Mathematical Methods in the Physical Sciences · Problem 11.22

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · contour-integration-verify-formula

Problem

Verify the formula by contour integration or as indicated. Assume

a > 0

m > 0

\int_0^{2\pi} \frac{d\theta}{(a+b\sin\theta)^2} = \int_0^{2\pi} \frac{d\theta}{(a+b\cos\theta)^2} = \frac{2\pi a}{(a^2-b^2)^{3/2}}, \qquad |b| < a

Hint: You can do this directly by contour integration, but it is easier to differentiate Problem 21 with respect to

a