Evaluate using contour Integration
J(a) = integral from zero to infinity of (dx/(1 + x^a)) ; a> 1© BrainMass Inc. brainmass.com March 5, 2021, 12:44 am ad1c9bdddf
We can rewrite the integral using the substitution x^a = u as:
J(a) = 1/a Integral from 0 to infinity of u^(1/a -1)/(1+u) du
Putting p = 1/a - 1, we thus have to evaluate:
Integral from 0 to infinity of x^p/(1+x) dx
for -1 < p < 0
To compute the integral using contour integration, we define the function z^p in the complex plane as follows. For any given z we define polar coordinates r and theta, such that z = r exp(i theta) and we choose theta such that 0 < theta < 2 pi. We then define z^p = r^p exp(i p theta). We thus put the branch cut along the positive real axis. Then we consider the contour integral of z^p/(1+z) along the following contour. We start at the point epsilon + i delta and move parallel to the real axis to i delta + R (we need to stay clear of the ...
I show how the integral can be computed using the methods of complex analysis.