Mathematical Methods in the Physical Sciences · Problem 10.22

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · miscellaneous-poisson-formula-series

Problem

Use Poisson's formula (Problem 21b) and Problem 20 to show that

\sum_{-\infty}^{\infty} \frac{\sin^2 n\theta}{n^2} = \pi\theta, \quad 0 < \theta < \pi.

(This sum is needed in the theory of scattering of light in a liquid.) Hint: Consider

f(x)

and

g(\alpha)

as in Problem 20. Note that

f(2k\pi) = 0

except for

k = 0

a < \pi

. Put

\alpha = n

a = \theta