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Re: barfing on an integral

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

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.
>Birthdays are good for you:  A federal funded project has recently determined
>that people with the most number of birthdays will live the longest.....

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]]

Out[3]=      t
           -E  + t x
Integrate[E         , {t, 0, 1}]

but with some sleezy substitution :


Out[5]= Exp[-u + x Log[u]]
and massaging the integration:

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:

           -E  + t x
Integrate[E         , {t, 0, 1}] = -Gamma[x, Exp[t]]

************ CHECK : *****************

D[-Gamma[x, Exp[t]],t]//PowerExpand//FullSimplify//InputForm
E^(-E^t + t*x)


as the old ones say,
nothing beats the good old Compaq & Mma3 methods (;-)#
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