Complex form of fourier seriesPermalink

The complex form of fourier series is obtained by exaprassiong $cos\frac{n\pi.x}{l}$ and $sin\frac{n\pi.x}{l}$ in expontial form that is, $f(x) = \frac{a_0}{2} + \sum_{i=1}^\infty[a_ncos\frac{n.\pi.x}{l}+b_nsin\frac{n.\pi.x}{l}]\\ f(x) = \frac{a_0}{2} + \sum_{i=1}^\infty[\frac{\epsilon^(\frac{n.\pi.x}{l}+\epsilon^(\frac{-n.\pi.x}{l}}{2}]a_n+ \sum_{i=1}^\infty[\frac{\epsilon^(\frac{n.\pi.x}{l}-\epsilon^(\frac{-n.\pi.x}{l}}{2i}]b_n$

Even Odd function:-Permalink

In simple we know that function is even if

$f(-x) = f(x)$.

and function is odd if

$f(-x) = -f(x)$.

example of even is cos function and example of odd functio is sin function.The fourier coeffient $b_n$ zero it have value for $a_0$ and $a_n$ is odd and fourier cofficent $a_0$ and $a_n$ zero have value for $b_n$ is even.