Question

complex analysis

Answer 1

\[\int\limits_{0}^{\infty} (x^p dx)/(x^4+1) \] -1<p<3

Answer 2

I get \[\frac{2\pi i}{1-i^{p+1}}\frac{1}{4}e^{i\pi(p-3)/4}\]

Answer 3

which simplifies to

\[\frac{1}{4}\pi \cdot csc\left[\frac{1}{4}\pi p+\frac{1}{4}\pi\right]\]

Answer 4

the contour I used was the quarter circle in the first quadrant and the positive x and y axis.

Answer 5

can you show me step by step? or attach it in pdf?

Answer 6

thank you. I have few questions more:
why do I choosethe quarter circle in quadrant 1 for C? shouldn't it be a circle with radius R, excluding positive part of x-axis, that contains all the poles?

Answer 7

the original problem is an integral over the positive real axis...we need to include that