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Asymptotic expansions for the generalised trigonometric integral and its zeros

Gergő Nemes

TL;DR

The paper analyzes the asymptotic behavior of the generalised trigonometric integral $ti(a,z,\alpha)$ and its modulus and phase for large complex $z$, deriving explicit, computable error bounds for the expansions of $f(a,z)$, $g(a,z)$, and $M(a,z)$, and for the phase $\varphi(a,z)$. It shows that $ti(a,z,\alpha)$ can be written as $ti(a,z,\alpha) = M(a,z)\cos(\varphi(a,z)-\pi\alpha)$ with accurate large-$z$ expansions and remainder terms expressed via the basic terminant $\Pi_p(z)$; for real $a<1$, it proves that $ti(a,z,\alpha)$ has infinitely many positive real zeros and gives the large-$\kappa$ expansion of these zeros through the inverse $X(a,w)$ of $\varphi(a,z)-\frac{\pi}{2}$, with enveloping bounds. The phase and inverse functions are analytically extended to $\Re(z)>0$, and corresponding remainder bounds $R_N^{(\varphi)}$ and $R_N^{(X)}$ are established, yielding enveloping expansions for the zeros. The work also discusses extensions to $a>1$, optimal truncation for exponential accuracy, and connections to Fresnel integrals, providing a comprehensive framework for precise asymptotics with rigorous error control.

Abstract

In this paper, we investigate the asymptotic properties of the generalised trigonometric integral $\operatorname{ti}(a, z, α)$ 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 $\operatorname{ti}(a, z, α)$ 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.

Asymptotic expansions for the generalised trigonometric integral and its zeros

TL;DR

The paper analyzes the asymptotic behavior of the generalised trigonometric integral and its modulus and phase for large complex , deriving explicit, computable error bounds for the expansions of , , and , and for the phase . It shows that can be written as with accurate large- expansions and remainder terms expressed via the basic terminant ; for real , it proves that has infinitely many positive real zeros and gives the large- expansion of these zeros through the inverse of , with enveloping bounds. The phase and inverse functions are analytically extended to , and corresponding remainder bounds and are established, yielding enveloping expansions for the zeros. The work also discusses extensions to , optimal truncation for exponential accuracy, and connections to Fresnel integrals, providing a comprehensive framework for precise asymptotics with rigorous error control.

Abstract

In this paper, we investigate the asymptotic properties of the generalised trigonometric integral and its associated modulus and phase functions for large complex values of . We derive asymptotic expansions for these functions, accompanied by explicit and computable error bounds. For real values of , the function possesses infinitely many positive real zeros. Assuming , 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.
Paper Structure (7 sections, 13 theorems, 110 equations, 2 figures, 2 tables)

This paper contains 7 sections, 13 theorems, 110 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Proof of Proposition \ref{['prop1']} and Theorem \ref{['thm1']}
  3. Proof of Theorem \ref{['thm2']}
  4. Proof of Theorem \ref{['thm3']}
  5. Discussion
  6. Properties and computation of the coefficients
  7. Bounds for the basic terminant

Key Result

Proposition 1.1

For each non-negative integer $N$, the auxiliary functions $\operatorname{f}(a,z)$ and $\operatorname{g}(a,z)$ can be expressed as and where the remainder terms are given by the following expressions in terms of the basic terminant: provided $2N + 1 > \operatorname{Re}(a)$ for $R_N^{(\operatorname{f})}(a,z)$ and $2N + 2 > \operatorname{Re}(a)$ for $R_N^{(\operatorname{g})}(a,z)$.

Figures (2)

  • Figure 1: The integration contour $\Gamma$.
  • Figure 2: The range of the function $w = \Phi(a,z)$ (unshaded), bounded by the curve $\mathcal{G}_a$, which intersects the real axis at $w = \frac{\pi}{4}(a + |a| - 2)$.

Theorems & Definitions (21)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof : Proof of Proposition \ref{['prop1']}
  • proof : Proof of Theorem \ref{['thm1']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 11 more