From: israel@math.ubc.ca (Robert Israel)
Subject: Re: FresnelS
Date: 16 Mar 2000 18:12:07 GMT
Newsgroups: sci.math.symbolic
Summary: [missing]

In article <8alnqj$ta1$1@nnrp1.deja.com>,
 gagibson@ncat.edu writes:
> 
> 
>   What is FresnelS used for?  I've seen it as a result in Maple.
> int(sin(x^2),x) gives a result in terms of this function. What
> application area?  What about the limit of FresnelS(x) as x -> infinity.
>  How can this limit be found by hand? I'm interested in using this as an
> example in class of a function that has a horizontal asymptote even
> though its derivative does not have a horizontal asymptote.

The classic application, I think, is to diffraction of plane waves by
a half-plane barrier.  See e.g. Jackson, "Classical Electrodynamics".
The integrals int(sin(x^2),x=0..infinity) and int(cos(x^2),x=0..infinity)
can be evaluated by integrating the function f(z) = exp(-z^2) around the
boundary of the sector 0 <= arg z <= pi/4, 0 <= |z| <= R, using Cauchy's
Theorem and the fact that int(exp(-x^2),x=0..infinity) = sqrt(pi)/2.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2
==============================================================================

From: gagibson@ncat.edu
Subject: Re: FresnelS
Date: Thu, 16 Mar 2000 21:08:21 GMT
Newsgroups: sci.math.symbolic

Thanks to everyone who answered regarding FresnelS.
I found a few references in the American Mathematical Monthly
on computing the limit of this function as x goes to infinity.

  Olds, Mar 1968(75)
  Yzeren, Oct 1979(86)
  Flanders, Apr 1982(89)
  Leonard May 1988(95)
  Weinstock Jan 1990(97)

 Most of these where either clever applications of complex variables or
techniques with functions that resembled laplace transforms.  The later
where introduced to avoid complex integration.  There where also a few
references in the journal Mathematics of Computation referring to
methods for estimating the value of these integrals for finite x.
Clearly this would not work as an example for calculus II students but
some of our seniors might get something out of some of this material.

Thanks Again,

    - Greg Gibson

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