Graduate Analytical Integration of a Difficult Function

[junt]

Is it possible to integrate the following function analytically?

$\int_{0}^{\infty} \frac{\exp{-(\frac{A}{\tau}+B\tau+\frac{A}{\beta-\tau})}}{\sqrt{\tau(\beta-\tau)}}d\tau,$

where $A$, $B$ and $\beta$ are real numbers. What sort of coordinate transformation makes the integral bounded? Is it even bounded? Are these poles integrable?

Any help is much appreciated!

 

[jedishrfu]

Can you provide a context of where you got this problem?

 

[junt]

Can you provide a context of where you got this problem?

Integrals like this appear when one is looking at chemical reaction rates. The exponent is basically the classical action. A and B contains space coordinates, which will be integrated after integral over $\tau$ has been performed.

 

[Buzz Bloom]

Hi junt:

I think I understand that you are asking about whether the integral is finite. I think it is easy to see that the integrand behaves OK at infinity. It is a bit trickier to consider behavior at zero. Can you simplify the integrand behavior near zero and see if the integral of the simplification is OK? That is , consider the integral from zero to ε<<1 of a simplified integrand between zero and ε.

Regards,
Buzz

 

[SlowThinker]

I've played a bit with WolframAlpha and it suggests that between 0 and $\beta$ it should be fine but the tail from $\beta$ to $\infty$ is purely imaginary and also infinite.

 

[Irene Kaminkowa]

is purely imaginary

Sure, since the expression under the root becomes negative