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 $[0,L]$ .

Sine series

The Fourier sine series of f is defined to be

\sum _{n=1}^{\infty }c_{n}\sin {\frac {n\pi x}{L}}

where

c_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin {\frac {n\pi x}{L}}\,dx,n\in \mathbb {N}

.

If f is continuous and $f(0)=f(L)=0$ , then the Fourier sine series of f is equal to f on $[0,L]$ , odd, and periodic with period $2L$ .

Cosine series

The Fourier cosine series is defined to be

{\frac {c_{0}}{2}}+\sum _{n=1}^{\infty }c_{n}\cos {\frac {n\pi x}{L}}

where

c_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\cos {\frac {n\pi x}{L}}\,dx,n\in \mathbb {N} _{0}

.

If f is continuous, then the Fourier sine series of f is equal to f on $[0,L]$ , even, and periodic with period $2L$ .

Remarks

This notion can be generalized to functions which are not continuous.

References

Haberman, Richard. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (4th ed.). Pearson. pp. 97–113. ISBN 978-0130652430.

@@ Line 1: / Line 1: @@
-{{unreferenced|date=October 2012}}
 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, ''f'' denotes a real valued function on <math>[0,L]</math>.
 ==Sine series==
-If f(x) is an [[odd function]], then the Fourier sine series of f is defined to be
+The Fourier sine series of f is defined to be
-:<math> f(x) = \sum_{n=1}^\infty c_n \sin \frac{n\pi x}{L}</math>
+:<math> \sum_{n=1}^\infty c_n \sin \frac{n\pi x}{L}</math>
 where
 :<math>c_n=\frac{2}{L}\int_0^L f(x) \sin \frac{n\pi x}{L} \, dx, n\in \mathbb{N}</math>.
+If f is continuous and <math>f(0)=f(L)=0</math>, then the Fourier sine series of f is equal to f on <math>[0,L]</math>, odd, and periodic with period <math>2L</math>.
 ==Cosine series==
-If f(x) is an [[even function]], then the Fourier cosine series is defined to be
+The Fourier cosine series is defined to be
-:<math> f(x)=\frac{c_0}{2} + \sum_{n=1}^\infty c_n \cos \frac{n\pi x}{L}</math>
+:<math> \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>.
+If f is continuous, then the Fourier sine series of f is equal to f on <math>[0,L]</math>, even, and periodic with period <math>2L</math>.
 ==Remarks==
-This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.
+This notion can be generalized to functions which are not continuous.
 ==See also==
@@ Line 26: / Line 29: @@
 *[[Fourier analysis]]
+== References ==
+{{cite book|last1=Haberman|first1=Richard|title=Applied Partial Differential Equations with Fourier Series and Boundary Value Problems|publisher=Pearson|isbn=978-0130652430|pages=97-113|edition=4th}}
 [[Category:Fourier series]]