- #1
Fraggler
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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.
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.