Dirichlet Conditions/Examples/Sine of Reciprocal of x - 1

Example of Dirichlet Conditions

The function:

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

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

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

Around the point $x = 1$, $\map f x$ has an infinite number of local maxima and local minima.

Hence it does not satisfy Dirichlet condition $(\text D 2)$.

$\blacksquare$

Sources

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