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.

Notation[edit]

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

where

.

Cosine series[edit]

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

where

.

Remarks[edit]

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

See also[edit]

Bibliography[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.