# Finding Trigonometric Fourier Series of Functions of Period 2L

So far, all of the formulas we have looked at regarding trigonometric Fourier series have been for functions $f$ that are defined on an interval of length $2\pi$ and/or are $2\pi$-periodic. We would like to generalize obtaining a trigonometric Fourier series for functions that are instead of period $2L$. Fortunately, this is rather simple.

Suppose that $f$ is $2L$-periodic ($L > 0$), i.e., $f(x + 2L) = f(x)$ for all $x \in \mathbb{R}$. Then the trigonometric Fourier series of $f$ is given by:

(1)Where $\displaystyle{c_n = \frac{1}{L} \int_{-L}^{L} f(t) \cos \left ( \frac{n \pi t}{L} \right ) \: dt}$ and $\displaystyle{d_n = \frac{1}{L} \int_{-L}^{L} f(t) \sin \left ( \frac{n \pi t}{L} \right ) \: dt }$.

For example, suppose that we want to find the Fourier series of the function $f(x) = \mid x \mid$ for $-1 \leq x \leq 1$ and $f(x + 2) = f(x)$ for all $x \in \mathbb{R}$ ($f$ is $2$-periodic). Then $L = 1$ and:

(2)We use integration by parts to evaluate the integrals above. Let $u = t$ and let $dv = \cos n \pi t$. Then $du = dt$ and $\displaystyle{v = \frac{\sin n \pi t}{n \pi}}$, so:

(4)Evaluating the righthand side of the equation above at $t = 0$ and $t = 1$ yields:

(5)We similarly find that:

(6)Thus, by plugging $(**)$ and $(***)$ into $(*)$ we see that:

(7)Furthermore we have that:

(8)Note that $\mid t \mid$ is an even function and $\sin n \pi t$ is an odd function on $[-L, L] = [-1, 1]$, so $\mid t \mid \sin n \pi t$ is an odd function on the symmetric interval $[-1, 1]$ so the integral $(****)$ above is equal to $0$ and the trigonometric Fourier series of $f$ is therefore:

(9)The graph of the first three approximations of $f$ (in red, yellow, and green) are graphed below alongside with $f$ itself (in blue):

Clearly the Fourier series of $f$ converges to $f$ on all of $\mathbb{R}$. In particular, the series converges at $x = 1$ to $f(1)$, and consequentially:

(10)Therefore:

(11)Finding the sum of the series above would have been much more difficult without the use of Fourier series.