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 smaller. 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 http://www.efunda.com/math/miscellaneousfun/SCPlot.cfm I hope this helps. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ |
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