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A Fourier series is a method of approximating a periodic function as a (possibly infinite) sum of sinusoidal functions by decomposing it into individual signals. Any odd function can be purely constructed from sines, and likewise any even function can be constructed purely from cosines. By combining them, a series representation can be found for any periodic function. Unlike a Taylor series, the Fourier series does not require all the derivatives of a function to exist.

The formula for the Fourier series of a periodic function is

f(x) = \sum_{n = 0}^{\infty} A_n \cos(\tfrac{n \pi x}{P}) + \sum_{n = 1}^{\infty} B_n \sin(\tfrac{n \pi x}{P})
A_n =
\begin{cases}
\tfrac{1}{2P} \int^P_{-P} f(x) dx, & n = 0 \\ 
\tfrac{1}{P} \int^P_{-P} f(x) \cos(\tfrac{n \pi x}{P}) dx, & n > 0
\end{cases}
B_n = \tfrac{1}{P} \int^P_{-P} f(x) \sin(\tfrac{n \pi x}{P}) dx

P being the radius of convergence and n being a positive integer.

By Euler's formula, these can be combined into the single series.

f(x) = \sum_{n = - \infty }^{\infty} C_n e^{\frac{2 \pi i n x}{T}}
C_n = \tfrac{1}{P} \int^P_0 f(x) e^{-\frac{2 \pi i n x}{T}} dx

If

C_n^* = C_{-n}

Cn* being the complex conjugate of Cn, f(x) and its Fourier series will be entirely real.

See also

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