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In mathematics, a Fourier series is a way of representing a periodic function as the sum of a series of sines and cosines. It is named after Jean-Baptiste Joseph Fourier, who introduced the concept in the early 19th century.

A Fourier series can be used to represent any periodic function, no matter how complicated, as long as it is defined over a finite interval. The function is represented as an infinite sum of sines and cosines with different frequencies and amplitudes.

The coefficients of the sines and cosines in the series are called the Fourier coefficients, and they can be calculated using integration.

Fourier series have many applications in mathematics, physics, engineering, and other fields. They are used to analyze and represent periodic signals in signal processing, to solve differential equations, and to analyze the vibrations of mechanical systems, among other things.

A Fourier series is usually written in the form:

f(x) = a0 + ∑(ancos(nx) + bnsin(nx))

where a0 is the constant term, and the series is the sum of the cosine and sine terms with coefficients an and bn, respectively. The coefficients an and bn are found by integrating the function over a period and applying certain formulas.

where a0 is the constant term, and the series is the sum of the cosine and sine terms with coefficients an and bn, respectively. The coefficients an and bn are found by integrating the function over a period and applying certain formulas.