1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Approximation of integral for small boundary

  1. Jun 4, 2012 #1
    This problem arises in a paper on population genetics (Kimura 1962).

    1. The problem statement
    Let [itex]f(p) = \int_0^p ((1 - x)/x)^k dx[/itex].

    For a small value of p, we have approximately
    f(p) = (p ^ (1-k)) / (1-k)

    How is this obtained?

    2. My attempt at a solution
    I tried to expand the f(p) around p = 0. However, f'(p) = ((1 - p)/p)^k is undefined at p=0. Furthermore, it does not seem that this approach can yield the form p^(1-k) / (1-k). I must be missing something.

    I would appreciate any insights. Thanks.
  2. jcsd
  3. Jun 4, 2012 #2
    [itex]\frac{d}{dp} \frac{p^{1-k}}{1-k} = \frac{1}{p^{k}}[/itex]

    And the derivative of your original function you said was [itex](\frac{1-p}{p})^{k}[/itex]

    Does that help at all?
  4. Jun 4, 2012 #3
    Thanks for your reply Villyer. I've just solved it, with some help from a friend.

    Here's the solution.

    For [itex] x \approx 0 [/itex], [itex] (\frac{1 - x}{x})^k = x^{-k} [/itex]. Therefore, [itex] f(p) = \int_0^p x^{-k} dx = \frac{1}{1-k} p^{1 - k} [/itex], as required.

  5. Jun 4, 2012 #4
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook