[Paper Review] Generalized Fresnel integrals as oscillatory integrals with positive real power phase functions and applications to asymptotic expansions

Toshio Nagano, Naoya Miyazaki|arXiv (Cornell University)|May 24, 2020

Mathematical functions and polynomials11 references4 citations

TL;DR

This paper generalizes Fresnel integrals to oscillatory integrals with phase functions of the form $x^p$ for $p > 0$, using complex analysis and analytic continuation to derive asymptotic expansions for oscillatory integrals with degenerate critical points. The key contribution is a refined asymptotic expansion for integrals with positive real power phase functions, extending the classical stationary phase method to non-degenerate and degenerate critical points via generalized Fresnel-type integrals.

ABSTRACT

In this paper, we first generalize the Fresnel integrals by changing of a path for integration in the proof of the Fresnel integrals by Cauchy's integral theorem. Next, according to oscillatory integral, we also obtain further generalization of the extended Fresnel integrals. Moreover by using this result, we have an asymptotic expansion of an oscillatory integral with a positive real parameter, for a phase function with a degenerate critical point expressed by positive real power, including a moderate oscillation, and for a suitable amplitude function. This result gives a finer extension of the stationary phase method in one variable, which is known as a method for an asymptotic expansion of an oscillatory integral of a phase function with a non-degenerate critical point.

Motivation & Objective

• To generalize the classical Fresnel integrals by extending the phase function from $x^2$ to $x^p$ with $p > 0$ using contour integration and analytic continuation.
• To establish a rigorous framework for oscillatory integrals with positive real power phase functions and degenerate critical points via generalized Fresnel integrals.
• To extend the stationary phase method in one variable to phase functions with degenerate critical points, including moderate oscillations.
• To provide an asymptotic expansion for oscillatory integrals with amplitude functions in a broader class than the Schwartz space, specifically $\mathcal{A}^{\tau}_{\delta}(\mathbb{R})$.
• To generalize the asymptotic behavior of Fourier transforms to phase functions with singular critical points, such as $A_k$, $E_6$, $E_8$ types.

Proposed method

• Use of Cauchy’s integral theorem with a holomorphic function $e^{-iz^p}z^{q-1}$ over a fan-shaped contour in the complex plane to generalize the Fresnel integral.
• Definition of generalized Fresnel integrals $\tilde{I}_{p,q}^{\pm}$ as limits involving a cutoff function $\chi(\varepsilon x)$, ensuring convergence via oscillatory integral theory.
• Analytic continuation of $\tilde{I}_{p,q}^{\pm}$ to a meromorphic function on $\mathbb{C}$ for $p > 0$, $q > 0$.
• Derivation of asymptotic expansion for $\int_0^\infty e^{\pm i\lambda x^p} a(x) \chi(\varepsilon x) dx$ as $\lambda \to \infty$, using Taylor expansion of $a(x)$ around $x=0$ and generalized Fresnel integrals.
• Extension to full real line integrals for even $m \in \mathbb{N}$ by combining contributions from $x^m$ and $(-x)^m$, leading to coefficients $\tilde{c}_k^{\pm}$ involving $\tilde{I}_{m,k+1}^{\pm}$ and phase corrections.
• Application of the method to phase functions with degenerate critical points, including $A_k$, $E_6$, $E_8$ types, via extension to several variables.

Experimental results

Research questions

• RQ1 How can the classical Fresnel integral $\int_0^\infty e^{\pm i x^2} dx$ be generalized to phase functions of the form $x^p$ with $p > 0$?
• RQ2 What is the asymptotic behavior of oscillatory integrals $\int_0^\infty e^{\pm i\lambda x^p} a(x) dx$ as $\lambda \to \infty$ when the phase function has a degenerate critical point at $x=0$?
• RQ3 Can the stationary phase method in one variable be extended to phase functions with non-isolated or degenerate critical points, such as $x^m$ for $m \geq 2$?
• RQ4 What is the role of the generalized Fresnel integral $\tilde{I}_{p,q}^{\pm}$ in constructing asymptotic expansions for such oscillatory integrals?
• RQ5 How does the class of amplitude functions $\mathcal{A}^{\tau}_{\delta}(\mathbb{R})$ compare to the Schwartz space in enabling such asymptotic expansions?

Key findings

• The generalized Fresnel integral is given by $\tilde{I}_{p,q}^{\pm} = p^{-1} e^{\pm i \frac{\pi}{2} \frac{q}{p}} \varGamma\left(\frac{q}{p}\right)$ for $p > 0$, $q > 0$, derived via contour integration and analytic continuation.
• For $p = m \in \mathbb{N}$, the asymptotic expansion of $\int_{-\infty}^\infty e^{\pm i\lambda x^m} a(x) dx$ as $\lambda \to \infty$ is $\sum_{k=0}^{N-m-1} \tilde{c}_k^{\pm} \frac{a^{(k)}(0)}{k!} \lambda^{-\frac{k+1}{m}} + O\left(\lambda^{-\frac{N-m+1}{m}}\right)$, where $\tilde{c}_k^{\pm} = \tilde{I}_{m,k+1}^{+} + (-1)^k \tilde{I}_{m,k+1}^{\pm\pm_m}$.
• For $m=2$, the expansion reduces to $\sqrt{\pi} \sum_{k=0}^{N-1} e^{\pm i \frac{\pi}{2}(k+\frac{1}{2})} \frac{a^{(2k)}(0)}{4^k k!} \lambda^{-k-\frac{1}{2}} + O(\lambda^{-N})$, matching known results for quadratic phase functions.
• When $m=1$, the asymptotic expansion vanishes to $O(\lambda^{-2N})$ for any $N$, consistent with the rapid decay of the Fourier transform of smooth functions in $\mathcal{A}^{\tau}_{\delta}(\mathbb{R})$.
• The method extends beyond one variable and applies to singular phase functions such as $A_k$, $E_6$, and $E_8$ types, indicating broader applicability in singularity theory and asymptotic analysis.
• The class $\mathcal{A}^{\tau}_{\delta}(\mathbb{R})$ allows for broader amplitude functions than the Schwartz space, enabling the asymptotic expansion to hold under weaker decay and smoothness conditions.