Fourier sine and cosine series

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In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.


In this article, f denotes a real valued function on which is periodic with period 2L.

Sine series[edit]

If f(x) is an odd function, then the Fourier sine series of f is defined to be



Cosine series[edit]

If f(x) is an even function, then the Fourier cosine series is defined to be




This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.

See also[edit]


  • Byerly, William Elwood (1893). "Chapter 2: Development in Trigonometric Series". An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics (2 ed.). Ginn. p. 30. 
  • Carslaw, Horatio Scott (1921). "Chapter 7: Fourier's Series". Introduction to the Theory of Fourier's Series and Integrals, Volume 1 (2 ed.). Macmillan and Company. p. 196.