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
In this article, f denotes a real-valued function on [math]\displaystyle{ \mathbb{R} }[/math] which is periodic with period 2L.
Sine series
If f is an odd function with period [math]\displaystyle{ 2L }[/math], then the Fourier Half Range sine series of f is defined to be [math]\displaystyle{ f(x) = \sum_{n=1}^\infty b_n \sin \frac{n\pi x}{L} }[/math] which is just a form of complete Fourier series with the only difference that [math]\displaystyle{ a_0 }[/math] and [math]\displaystyle{ a_n }[/math] are zero, and the series is defined for half of the interval.
In the formula we have [math]\displaystyle{ b_n = \frac{2}{L} \int_0^L f(x) \sin \frac{n\pi x}{L} \, dx, \quad n \in \mathbb{N} . }[/math]
Cosine series
If f is an even function with a period [math]\displaystyle{ 2L }[/math], then the Fourier cosine series is defined to be [math]\displaystyle{ f(x) = \frac{c_0}{2} + \sum_{n=1}^{\infty} c_n \cos \frac{n \pi x}{L} }[/math] where [math]\displaystyle{ c_n = \frac{2}{L} \int_0^L f(x) \cos \frac{n\pi x}{L} \, dx, \quad n \in \mathbb{N}_0 . }[/math]
Remarks
This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.
See also
Bibliography
- 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. https://books.google.com/books?id=BMQ0AQAAMAAJ&pg=PA30.
- 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. https://books.google.com/books?id=JNVAAAAAIAAJ&pg=PA196.
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