Definition:Dirichlet Conditions

Definition
Let $\alpha \in \R$ be a real number.

Let $f \left({x}\right)$ be a real function which is defined and bounded on the interval $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$.

The Dirichlet conditions on $f$ are sufficient conditions that $f$ must satisfy on $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$ in order for:
 * the Fourier series of $f$ at every $x$ in $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$ to equal $f \left({x}\right)$
 * the behaviour of a Fourier series to be determined at finite discontinuities of $f$ in $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$:

They are as follows:
 * $(\mathrm D 1): \quad$ Let $f$ be absolutely integrable.


 * $(\mathrm D 2): \quad$ Let $f$ have a finite number of local maxima and local minima.


 * $(\mathrm D 3): \quad$ Let $f$ have a finite number of finite discontinuities.