Exponential Form of Fourier Series

We have Trigonometric fourier series,

$$\begin{array}{l} f(t) = {a_0} + {a_1}\cos {\omega}t + {a_2}\cos 2{\omega}t + .... + {a_n}\cos n{\omega}t + {b_1}\sin {\omega}t + {b_2}\sin 2{\omega}t + .... + {b_n}\sin n{\omega}t\\  = {a_0} + \sum\limits_{n = 1}^\infty  {\left( {{a_n}\cos n{\omega}t + {b_n}\sin n{\omega}t} \right)} \end{array}$$

We know that,

                       $\sin n\omega t = \frac{{{e^{jn\omega t}} - {e^{ - jn\omega t}}}}{{2j}}$

and                  $\cos n\omega t = \frac{{{e^{jn\omega t}} + {e^{ - jn\omega t}}}}{2}$

Thus,

$$\begin{array}{l} f(t) = {a_0} + \sum\limits_{n = 1}^\infty  {\left[ {{a_n}\frac{{({e^{jn\omega t}} + {e^{ - jn\omega t}})}}{2} + {b_n}\frac{{({e^{jn\omega t}} - {e^{ - jn\omega t}})}}{{2j}}} \right]} \\  = {a_0} + \sum\limits_{n = 1}^\infty  {} \left[ {(\frac{{{a_n} - j{b_n}}}{2}){e^{jn\omega t}} + (\frac{{{a_n} + j{b_n}}}{2}){e^{ - jn\omega t}}} \right] \end{array}$$

Let,

$$\begin{array}{l} {C_0} = {a_0}\\ {C_n} = \frac{{{a_n} - j{b_n}}}{2}\\ {C_{ - n}} = \frac{{{a_n} + j{b_n}}}{2} \end{array}$$

And the series becomes,

$f(t) = {C_0} + \sum\limits_{n = 1}^\infty  {\left[ {{C_n}{e^{jn\omega t}} + {C_{ - n}}{e^{ - jn\omega t}}} \right]}$

This is the exponential form of the Fourier series.

Now ,

$$\begin{array}{l} {C_n} = \frac{{{a_n} - j{b_n}}}{2} = \frac{1}{T}\int\limits_0^T {f(t)(\cos n\omega t - j\sin n\omega t)dt} \\  = \frac{1}{T}\int\limits_0^T {f(t)} {e^{ - jn\omega t}}dt \end{array}$$

This equation is valid for both positive, negative and zero values of n.