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
>
>--
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>
>
>
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 (;-)#
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