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.

Notation

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

Sine series

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If f is an odd function with period  , 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   are zero, and the series is defined for half of the interval.

In the formula we have  

Cosine series

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If f is an even function with a period  , then the Fourier cosine series is defined to be   where  

Remarks

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

See also

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Bibliography

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  • 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.