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

Integration with exponential and inverse power

  1. Nov 8, 2013 #1
    I confront an integration with the following form:

    [itex] \int d^2{\vec q} \exp(-a \vec{q}^{2}) \frac{\vec{k}^{2}-\vec{k}\cdot
    \vec{q}}{((\vec q-\vec k)^{2})(\vec{q}^{2}+b)}
    [/itex]
    where [itex]a[/itex] and [itex]b[/itex] are some constants, [itex]\vec{q} = (q_1, q_2)[/itex] and [itex]\vec{k} = (k_1, k_2)[/itex] are two-components vectors.

    In the case of [itex]a\rightarrow \infty [/itex] in which the exponential becomes 1, I can perform the integration using Feynman parameterization.

    In the general case I have now idea to calculate it. I know the answer is

    [itex]\pi \exp(ab)\left(\Gamma(0,ab)-\Gamma(0,a(\vec{k}^2+b))\right)[/itex]

    where [itex]\Gamma(0,x)=\int_x^\infty t^{-1} e^{-t}\,dt [/itex] is the incomplete gamma function.

    But i don't know how to arrive at this result. can someone give any clue to perform this kind of integration? thanks a lot.
     
    Last edited: Nov 9, 2013
  2. jcsd
  3. Nov 10, 2013 #2
    I would try using Feynman parametrization anyways, and than convert the integration to polar coordinates (The fact that you end up with an incomplete gamma function is a clue that polar coordinates were used).
     
  4. Nov 10, 2013 #3

    Thanks. I just found the solution from another paper. So first one should perform the integration to polar coordinates using the formula:
    [itex] \int_0^\pi d\theta \cos(n\theta)/( 1+a\cos(\theta))=\left(\pi/\sqrt{1-a^2}\right)\left((\sqrt{1-a^2}-1)/ a\right)^n,~~~a^2<1,~~n\geq0[/itex]
    then perform the integration on [itex]p^2[/itex] will yield the above result.
     
    Last edited: Nov 10, 2013
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Integration with exponential and inverse power
  1. Power counting (Replies: 1)

  2. Fusion Power (Replies: 5)

  3. Inverse beta decay (Replies: 5)

Loading...