Fourier sine and cosine series

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. The value of term c is L which is always taken as half of the total interval function f is defined in or we are concerned in .. In here we are using L instead of c .

Sine seriesEdit

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


which is just a form of complete Fourier series with the only difference that   and   is zero, and the series is defined for half of the interval.

In the formula we have ....


Cosine seriesEdit

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 alsoEdit


  • 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.