A Analytical Integration of a Difficult Function

junt
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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!
 
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Can you provide a context of where you got this problem?
 
jedishrfu said:
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.
 
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
 
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.
 
SlowThinker said:
is purely imaginary
Sure, since the expression under the root becomes negative
 

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