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Fresnel Integral of sin(x^2)

Date: 03/28/2002 at 02:25:51
From: robert Ihnot
Subject: Fresnel Integral of sin(x^2)

I can not see how this integral from 0 to infinity can have any limit 
at all.  Certainly, if we used infinite series, we would not get a 
value for x = infinity.  

Thanks, Bob

Date: 03/30/2002 at 11:57:35
From: Doctor Douglas
Subject: Re: Fresnel Integral of sin(x^2)

Hi, Bob,

Before just plugging Infinity into series formulas for the sine and
trying to integrate term-by-term, let us remember that remember that 
the Fresnel sine integral is a function of t, namely

  I(t) = Integral{x,0,t} sin(x^2).

Certainly we can evaluate this for any finite t, and now we have to 
ask ourselves what is its limit (if any) as t->Infinity.

As t->Infinity, the positive and negative areas traced out by the 
integrand become smaller and smaller. This is because x^2 runs out to 
Infinity much faster than x does, so there are more and more "periods" 
squeezed in, so the area of each "loop" (or half-period) has to get 

Thus, as t->Infinity, these areas can cancel each other out and the 
deviation from the limiting value is bounded above by a quantity (say 
the area of one of the loops) that is known to approach zero. Thus the 
integral I(t->Infinity) has a limit.

There's no problem with using an infinite series to represent I(t) 
also, since in

   I(t) = Integral{x,0,t} sin(x^2)

        = Integral{x,0,t} [x^2 - x^6/3! + x^10/5! - ...]  

        = t^3/3 - t^7/(7*3!) + t^11/(11*5!) - ...            

This can be evaluated for any real t, because the factorials 
eventually win out over any finite t. And there's no reason why I(t) 
could not have a limit as t->Infinity.

You can see a graph of the function I(t) at the following web page:

  Fresnel sine and cosine integral plots - Engineering Fundamentals   

I hope this helps.

- Doctor Douglas, The Math Forum   
Associated Topics:
High School Calculus
High School Sequences, Series

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