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Konrad Viltersten
science forum addict

Joined: 27 Jun 2005
Posts: 79

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

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?

--
Vänligen
Konrad
---------------------------------------------------

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

Ambition - a poor excuse for not having
enough sence to be lazy
---------------------------------------------------
Jannick Asmus
science forum Guru

Joined: 25 Mar 2005
Posts: 312

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

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.
Konrad Viltersten
science forum addict

Joined: 27 Jun 2005
Posts: 79

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

 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?

--
Vänligen
Konrad
---------------------------------------------------

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

Ambition - a poor excuse for not having
enough sence to be lazy
---------------------------------------------------
Dave L. Renfro
science forum Guru

Joined: 29 Apr 2005
Posts: 570

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

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
Alois Steindl
science forum beginner

Joined: 10 May 2005
Posts: 25

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

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
Duncan Muirhead
science forum addict

Joined: 08 Oct 2005
Posts: 70

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

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
C6L1V@shaw.ca
science forum Guru

Joined: 23 May 2005
Posts: 628

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

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? -- Vänligen Konrad --------------------------------------------------- Sleep - thing used by ineffective people as a substitute for coffee Ambition - a poor excuse for not having enough sence to be lazy ---------------------------------------------------
Ronald Bruck
science forum Guru

Joined: 05 Jun 2005
Posts: 356

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

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.)

--
Ron Bruck
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