PlanetMath: Fourier sine and cosine series

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Fourier sine and cosine series

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One sees from the formulae

$\displaystyle a_n$	$\displaystyle = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos{nx}\,dx,$
$\displaystyle b_n$	$\displaystyle = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin{nx}\,dx$

of the coefficients

and

for the Fourier series expansion

$\displaystyle f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty(a_n\cos{nx}+b_n\sin{nx})$

of the Riemann integrable real function

on the interval $[-\pi,\,\pi]$ , that

$\displaystyle a_n = \frac{2}{\pi}\int_0^\pi\!f(x)\cos{nx}\,dx$ , $\forall n$ if is an even function;
$\displaystyle b_n = \frac{2}{\pi}\int_0^\pi\!f(x)\sin{nx}\,dx$ , $\forall n$ if is an odd function.

Thus the Fourier series of an even function contains mere cosine terms and of an odd function mere sine terms. This concerns the whole interval $[-\pi,\,\pi]$ . So e.g. one has on this interval

$\displaystyle x \,\equiv\, 2\!\left(\frac{\sin{x}}{1}\!-\!\frac{\sin{2x}}{2}\!+\!\frac{\sin{3x}}{3}\! -+\cdots\right).$

Remark 1. On the half-interval $[0,\,\pi]$ one can in any case expand each Riemann integrable function both to a cosine series and to a sine series, irrespective of how it is defined for the negative half-interval or is it defined here at all.

Remark 2. On an interval $[-p,\,p]$ , instead of $[-\pi,\,\pi]$ , the Fourier coefficients of the series

$\displaystyle f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty\left(a_n\cos\frac{n\pi x}{p} +b_n\sin\frac{n\pi x}{p}\right)$

have the expressions

$\displaystyle a_n = \frac{2}{p}\int_0^p\!f(x)\cos\frac{n\pi x}{p}\,dx$ , $\forall n$ if is an even function;
$\displaystyle b_n = \frac{2}{p}\int_0^p\!f(x)\sin\frac{n\pi x}{p}\,dx$ , $\forall n$ if is an odd function.

Example. Expand the identity function $x\mapsto x$ to a Fourier cosine series on $[0,\,\pi]$ .

This odd function may be replaced with the even function $f: x\mapsto \vert x\vert$ . Then we get

$\displaystyle a_0 = \frac{2}{\pi}\int_0^\pi x\,dx = \pi$

and, integrating by parts,

$\displaystyle a_n = \frac{2}{\pi}\int_0^\pi\!x\cos{nx}\,dx = \frac{2}{\pi}\left... ...\!\!\!0}^{\,\quad\pi}\!\frac{\cos{nx}}{n^2} = \frac{2}{\pi n^2}((-1)^n\!-\!1));$

this equals to $\displaystyle-\frac{4}{\pi n^2}$ if

is an odd integer, but vanishes for each even

. Thus we obtain on the interval $[0,\,\pi]$ the cosine series

$\displaystyle x \,\equiv\, \frac{\pi}{2}\!-\!\frac{4}{\pi}\!\left(\frac{\cos{x}}{1^2}\!+\!\frac{\cos{3x}}{3^2} +\!\frac{\cos{5x}}{5^2}\!+\cdots\right).$

Fourier double series. The Fourier sine and cosine series introduced in Remark 1 on the half-interval $[0,\,\pi]$ for a function of one real variable may be generalized for e.g. functions of two real variables on a rectangle $\{(x,\,y)\in \mathbb{R}^2\,\vdots\,\, 0\le x \le a,\,0\le y \le b\}$ :

$\displaystyle f(x,\,y) = \sum_{m=1}^\infty\sum_{n=1}^\infty c_{mn}\sin\frac{m\pi x}{a} \sin\frac{n\pi y}{b},$

(1)

$\displaystyle f(x,\,y) = \frac{d_{00}}{4}+\frac{1}{2}\sum_{l=1}^\infty \left(d_... ...m_{m=1}^\infty\sum_{n=1}^\infty d_{mn}\cos\frac{m\pi x}{a} \cos\frac{n\pi y}{b}$

(2)

The coefficients of the Fourier double sine series (1) are

$\displaystyle c_{mn} = \frac{4}{ab} \int_0^a\int_0^b f(x,\,y)\,\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{b}\,dx\,dy$

where $m = 1,\,2,\,3,\,\ldots$ and $n = 1,\,2,\,3,\,\ldots$ The coefficients of the Fourier double cosine series (2) are

$\displaystyle d_{mn} = \frac{4}{ab} \int_0^a\int_0^b f(x,\,y)\,\cos\frac{m\pi x}{a}\cos\frac{n\pi y}{b}\,dx\,dy$

where $m = 0,\,1,\,2,\,\ldots$ and $n = 0,\,1,\,2,\,\ldots$

Note. One can use in the double series of (1) and (2) also the diagonal summing, e.g. for the double sine series as follows:
$c_{11}\sin\!\frac{\pi x}{a}\sin\!\frac{\pi y}{b}\!+\! \left(c_{12}\sin\!\frac{... ...pi y}{b}\!+\! c_{31}\sin\!\frac{3\pi x}{a}\sin\!\frac{\pi y}{b}\right)\!+\ldots$

Bibliography

1: K. V¨AISÄLÄ: Matematiikka IV. Hand-out Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).

"Fourier sine and cosine series" is owned by pahio.

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Also defines:

Fourier sine series, Fourier cosine series, sine series, cosine series, half-interval, Fourier double sine series, Fourier double cosine series

Keywords:

odd function, even function

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Cross-references: diagonal summing, rectangle, variable, real, double series, even, vanishes, odd integer, expressions, series, Fourier coefficients, negative, function, expand, sine, cosine, odd function, even function, interval, real function, Riemann integrable, Fourier series, coefficients
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This is version 18 of Fourier sine and cosine series, born on 2006-02-23, modified 2007-12-13.
Object id is 7650, canonical name is FourierSineAndCosineSeries.
Accessed 9388 times total.

Classification:

AMS MSC:

11F30 (Number theory :: Discontinuous groups and automorphic forms :: Fourier coefficients of automorphic forms)

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