Fourier sine and cosine series

One sees from the formulae

$a_n$$={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\cos nxdx,$

$b_n$$={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\sin nxdx$

of the coefficients $a_n$ and $b_n$ for the Fourier series

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

of the Riemann integrable real function $f$ on the interval  $[-\pi ,\pi ]$,  that

• $a_n={\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\pi }}f(x)\cos nxdx$,  $b_n=0$  $\forall n$  if $f$ is an even function;

• $b_n={\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\pi }}f(x)\sin nxdx$,  $a_n=0$  $\forall n$  if $f$ is an odd function.

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

$$x\equiv \mathrm{\hspace{0.17em}2}(\frac{\sin x}{1}-\frac{\sin 2x}{2}+\frac{\sin 3x}{3}-+\mathrm{\cdots }).$$

Remark 1.  On the half-interval  $[0,\pi ]$  one can in any case expand each Riemann integrable function $f$ 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

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

have the expressions

• $a_n={\displaystyle \frac{2}{p}}{\displaystyle {\int }_0^p}f(x)\cos {\displaystyle \frac{n\pi x}{p}}dx$,  $b_n=0$  $\forall n$  if $f$ is an even function;

• $b_n={\displaystyle \frac{2}{p}}{\displaystyle {\int }_0^p}f(x)\sin {\displaystyle \frac{n\pi x}{p}}dx$,  $a_n=0$  $\forall n$  if $f$ is an odd function.

Example.  Expand the identity function (http://planetmath.org/IdentityMap)  $x\mapsto x$  to a Fourier cosine series on  $[0,\pi ]$.

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

$$a_0=\frac{2}{\pi }{\int }_0^{\pi }x𝑑x=\pi$$

and, integrating by parts,

$$a_n=\frac{2}{\pi }{\int }_0^{\pi }x\cos nxdx=\frac{2}{\pi }[\underset{0}{\overset{\pi }{/}}x\frac{\sin nx}{n}-{\int }_0^{\pi }\frac{\sin nx}{n}dx]=\frac{2}{\pi }\underset{0}{\overset{\pi }{/}}\frac{\cos nx}{n^2}=\frac{2}{\pi n^2}({(-1)}^n-1));$$

this equals to $-{\displaystyle \frac{4}{\pi n^2}}$ if $n$ is an odd integer, but vanishes for each even $n$.  Thus we obtain on the interval  $[0,\pi ]$  the cosine series

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

Chosing here  $x:=0$  one gets the result

$$\frac{{\pi }^2}{8}=\mathrm{\hspace{0.33em}1}+\frac{1}{3^2}+\frac{1}{5^2}+\mathrm{\dots }$$

(cf. the entry on http://planetmath.org/node/11010Dirichlet eta function at 2).

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  $\{(x,y)\in {ℝ}^2\mathrm{⋮}\mathrm{\hspace{0.17em}\hspace{0.17em}0}\le x\le a,\mathrm{\hspace{0.17em}0}\le y\le b\}$:

$f(x,y)={\displaystyle \sum _{m=1}^{\mathrm{\infty }}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{c}_{mn}\sin {\displaystyle \frac{m\pi x}{a}}\sin {\displaystyle \frac{n\pi y}{b}},$(1)

$f(x,y)={\displaystyle \frac{d_{00}}{4}}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{l=1}^{\mathrm{\infty }}}\left(d_{l0}\cos {\displaystyle \frac{l\pi x}{a}}+d_{0l}\cos {\displaystyle \frac{l\pi y}{b}}\right)+{\displaystyle \sum _{m=1}^{\mathrm{\infty }}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{d}_{mn}\cos {\displaystyle \frac{m\pi x}{a}}\cos {\displaystyle \frac{n\pi y}{b}}$(2)

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

$$c_{mn}=\frac{4}{ab}{\int }_0^a{\int }_0^{b}f(x,y)\sin \frac{m\pi x}{a}\sin \frac{n\pi y}{b}dxdy$$

where  $m=1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots }$  and  $n=1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots }$  The coefficients of the Fourier double cosine series (2) are correspondingly

$$d_{mn}=\frac{4}{ab}{\int }_0^a{\int }_0^{b}f(x,y)\cos \frac{m\pi x}{a}\cos \frac{n\pi y}{b}dxdy$$

where  $m=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }$  and  $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }$

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:
$c_{11}\sin \frac{\pi x}{a}\sin \frac{\pi y}{b}+\left(c_{12}\sin \frac{\pi x}{a}\sin \frac{2\pi y}{b}+c_{21}\sin \frac{2\pi x}{a}\sin \frac{\pi y}{b}\right)+\left(c_{13}\sin \frac{\pi x}{a}\sin \frac{3\pi y}{b}+c_{22}\sin \frac{2\pi x}{a}\sin \frac{2\pi y}{b}+c_{31}\sin \frac{3\pi x}{a}\sin \frac{\pi y}{b}\right)+\mathrm{\dots }$