Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Analytical Integration of a Difficult Function

  1. Apr 27, 2017 #1
    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!
     
  2. jcsd
  3. Apr 27, 2017 #2

    jedishrfu

    Staff: Mentor

    Can you provide a context of where you got this problem?
     
  4. Apr 28, 2017 #3
    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.
     
  5. Apr 29, 2017 #4
    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
     
  6. Apr 30, 2017 #5
    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.
     
  7. May 1, 2017 #6
    Sure, since the expression under the root becomes negative
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted