Fourier sine and cosine series: Difference between revisions

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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

In this article, $f$ denotes a real-valued function on $\mathbb {R}$ which is periodic with period 2L.

Sine series

If $f$ is an odd function with period $2L$ , then the Fourier Half Range sine series of f is defined to be $f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n\pi x}{L}}\right)$ which is just a form of complete Fourier series with the only difference that $a_{0}$ and $a_{n}$ are zero, and the series is defined for half of the interval.

In the formula we have $b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n\pi x}{L}}\right)\,dx,\quad n\in \mathbb {N} .$

Cosine series

If $f$ is an even function with a period $2L$ , then the Fourier cosine series is defined to be $f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }a_{n}\cos \left({\frac {n\pi x}{L}}\right)$ where $a_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\cos \left({\frac {n\pi x}{L}}\right)\,dx,\quad n\in \mathbb {N} _{0}.$

Remarks

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

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.
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.

@@ Line 1: / Line 1: @@
+{{Short description|Special cases of the Fourier 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]].
+{{distinguish-redirect|Sine and cosine series|Sine and cosine#Series definitions}}
+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>\mathbb{R}</math> which is periodic with period 2''L''.
+In this article, {{math|''f''}} denotes a [[real number|real]]-valued [[function (mathematics)|function]] on <math>\mathbb{R}</math> which is [[periodic function|periodic]] with period 2''L''.
 ==Sine series==
-If f(x) is an [[odd function]],  then the Fourier sine series of f is defined to be
+If {{math|''f''}} is an [[odd function]] with period <math>2L</math>, then the Fourier Half Range sine series of ''f'' is defined to be
+<math display="block">f(x) = \sum_{n=1}^\infty b_n \sin \left(\frac{n\pi x}{L}\right) </math>
+which is just a form of complete [[Fourier series]] with the only difference that <math>a_0</math> and <math>a_n</math> are zero, and the series is defined for half of the [[interval (mathematics)|interval]].
+In the formula we have
-:<math> f(x) = \sum_{n=1}^\infty b_n \sin \frac{n\pi x}{L}</math>
+<math display="block">b_n = \frac{2}{L} \int_0^L f(x) \sin \left(\frac{n\pi x}{L}\right) \, dx, \quad n \in \mathbb{N} .</math>
-which ia juat a form of completw fourier series with the only difference that <math> a_0 </math> and <math> a_nn </math> is zero, and the series is defined for half of the interval.
-In the formula we have ....
-:<math>b_n=\frac{2}{L}\int_0^L f(x) \sin \frac{n\pi x}{L} \, dx, n\in \mathbb{N}</math>.
 ==Cosine series==
-If f(x) is an [[even function]], then the Fourier cosine series is defined to be
+If {{math|''f''}} is an [[even function]] with a period <math>2L</math>, then the Fourier cosine series is defined to be
+<math display="block">f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos \left(\frac{n \pi x}{L}\right)  </math>
-:<math> f(x)=\frac{c_0}{2} + \sum_{n=1}^\infty c_n \cos \frac{n\pi x}{L}</math>
 where
-:<math>c_n=\frac{2}{L}\int_0^L f(x) \cos \frac{n\pi x}{L} \, dx, n\in \mathbb{N}_0</math>.
+<math display="block">a_n = \frac{2}{L} \int_0^L f(x) \cos \left(\frac{n\pi x}{L}\right) \, dx, \quad n \in \mathbb{N}_0 .</math>
 ==Remarks==
@@ Line 26: / Line 27: @@
 *[[Fourier series]]
 *[[Fourier analysis]]
+*[[Least-squares spectral analysis]]
 ==Bibliography==