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→Non-smooth functions: I edited the formula for cycloid to yield a closed form expression given the inverse of another function. Please note that some viewers might assume f(x) is an official mathematical function f(x)=x-sin(x), so if there's anything future editors could do to make it clear that that is not the case, that'd be great! |
→Non-smooth functions: The Square Wave was missing its Fourier Series, The Sawtooth Wave had the wrong Fourier Series, and the Triangle Wave's Fourier Series confused two variables i and n, and had incorrect bounds for its summation. I also forgot to mention in the section on Cycloids that r was the radius of the circle. |
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! Name !! Formula !! Fourier Series !! Notes
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|| [[Triangle wave]] || <math> \frac{4}{p} \left (x-\frac{p}{2} \left \lfloor\frac{2 x}{p}+\frac{1}{2} \right \rfloor \right )(-1)^\left \lfloor\frac{2 x}{p}+\frac{1}{2} \right \rfloor</math> ||
</math>
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|| [[Sawtooth wave]] || <math>2 \left( {\frac x p} - \left \lfloor {\frac 1 2} + {\frac x p} \right \rfloor \right)</math> ||
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|| [[Square wave]] || <math> \sgn\left(\sin \frac{2\pi x}{p} \right) </math> ||<math>
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|| [[Cycloid]] ||<math>r(1-\cos(f^{(-1)}\Bigl(\frac{x}{r}\Bigr)))</math>
<small>where <math>f(x)=x-\sin(x)</math>
<small>its real valued inverse
<small>circle.</small>
| - || non-continuous first derivative
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