when does this integral exist?

Apr 2010
3
0
Hi there,

I am new here, this is the first post I make. I am statistician who is rusty in calculus. Also I know I must check how to write in some kind of math (or latex) environment in this forum and any hints will be welcome. I should check the site some more of course.

So, let following integral

int ( exp(tx-g (lx)^a ) * x^(ag-1) ,from 0 to Inf) dx

where -Inf<a<Inf, l>0, g>0, -Inf<t<Inf.


I don't want to solve it, I just want to know when does it exist (and I think it is not easy).

I derived that it exists for all t<0. I think that the whole "game" will take place with the possible values of "a" and "t". There might be some problems if a>0 and t>0? or if 0<a<1 maybe.

I am a little confused and tried some possible values in the computer to see when it does exists but I haven't came up with any general constraints.


Thanx in advance for any help!
 
Apr 2010
3
0
I restate my problem in latex typesetting.

\(\displaystyle \int_{0}^{\infty}exp(tx-\gamma\lambda^{\alpha}x^{\alpha})x^{\alpha\gamma-1}dx \)

for \(\displaystyle -\infty<t<\infty, -\infty<\alpha<\infty, \gamma>0, \lambda>0 \).

I just want to know when the above intgral exists.

What I think is that the integral can be written as:

\(\displaystyle \int_{0}^{\infty}exp(-x(-t+\gamma\lambda^{\alpha}x^{\alpha-1}))x^{\alpha\gamma-1}dx \). So,

For \(\displaystyle \alpha=1 \ and \ \gamma\lambda>t \) it exists.
For \(\displaystyle \alpha<1 \) it does not exist since the term \(\displaystyle x^{\alpha-1} \) will go to infinity.
For \(\displaystyle t<0 \ and \ \alpha>1 \) it exists.
For \(\displaystyle \alpha<1 \) it does not exists.


I don't know if my thinking is correct. The above constraints do not take under consideration all cases...

It might help to "break" the last integral into the sum of two integrals. The first would be from 0 to \(\displaystyle (\frac{t}{\gamma\lambda^{\alpha}})^{\frac{1}{a-1}} \) and the second from \(\displaystyle (\frac{t}{\gamma\lambda^{\alpha}})^{\frac{1}{a-1}} \) to infinity.