Re: NIntegrate of a Decaying Exponential

*To*: mathgroup at smc.vnet.net*Subject*: [mg15176] Re: [mg15137] NIntegrate of a Decaying Exponential*From*: Robert Pratt <rpratt at math.unc.edu>*Date*: Thu, 17 Dec 1998 00:27:53 -0500*Sender*: owner-wri-mathgroup at wolfram.com

The exact answer can be obtained by hand by using the substitution u = -v^2. The integrand then becomes E^u. Applying the Fundamental Theorem of Calculus, we get E^(-4) - E^(-1) as the exact answer. Integrate[-2v Exp[-v^2],{v,-1,2}] yields the same result. Note that since -2v Exp[-v^2] is an odd function, the integral from -1 to 1 is 0. Hence it should be no surprise that Integrate[-2v Exp[-v^2],{v,1,2}] also gives the exact result obtained above. Rob Pratt Department of Mathematics The University of North Carolina at Chapel Hill CB# 3250, 331 Phillips Hall Chapel Hill, NC 27599-3250 rpratt at math.unc.edu http://www.math.unc.edu/Grads/rpratt/ On Wed, 16 Dec 1998, Wretch wrote: > Greetings -- I'm trying to get Mathematica to do an integral that > appears regularly in plasma physics problems. It appears to give the > right answer, but it's always accompanied with an annoying error > message whose roots are a bit vague. Here is the function to be > integrated: > > f = -2 v Exp[-v^2] over the v-interval (-1,2) > > The answer should be about -.34956 (according to an IMSL subroutine) > > Here's the mathematica input and output: > > In[1]:=NIntegrate[-2 v Exp[-v^2],{v,-1,2}] Out[1]:=Out[99]=-0.349564 > > NIntegrate::"ploss": > "Numerical integration stopping due to loss of precision. Achieved \ > neither the requested PrecisionGoal nor AccuracyGoal; suspect highly \ > oscillatory integrand, or the true value of the integral is 0. If your > \ integrand is oscillatory try using the option Method->Oscillatory in > \ NIntegrate." > > So, Mathematica gets it right, but with the mysterious warning. The > error/warning message isn't surprising since the integrand has such a > sharp peak at v=0, but none of the options specified in the help menu, > such as MinRecursion, MaxRecursion, Method->, etc. were of any use in > suppressing this error message. I want to suppress messages of this > sort not only so that I don't have to look at them, but also to have an > extra measure of confidence that the answer is actually right! > > Any help is greatly appreciated. > > Thanks, > > AC