Definition

The complex Fourier Series of a periodic function 𝑓(𝑥) of period 𝑃:

$f (x) = \sum_{n = - \infty}^{\infty} C_{n} e^{i 2 π \frac{n}{P} x}$
$f (x) = \sum_{n = - \infty}^{\infty} (𝛼_{n} + i 𝛽_{n}) (cos (2 π \frac{n}{P} x) + i s in (2 π \frac{n}{P} x))$

where:

𝐶_𝑛 is defined by the inner product of functions:
- $C_{n} = ⟨ f (x), e^{i 2 π \frac{n}{P} x} ⟩$
𝑒^𝑖𝑥 is Euler’s Formula

Note that:

$⟨ e^{i 2 π \frac{a _{1}}{P} x}, e^{i 2 π \frac{a _{2}}{P} x} ⟩ = \int_{0}^{P} e^{i 2 π \frac{a _{1}}{P} x} e^{- i 2 π \frac{a _{2}}{P} x} d x$
$⟨ e^{i 2 π \frac{a _{1}}{P} x}, e^{i 2 π \frac{a _{2}}{P} x} ⟩ = \int_{0}^{P} e^{i 2 π \frac{1}{P} x (a_{1} - a_{2})} d x$
$⟨ e^{i 2 π \frac{a _{1}}{P} x}, e^{i 2 π \frac{a _{2}}{P} x} ⟩ = \int_{0}^{P} e^{\frac{i 2 π ( a _{1} - a _{2} )}{P} x} d x$
$⟨ e^{i 2 π \frac{a _{1}}{P} x}, e^{i 2 π \frac{a _{2}}{P} x} ⟩ = \frac{P}{i 2 π ( a _{1} - a _{2} )} [e^{\frac{i 2 π ( a _{1} - a _{2} )}{P} x}]_{0}^{P}$
$⟨ e^{i 2 π \frac{a _{1}}{P} x}, e^{i 2 π \frac{a _{2}}{P} x} ⟩ = {01 if a_{1} \neq = a_{2} if a_{1} = a_{2}$

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