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[edit]
In this article, f denotes a real-valued function on which is periodic with period 2L.
Sine series[edit]
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[edit]
If f is an even function with a period , 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.