Fourier Sine Series Example

Fourier Sine series example demonstrating the orthogonality of functions $\displaystyle{f(x) = \sin(n\pi x)}$ and $\displaystyle{g(x) = \sin(m\pi x)}$ for $m \neq n$.

The inner product of two vectors (real integrable functions on the interval [0, 1]) is defined as $\displaystyle{f \cdot g = \int_0^1} f(x)g(x) dx$

For the case $m \ne n$ and using the trigonometric identity $2\sin\alpha \sin\beta = \cos(\alpha - \beta) + \cos(\alpha + \beta)$, the following inner product results

$\begin{align*} 2\int_0^1 \sin[m\pi x] \sin[n\pi x]dx &= \int_0^1 \cos[m\pi x - n\pi x]dx - \int_0^1 \cos[m\pi x+ n\pi x] dx \\ &= \int_0^1 \cos[(m - n)\pi x]dx - \int_0^1 \cos[(m + n)\pi x] dx \\ &= \left.\frac{1}{(m-n)\pi}\sin[(m-n)\pi x]\right|_0^1 - \left.\frac{1}{(m+n)\pi}\sin[(m+n)\pi x]\right|_0^1\\ &= \frac{1}{(m-n)\pi}\sin[(m-n)\pi \cdot 1] - \frac{1}{(m-n)\pi}\sin[(m-n)\pi \cdot 0] - \frac{1}{(m+n)\pi}\sin[(m+n)\pi \cdot 1] + \frac{1}{(m+n)\pi}\sin[(m+n)\pi \cdot 0] \hspace{0.25in} (m-n \mbox{ and } m+n \mbox{ are integer values}) \\ &= 0 - 0 - 0 + 0 \end{align*}$

Coefficient Array for the case for m = 2: