Dated: 30-10-2024
Examples
Example
Find the fourier series1 for the following function2
\[f(x) = -x \quad - \pi < x < 0\]
\[f(x) = 0 \quad 0 < x < \pi\]
\[f(x) = f(x + 2\pi)\]
Solution
\(a_0\)
\[a_{0}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)dx\]
\[=\frac{1}{\pi}\left(\int_{-\pi}^{0}(-x)dx+\int_{0}^{\pi}0~dx\right)\]
\[=\frac{1}{\pi}\int_{-\pi}^{0}(-x)dx\]
\[=\frac{1}{\pi}\left[-\frac{x^{2}}{2}\right]_{-\pi}^{0}\]
\[=\frac{\pi}{2}\]
\(a_n\)
\[a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx\]
\[=\frac{1}{\pi}\int_{-\pi}^{0}(-x)\cos(nx)dx+\frac{1}{\pi}\int_{0}^{\pi}0~dx\]
\[=-\frac{1}{\pi}\int_{-\pi}^{\pi}x \cdot \cos(nx)dx\]
\[=-\frac{1}{\pi}\left(\left[x\frac{\sin(nx)}{n}\right]_{-\pi}^{0}-\frac{1}{n}\int_{-\pi}^{0}\sin(nx)dx\right)\]
\[=-\frac{1}{\pi}\left((0-0)-\frac{1}{n}\left[\frac{-\cos(nx)}{n}\right]_{-\pi}^{0}\right)\]
\[=-\frac{1}{\pi}\left(\frac{1}{n}\left[\frac{\cos(nx)}{n}\right]_{-\pi}^{0}\right)\]
\[=-\frac{1}{\pi n^{2}}\left(\left[\frac{\cos(nx)}{n}\right]_{-\pi}^{0}\right)\]
\[=-\frac{1}{\pi n^{2}}\left(\cos(0)-\cos(n)\pi\right)\]
\[= - \frac 1 {\pi n^2} (1 - \cos (n \pi))\]
If \(n\) is even
\[a_n = - \frac 2 {\pi n^2}\]
If \(n\) is Odd
\[a_n = 0\]
\(b_n\)
\[b_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)dx\]
\[=\frac{1}{\pi}\int_{-\pi}^{0}(-x)\sin(nx)dx\]
\[=-\frac{1}{\pi}\int_{-\pi}^{\pi}x~\sin(nx)dx\]
\[=-\frac{1}{\pi}\left(\left[x\frac{\cos(nx)}{n}\right]_{-\pi}^{0}+\frac{1}{n}\int_{-\pi}^{0}\cos(nx)dx\right)\]
\[=-\frac{1}{\pi}\left(\frac{\pi \cdot \cos(n\pi)}{n}+\frac{1}{n}\left[\frac{\sin(nx)}{n}\right]_{-\pi}^{0}\right)\]
\[=\frac{\cos(n\pi)}{n}\]
If \(n\) is even
\[b_n = \frac 1 n\]
If \(n\) is Odd
\[b_n = - \frac 1 n\]
\[\therefore f(x) = \frac \pi 4 - \frac 2 \pi \left(\cos (x) + \frac 1 9 \cos(3x) + \frac 1 {25} \cos(5x) + \ldots \right) + \left(\sin(x) - \frac 1 2 \sin(2x) + \frac 1 3 \sin(3x) + \ldots \right)\]
Products of odd and even functions2
Let's say we have 2 functions,2 \(g(x)\) and \(f(x)\).
Both Even
If both of them are even functions2 then the following is also even function.2
\[h(x) = f(x) \cdot g(x)\]
Both Odd
If both of them are odd functions2 then the following is an even function.2
\[h(x) = f(x) \cdot g(x)\]
One Odd and other Even
If both functions2 differ from each other in their nature then the following is an odd function.2
\[h(x) = f(x) \cdot g(x)\]
Useful Facts
For Even Functions
\[\int_{-a}^a f(x) dx = 2 \int_{- a}^a f(x) dx\]
For Odd Functions
\[\int_{-a}^a f(x) dx = 0\]
Theorems
- If the \(f(x)\) is an
even function2 over theinterval3 \([- \pi, \pi]\) then thefourier series1 consists of only \(\cos\) terms. - If the \(f(x)\) is an
odd function2 over theinterval3 \([- \pi, \pi]\) then thefourier series1 consists of only \(\sin\) terms.
References
Read more about notations and symbols.