Dirichet’s Theorem as applied to a Fourier Series expansion of a function indicates under what conditions a Fourier Series expansion converges.

In Boas notation, a function f(x) can be expanded in a Fourier series over the interval a \le x \le b as f(x) = \sum_{n = -\infty}^{+\infty} c_n \exp(+i n x) where n is an integer and c_n = \frac{1}{b-a} \int_{a}^{b} \exp \left [ -\frac{2 \pi i n x }{b-a} \right ] f(x)\ dx .

Dirichlet’s Theorem indicates that this Fourier Series expansion converges, namely the infinite sum is equal to a finite value, if and only if the following conditions are met:

- the function f(x) is periodic with period (b-a)
- the function f(x) is single valued when a \le x \le b
- the function f(x) has a finite number of maxima and minima when a \le x \le b
- the function f(x) is continuous when a \le x \le b

If condition 4. above is not met at some discontinuity for f(x) at x=x_d, then the Fourier series still converges at the midpoint of the jump at x=x_d.