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Here's an demonstration of using the expontential function in a Fourier Series example.
One way of representing a basis function is with cosine
.
Where the Fourier series is
However, a more convient way is using an exponential funtion .
To solve a Fourier series equation for the coefffients using the above expressions result in similar solutions but using the eponetial basis function is simplier to solving. To find the coefficients perform the dot product ' . ' as follows.
Using exponential basis function
.
Then the result is
Using cosine basis function
. At this point you should use a trig identity
applying this trig identity gives
Then the result is