeros. Assuming $a < 1$, we establish asymptotic expansions for the large zeros, accompanied by precise error estimates. The error bounds for the asymptotics of the phase function and its zeros will be derived by studying the analytic properties of both the phase function and its inverse. Additionally, we demonstrate that for real variables, the derived asymptotic expansions are enveloping, meaning that successive partial sums provide upper and lower bounds for the corresponding functions.