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Integrating t^(-3/2) * e^(1/t)
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Konrad Viltersten
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Joined: 27 Jun 2005
Posts: 79

PostPosted: Wed Jul 05, 2006 2:27 pm    Post subject: Integrating t^(-3/2) * e^(1/t) Reply with quote

How do i go about integrating this function
t^(-3/2) * e^(1/t)

from 0 to, say, S?

I have tried with variable substitution t = 1/k but
that gave me nothing but headache. What do i miss?

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Konrad
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Jannick Asmus
science forum Guru


Joined: 25 Mar 2005
Posts: 312

PostPosted: Wed Jul 05, 2006 2:43 pm    Post subject: Re: Integrating t^(-3/2) * e^(1/t) Reply with quote

On 05.07.2006 16:54, Konrad Viltersten wrote:
Quote:
How do i go about integrating this function
t^(-3/2) * e^(1/t)

from 0 to, say, S?

I have tried with variable substitution t = 1/k but
that gave me nothing but headache. What do i miss?

You could use the Taylor expansion of exp ...

Best wishes,
J.
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Konrad Viltersten
science forum addict


Joined: 27 Jun 2005
Posts: 79

PostPosted: Wed Jul 05, 2006 3:07 pm    Post subject: Re: Integrating t^(-3/2) * e^(1/t) Reply with quote

Quote:
How do i go about integrating this function
t^(-3/2) * e^(1/t)

from 0 to, say, S?

I have tried with variable substitution t = 1/k but
that gave me nothing but headache. What do i miss?

You could use the Taylor expansion of exp ...

That would lead to one of the following.
1. Truncation, which is to be avoided if possible.
2. Fooling around with ordo, which is simply unpleasant.
3. Write up a lots of lots of terms, which is uncomfortable
since my pile of infitely many sheets has just run out. :)

Any suggestions?

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Konrad
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Sleep - thing used by ineffective people
as a substitute for coffee

Ambition - a poor excuse for not having
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Dave L. Renfro
science forum Guru


Joined: 29 Apr 2005
Posts: 570

PostPosted: Wed Jul 05, 2006 3:13 pm    Post subject: Re: Integrating t^(-3/2) * e^(1/t) Reply with quote

Konrad Viltersten wrote:

Quote:
How do i go about integrating this function
t^(-3/2) * e^(1/t)

from 0 to, say, S?

I have tried with variable substitution t = 1/k but
that gave me nothing but headache. What do i miss?

I did this fast so you might want to double check it:

u = 1/t leads to integral of [-1/sqrt(u)] * e^u

w = sqrt(u) leads to integral of -2*e^(w^2)

The w-integral isn't expressible in terms of
elementary functions, so the t-integral isn't
expressible in terms of elementary functions.

Dave L. Renfro
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Alois Steindl
science forum beginner


Joined: 10 May 2005
Posts: 25

PostPosted: Wed Jul 05, 2006 3:17 pm    Post subject: Re: Integrating t^(-3/2) * e^(1/t) Reply with quote

Hello,
the substitution t=1/u leads to an incomplete Gamma function. Did you
already learn about that subject?
Are you sure, that you want your lower boundary at 0? Don't you expect
convergence problems there?
Alois
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Duncan Muirhead
science forum addict


Joined: 08 Oct 2005
Posts: 70

PostPosted: Wed Jul 05, 2006 4:14 pm    Post subject: Re: Integrating t^(-3/2) * e^(1/t) Reply with quote

On Wed, 05 Jul 2006 16:54:08 +0200, Konrad Viltersten wrote:

Quote:
How do i go about integrating this function
t^(-3/2) * e^(1/t)

from 0 to, say, S?

I have tried with variable substitution t = 1/k but
that gave me nothing but headache. What do i miss?

How about t = s^-2?

If I = Integral{ a<=t<b | t^(-3/2) * e^(1/t)}
then I get
I = Integral{ 1/sqrt(a)<=s<=1/sqrt(b) | s^3 * e^(s^2) * -2*s^-3}
= 2*Integral{ 1/sqrt(b)<=s<=1/sqrt(a) | e^(s^2)}
which blows up as a->0. Did you perhaps mean t^(-3/2) * e^(-1/t)?
If so then the substitution would give you an incomplete gamma function.
Duncan
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C6L1V@shaw.ca
science forum Guru


Joined: 23 May 2005
Posts: 628

PostPosted: Wed Jul 05, 2006 4:28 pm    Post subject: Re: Integrating t^(-3/2) * e^(1/t) Reply with quote

Konrad Viltersten wrote:
Quote:
How do i go about integrating this function
t^(-3/2) * e^(1/t)

from 0 to, say, S?

For J = the integral from 1 to x, Maple 9.5 gets


J := -I*erf(I)*Pi^(1/2) + erf(1/x^(1/2)*I)*Pi^(1/2)*I

Here, I = sqrt(-1) and erf(x) = [2/sqrt(Pi)] * int(exp(-t^2), t=0..x).

Taking dJ/dx gives x^(-3/2)*e^(1/x), as it should.

R.G. Vickson


Quote:

I have tried with variable substitution t = 1/k but
that gave me nothing but headache. What do i miss?

--
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Konrad
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Sleep - thing used by ineffective people
as a substitute for coffee

Ambition - a poor excuse for not having
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Ronald Bruck
science forum Guru


Joined: 05 Jun 2005
Posts: 356

PostPosted: Sat Jul 08, 2006 5:53 am    Post subject: Re: Integrating t^(-3/2) * e^(1/t) Reply with quote

In article <4h20daF1pi3k4U1@individual.net>, Konrad Viltersten
<tmp1@viltersten.com> wrote:

Quote:
How do i go about integrating this function
t^(-3/2) * e^(1/t)

from 0 to, say, S?

I have tried with variable substitution t = 1/k but
that gave me nothing but headache. What do i miss?

k, indeed, for an integration variable! You DESERVE a headache.

Try t = 1/s^2 instead.

1/t = s^2

t^(-3/2) = s^3

dt = -2 s^-3 ds

\int t^(-3/2) e^(1/t) dt = -2 \int e^{s^2} ds.

The result is infinite on [0,S], of course. (The integrand is
Quote:
= t^(-3/2), which is not integrable near 0.)

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Ron Bruck
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