Fourier Series and Applications

Abstract

As we pointed out earlier with (3.30) and (3.31), a function f(z) analytic in a d containing the unit circle can be expressed on the unit circle by a Fourier series:

$$ f\left( {{e^{i\theta }}} \right) = F\left( \theta \right) = \sum\limits_{n = - \infty }^\infty {{a_n}{e^{in\theta }},} $$

((5.1))

where the Fourier coefficients a _n are given by

$$ {a_n} = \frac{1}{{2\pi }}\int_0^{2\pi } {F\left( \theta \right)} {e^{ - in\theta }}d\theta = {{\tilde F}_n}/2\pi . $$

((5.2))