Dirichlet Conditions/Examples/Reciprocal of 4 minus x squared
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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