Dirichlet Conditions/Examples/Reciprocal of 4 minus x squared

Example of Dirichlet Conditions

The function:

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

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

Proof

Recall the Dirichlet conditions:

\((\text D 1)\)	$:$							$f$ 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

At the point $x = 2$, $\dfrac 1 {4 - x^2}$ is not defined.

Let $\epsilon \in \R_{>0}$.

Then:

$\map f {2 - \dfrac \epsilon 2} > \map f {2 - \epsilon}$

and so $x = 2$ is not a finite discontinuity.

Hence $f$ does not satisfy Dirichlet condition $(\text D 3)$.

$\blacksquare$

Sources

• 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 2$. Fourier Series