Asymptotic expansions for the generalised trigonometric integral and its zeros

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.