A Double Integral Containing the Fresnel Integral Function  S   x   : Derivation and Computation

Robert Reynolds,Allan Stauffer,Serkan Araci

doi:10.1155/2022/7044988

Abstract

A two-dimensional integral containing S x is derived. S x is the Fresnel integral function, and the double integral is taken over the range 0 < x < ∞ and 0 < y < ∞ . A representation in terms of the Hurwitz–Lerch zeta function is derived, from which other special function representations can be evaluated. All the results in this work are new.

Highlights

In this paper, we derive the double definite integral given by3/2 − by2 m+(1/2) − 2m √ √ √ k− 1∞ ∞ αexyS( 2/π x α )log 􏼐ax/y 􏼑􏼐m log􏼐ax/y 􏼑 + k􏼑 􏽚􏽚 􏽰 dxdy, (1)α3x3 where the hypergeometric function representation for the Fresnel integral S(x) is given in Table (3.7) in [11] and parameters k, a, and m are general complex numbers and Re(b) > 0 and Re(m) < − 1/2 < Re(α). is definite integral will be used to derive special cases in terms of special functions and fundamental constants. e derivations follow the method used by us in [12, 13]. is method involves using a form of the generalized Cauchy’s integral formula given by Journal of Mathematics yk 1 ewy 􏽚 dw, (2)
Γ(k + 1) 2πi Cwk+1 where C is, in general, an open contour in the complex plane where the bilinear concomitant has the same value at the end points of the contour. We multiply both sides by a function of x and y and take a definite double integral of both sides. is yields a definite integral in terms of a contour integral. en, we multiply both sides of equation (2) by another function of y and take the infinite sum of both sides such that the contour integral of both equations are the same
We are able to switch the order of integration over x, y, and r using Fubini’s theorem for multiple integrals (see (9.112) in [15]), since the integrand is of bounded measure over the space C × [0, ∞) × [0, ∞)

Summary

Introduction

Α3x3 where the hypergeometric function representation for the Fresnel integral S(x) is given in Table (3.7) in [11] and parameters k, a, and m are general complex numbers and Re(b) > 0 and Re(m) < − 1/2 < Re(α). is definite integral will be used to derive special cases in terms of special functions and fundamental constants. e derivations follow the method used by us in [12, 13]. is method involves using a form of the generalized Cauchy’s integral formula given by Journal of Mathematics yk 1 ewy. Is definite integral will be used to derive special cases in terms of special functions and fundamental constants. Is method involves using a form of the generalized Cauchy’s integral formula given by Journal of Mathematics yk 1 ewy. Γ(k + 1) 2πi Cwk+1 where C is, in general, an open contour in the complex plane where the bilinear concomitant has the same value at the end points of the contour. We multiply both sides by a function of x and y and take a definite double integral of both sides. We multiply both sides by a function of x and y and take a definite double integral of both sides. is yields a definite integral in terms of a contour integral. en, we multiply both sides of equation (2) by another function of y and take the infinite sum of both sides such that the contour integral of both equations are the same

Definite Integral of the Contour Integral

The Hurwitz–Lerch Zeta Function and Infinite Sum of the Contour Integral

Definite Integral in terms of the Hurwitz–Lerch Zeta Function

Discussion