trigonometric series

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July 14, 2026

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A trigonometric series is a mathematical series composed of terms involving sines and cosines of multiples of a variable angle. These series are particularly useful for analyzing periodic functions, meaning functions whose values repeat over fixed intervals.12

The concept is closely related to Fourier series, which are infinite series of sines and cosines used to solve differential equations and analyze periodic functions. While trigonometry originally focused on triangles, its applications expanded significantly, influencing fields like engineering, physics, and the development of mechanical devices. The study of these series has roots in ancient mathematics, with contributions from Indian and Islamic scholars, eventually leading to modern mathematical analysis.234

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analysis
Trigonometric series solutions
In 1748, in response to d’Alembert’s work, the Swiss mathematician Leonhard Euler wrote a paper, Sur la vibration des cordes (“On the Vibrations of Strings”). In it he repeated d’Alembert’s derivation of the wave equation for a string, but he obtained a…
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In 1748, in response to d’Alembert’s work, the Swiss mathematician Leonhard Euler wrote a paper, Sur la vibration des cordes (“On the Vibrations of Strings”). In it he repeated d’Alembert’s derivation of the wave equation for a string, but he obtained a…
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Georg Cantor
Early life and training
…turned to the theory of trigonometric series, in which he extended the concept of real numbers. Starting from the work on trigonometric series and on the function of a complex variable done by the German mathematician Bernhard Riemann in 1854, Cantor in 1870 showed that such a function can be…
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…turned to the theory of trigonometric series, in which he extended the concept of real numbers. Starting from the work on trigonometric series and on the function of a complex variable done by the German mathematician Bernhard Riemann in 1854, Cantor in 1870 showed that such a function can be…
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mathematics
Riemann’s influence
His work on trigonometric series, for example, led to a deepening investigation of the question of when a function is integrable. Attention was concentrated on the nature of the sets of points at which functions and their integrals (when these existed) had unexpected properties. The conclusions that emerged…
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His work on trigonometric series, for example, led to a deepening investigation of the question of when a function is integrable. Attention was concentrated on the nature of the sets of points at which functions and their integrals (when these existed) had unexpected properties. The conclusions that emerged…
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