Re: barfing on an integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg6225] Re: [mg6160] barfing on an integral*From*: "Wouter.Meeussen. Vandemoortele CC R&D" <w.meeussen.vdmcc at vandemoortele.be>*Date*: Fri, 28 Feb 1997 03:21:44 -0500*Sender*: owner-wri-mathgroup at wolfram.com

At 02:52 27.02.97 -0500, you wrote: > >Hello All > >I'm trying to check my solution to approximations of integrals by the >method of steepest decent. The integral i'm approximating is: > > I(x) = int e(x*t - e^t) dt from t=0 to t=infty > >my approximation is: > > J(x) = sqrt(2 Pi/x) * e^{1.5 x log x} > >The trouble is that MMA is barfing on I(x). If I try to do it >analytically with Integrate[] it gives me a whole bunch of E^Infty >terms. Most of which go to zero, but I'm having trouble getting MMA to >actually evaluate the limit. N[%] doesn't seem to work, and MMA doesn't >seem to realize that e^infty / e^e^infty is truly zero. > >Then I tried using NIntegrate on I(x), and it gave me a whole bunch of >underflow errors, and said that the integrand is probably oscillatory >(which it most certainly is not). > >I'm in a very awkward position. My approximation is only valid for large >values of x (actually, I think anything over 10 will do). Yet, MMA barfs >for large values of x. > >Is there some way I can check my approximation?? > >Much thanks. > >Peter > >-- >Birthdays are good for you: A federal funded project has recently determined >that people with the most number of birthdays will live the longest..... >-=><=-+-+-=><=-+-+-=><=-+-+-=><=-+-+-=><=-+-+-=><=-+-+-=><=-+-+-=><=-+-+-=><=- > I BOYCOTT ANY COMPANY THAT USES MASS ADVERTISING ON THE INTERNET > > > > sorry, but you may Barphf at me if i don't get it: ok, straight on don't go : In[2]:=f[t_]:=Exp[x t -Exp[t]] In[3]:=Integrate[f[t],{t,0,1}] Out[3]= t -E + t x Integrate[E , {t, 0, 1}] but with some sleezy substitution : In[5]:=f[t]/.t->Log[u] Out[5]= Exp[-u + x Log[u]] and massaging the integration: In[7]:=Dt[t]/.t->Log[u] Out[7]=Dt[u]/u Out[5]=E^(-u + x*Log[u]) In[10]:=E^(-u ) u^x Out[10]=u^x / E^u In[11]:=Integrate[Exp[-u] u^x /u ,u] Out[11]=-Gamma[x, u] now put u back to E^t and your home: t -E + t x Integrate[E , {t, 0, 1}] = -Gamma[x, Exp[t]] ************ CHECK : ***************** In[15]:= D[-Gamma[x, Exp[t]],t]//PowerExpand//FullSimplify//InputForm Out[15]//InputForm= E^(-E^t + t*x) *************************************** as the old ones say, nothing beats the good old Compaq & Mma3 methods (;-)# NV Vandemoortele Coordination Center Group R&D Center Prins Albertlaan 79 Postbus 40 B-8870 Izegem (Belgium) Tel: +/32/51/33 21 11 Fax: +/32/51/33 21 75 Fax:+32/51/33 21 75 vdmcc at vandemoortele.be