# Definition:Dirichlet Conditions

## Definition

Let $\alpha, \beta \in \R$ be a real numbers such that $\alpha < \beta$.

Let $\map f x$ be a real function which is defined and bounded on the interval $\openint \alpha \beta$.

The Dirichlet conditions on $f$ are sufficient conditions that $f$ must satisfy on $\openint \alpha \beta$ in order for:

the Fourier series of $f$ at every $x$ in $\openint \alpha \beta$ to equal $\map f x$
the behaviour of a Fourier series to be determined at finite discontinuities of $f$ in $\openint \alpha \beta$:

They are as follows:

 $(\text D 1)$ $:$ $f$ is absolutely integrable $(\text D 2)$ $:$ $f$ has a finite number of local maxima and local minima $(\text D 3)$ $:$ $f$ has a finite number of discontinuities, all of them finite

## Examples

### Reciprocal of $4 - x^2$

The function:

$\map f x = \dfrac 1 {4 - x^2}$

does not satisfy the Dirichlet conditions on the real interval $\openint 0 {2 \pi}$.

### Sine of $\dfrac 1 {x - 1}$

The function:

$\map f x = \map \sin {\dfrac 1 {x - 1} }$

does not satisfy the Dirichlet conditions on the real interval $\openint 0 {2 \pi}$.

## Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.