Mathematical Methods in the Physical Sciences · Problem 10.19

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · miscellaneous-parseval-contour

Problem

(a) Use Parseval's theorem and Problem 4.11 to evaluate

\int_0^{\infty} \frac{\cos^2(\alpha\pi/2)}{(1-\alpha^2)^2}\,d\alpha.

(b) Evaluate the integral in (a) by contour integration. Hint:

\cos^2(\alpha\pi/2) = \frac{1}{2}(1 + \cos\alpha\pi)

. Evaluate

\oint_C \frac{1+e^{i\pi z}}{(z-1)^2(z+1)^2}\,dz,

where

C

is the upper half plane; note that the poles are actually simple poles (see Chapter 14, Section 7, Example 4).