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MathGroup Archive 1997

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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.....
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>     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



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