Complex form of fourier series

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:-

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.