One sees from the formulae

• $\displaystyle a_n = \frac{2}{\pi}\int_0^\pi\!f(x)\cos{nx}\,dx$ , $b_n = 0$ $\forall n$ if $f$ is an even function;
• $\displaystyle b_n = \frac{2}{\pi}\int_0^\pi\!f(x)\sin{nx}\,dx$ , $a_n = 0$ $\forall n$ if $f$ is an odd function.

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

• $\displaystyle a_n = \frac{2}{p}\int_0^p\!f(x)\cos\frac{n\pi x}{p}\,dx$ , $b_n = 0$ $\forall n$ if $f$ is an even function;
• $\displaystyle b_n = \frac{2}{p}\int_0^p\!f(x)\sin\frac{n\pi x}{p}\,dx$ , $a_n = 0$ $\forall n$ if $f$ 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 $f: x\mapsto |x|$ . Then we get $$a_0 = \frac{2}{\pi}\int_0^\pi x\,dx = \pi$$ and, integrating by parts, $$a_n = \frac{2}{\pi}\int_0^\pi\!x\cos{nx}\,dx = \frac{2}{\pi}\left[\sijoitus{0}{\quad\pi}\!x\frac{\sin{nx}}{n} -\int_0^\pi \frac{\sin{nx}}{n}\,dx\right] = \frac{2}{\pi}\sijoitus{0}{\quad\pi}\!\frac{\cos{nx}}{n^2} = \frac{2}{\pi n^2}((-1)^n\!-\!1));$$ this equals to $\displaystyle-\frac{4}{\pi n^2}$ if $n$ is an odd integer, but vanishes for each even $n$ . Thus we obtain on the interval $[0,\,\pi]$ the cosine series $$x \,\equiv\, \frac{\pi}{2}\!-\!\frac{4}{\pi}\!\left(\frac{\cos{x}}{1^2}\!+\!\frac{\cos{3x}}{3^2} \!+\!\frac{\cos{5x}}{5^2}\!+\cdots\right).$$ Chosing here $x := 0$ one gets the result $$\frac{\pi^2}{8} \;=\; 1+\frac{1}{3^2}+\frac{1}{5^2}+\ldots$$ (cf. the entry on Dirichlet 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 $\{(x,\,y)\in \mathbb{R}^2\,\vdots\,\, 0\le x \le a,\,0\le y \le b\}$ :

(1)

(2)

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

Bibliography

1

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