# 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.

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