Dated: 30-10-2024
Periodic Functions
A function1 is called periodic function if its values repeat after regular intervals of the independent variable.
This interval is called period of oscillations.
\[f(x + p) = f(x)\]
Graph of \(y = \sin(x)\)
This function1 repeats its values after \(x = 2 \pi\).
Its range1 is \([-1, 1]\), meaning it can only produce a maximum absolute value2 of \(1\).
Therefore, its amplitude is \(1\).
Example
| Functions | Amplitude | Period |
|---|---|---|
| \(y= 3 \sin(5x)\) | \(3\) | \(72^\circ\) |
| \(y = 2 \cos(\frac x 2)\) | \(2\) | \(720^\circ\) |
In general if we have something like \(y = a \sin (n \cdot x)\) then \(a\) is called the amplitude and period is \(\frac{360^\circ}{n}\).
Useful Integrals3
-
\[\int_{- \pi}^\pi \sin(n \cdot x) dx = \left[\frac{- \cos(n \cdot x)}{n}\right]_{- \pi}^\pi = \frac 1 n (- \cos (n \cdot \pi) + \cos(n \cdot x)) = 0\]
-
\[\int_{- \pi}^\pi \cos(n \cdot x) dx = \left[\frac{\sin(n \cdot x)}{n}\right]_{- \pi}^\pi = \frac 1 n (\sin (n \cdot \pi) + \sin(n \cdot x)) = 0\]
-
\[\int_{-\pi}^{\pi}\sin^{2}(n \cdot x)dx=\frac{1}{2}\int_{-\pi}^{\pi}(1-\cos(2n \cdot x))dx=\frac{1}{2}\left[x-\frac{\sin(2n \cdot x)}{2n}\right]_{-\pi}^{\pi}=\pi\]
-
\[\int_{-\pi}^{\pi}\cos^{2}(n \cdot x)dx=\frac{1}{2}\int_{-\pi}^{\pi}(1+\cos(2n \cdot x))dx=\frac{1}{2}\left[x+\frac{\sin(2n \cdot x)}{2n}\right]_{-\pi}^{\pi}=\pi\]
-
\[\int_{-\pi}^{\pi}\sin(n \cdot x)\cos(m \cdot x)dx=\frac{1}{2}\int_{-\pi}^{\pi}(\sin(n+m)x+\sin(n-m)x)dx=\frac{1}{2}(0+0) = 0\]
-
\[\int_{-\pi}^{\pi}\cos(n \cdot x)\cos(m \cdot x)dx=\frac{1}{2}\int_{-\pi}^{\pi}(\cos(n+m)x+\cos(n-m)x)dx=\frac{1}{2}(0+0)=0\]
References
Read more about notations and symbols.