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It was introduced by the French mathematician Joseph Fourier in the early 19th century and has since become a fundamental tool in the field of mathematical analysis. The basic idea behind a Fourier series is to decompose a periodic function into a series of sine and cosine functions with different frequencies and amplitudes. This allows for the representation of a wide range of periodic functions using a relatively small number of terms. The Fourier series has numerous applications in mathematics, physics, and engineering. It is used, for example, in solving differential equations, analyzing the behavior of vibrating systems, and representing periodic signals in electrical engineering and signal processing. The Wikipedia page on Fourier series provides a comprehensive overview of the subject, covering its history, mathematical formulation, properties, and applications. It also includes detailed explanations of various concepts related to Fourier series, such as Fourier coefficients, Fourier transforms, and the convergence of Fourier series. In addition, the page discusses various extensions and generalizations of the Fourier series, such as the Fourier transform, Fourier series for non-periodic functions, and the discrete Fourier transform. It also provides references to further reading and external resources for those interested in delving deeper into the topic. Overall, the Wikipedia page on Fourier series serves as a valuable resource for anyone seeking to understand the mathematical foundations and practical applications of this important mathematical tool.