From: ron@dorianresearch.com (Ron Levine)
Subject: Re: Fresnel Integrals
Date: Fri, 13 Aug 1999 00:16:01 GMT
Newsgroups: sci.math

On Thu, 12 Aug 1999 18:54:05 -0500, "Anthony J. Bruni"
<abruni@prodigy.net> wrote:

>What is a good reference for Fresnel Integrals?

A book on physical optics.  For example, the old one that I used in
college many years ago, Jenkins and  White, Fundamentals of Optics.

These are defined as
	integral(cos(pi v^2/2)dv)
and 	integral(sin(pi v^2/2)dv)

and give respectively the x and y coordinate as functions of arclength
of an amusing curve called "Cornu's Sprial", which comes up via
geometric construction in analyzing the phenomenon of Fresnel
diffraction.

Jenkins and White gives a table of  their values as functions of the
upper limit.

Ron
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[Maple treats these integrals as known functions. Here is an excerpt from
their help page. --djr]

Function: FresnelC - The Fresnel Cosine Integral 
Function: FresnelS - The Fresnel Sine Integral
Function: Fresnelf, Fresnelg - The Fresnel Auxiliary Functions

Description:

- The Fresnel cosine integral is defined as follows: 
        FresnelC(x) =  int(cos(Pi/2*t^2), t=0..x);

- The Fresnel sine integral is defined as follows: 
        FresnelS(x) =  int(sin(Pi/2*t^2), t=0..x);
                   
- The Fresnel auxiliary functions are defined as follows: 
        Fresnelf(x) = (1/2 - FresnelS(x))*cos(Pi/2*x^2) -
                                     (1/2 - FresnelC(x))*sin(Pi/2*x^2)
        Fresnelg(x) = (1/2 - FresnelC(x))*cos(Pi/2*x^2) -
                                     (1/2 - FresnelS(x))*sin(Pi/2*x^2)

For comparison:

- Dawson's integral is defined as follows: 
        dawson(x) = exp(-x^2) * int(exp(t^2), t=0..x)
- The error function is defined for all complex x by 
        erf(x) = 2/sqrt(Pi) * int((exp(-t^2), t=0..x);