Prove. If we substitute the Euler exponential equivalents

into the sine and cosine terms of the Fourier series

we obtain the Complex Fourier Series

where

or, equivalently,

Proof. Substituting

into 1) we obtain

Since 1/i = -i, 21) becomes

which becomes

If we now define

7) can be written as

or

Now C₀, C_n and C_-n are given by

We see from the above formulas that whether the index n is positive, negative, or zero, C_n is given by the single formula