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Important detail about the pulse wave Tags: Visual edit Mobile edit Mobile web edit |
→Non-smooth functions: I added the Fourier series to the Cycloid. I also made a slight adjustment to the Fourier series for the Square wave and Sawtooth wave, namely I replaced 2nπ with 2πn, just for the sake of consistency. |
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</math>|| non-continuous first derivative
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|| [[Sawtooth wave]] || <math>2 \left( {\frac x p} - \left \lfloor {\frac 1 2} + {\frac x p} \right \rfloor \right)</math> ||<math> \frac2\pi\sum_{n=1}^\infty\frac{(-1)^{n-1}}n\sin\left(\frac{
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|| [[Square wave]] || <math> \sgn\left(\sin \frac{2\pi x}{p} \right) </math> ||<math> \frac4\pi\sum_{n\,\mathrm{odd}}^\infty\frac1n\sin\left(\frac{
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|| [[Cycloid]] ||<math>\frac{p - p\cos \left( f^{(-1)}\left( \frac{2\pi x}{p} \right) \right)}{2\pi}</math>
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<small>its real-valued inverse</small><small>.</small>
| <math>\frac{p}{\pi} \biggl(\frac{3}4 + \sum_{n=1}^\infty \frac{\operatorname{J}_n(n)-\operatorname{J}_{n-1}(n)}n \cos\Bigl(\frac{2\pi nx}p\Bigr)\biggr)</math>
| - || non-continuous first derivative▼
<small>where <math>\operatorname{J}_n(x)</math> is the [[Bessel function|Bessel Function of the first kind]].</small>
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|| [[Pulse wave ]] ||<math>H \left( \cos \left( \frac{2\pi x}{p} \right) - \cos \left( \frac{\pi t}{p} \right) \right)</math>
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