Fourier Series Definition

Periodic Functions

A function $f(x)$ is periodic with period $2L$ if:

$f(x + 2L) = f(x)$

for all $x$.

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Example: Temperature Variation

Consider a function defined on $-L \leq x < L$ and extended periodically. Fourier series represent such functions as sums of sines and cosines.

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The Fourier Series

If $f(x)$ is periodic with period $2L$ and is “sufficiently nice” (piecewise continuous with finite discontinuities), then:

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

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Fourier Coefficients

The coefficients are found by integration:

$a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx$

$b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx$

For $n = 0$:

$a_0 = \frac{1}{L} \int_{-L}^{L} f(x)\,dx$

Interpretation: $a_0/2$ is the average value of $f$ over one period.

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Orthogonality of Trig Functions

The Fourier coefficient formulas come from orthogonality:

$\int_{-L}^{L} \cos\left(\frac{m\pi x}{L}\right) \cos\left(\frac{n\pi x}{L}\right) dx = \begin{cases} 0 & m \neq n \\ L & m = n \end{cases}$

$\int_{-L}^{L} \sin\left(\frac{m\pi x}{L}\right) \sin\left(\frac{n\pi x}{L}\right) dx = \begin{cases} 0 & m \neq n \\ L & m = n \end{cases}$

$\int_{-L}^{L} \cos\left(\frac{m\pi x}{L}\right) \sin\left(\frac{n\pi x}{L}\right) dx = 0$

This is analogous to orthogonal vectors — the sines and cosines form an orthogonal “basis”.

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Even and Odd Functions

Even Function: $f(-x) = f(x)$

• $b_n = 0$ for all $n$ (no sine terms)
• $f(x) = \frac{a_0}{2} + \sum a_n \cos\left(\frac{n\pi x}{L}\right)$

Odd Function: $f(-x) = -f(x)$

• $a_n = 0$ for all $n$ (no cosine terms)
• $f(x) = \sum b_n \sin\left(\frac{n\pi x}{L}\right)$

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Example: Triangle Wave

Find Fourier series for $f(x) = |x|$ on $-1 \leq x \leq 1$, period 2.

Since $f$ is even: $b_n = 0$.

$a_0 = \int_{-1}^{1} |x|\,dx = 2\int_0^1 x\,dx = 1$

$a_n = \int_{-1}^{1} |x|\cos(n\pi x)\,dx = 2\int_0^1 x\cos(n\pi x)\,dx$

Integration by parts: $a_n = \frac{2}{n^2\pi^2}[(-1)^n - 1]$

For $n$ even: $a_n = 0$. For $n = 2k+1$ odd: $a_n = -\frac{4}{n^2\pi^2}$.

Result:

$|x| = \frac{1}{2} - \frac{4}{\pi^2}\sum_{k=0}^{\infty} \frac{\cos((2k+1)\pi x)}{(2k+1)^2}$

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Example: Sawtooth Wave

Find Fourier series for $f(x) = x$ on $-1 < x < 1$, period 2.

Since $f$ is odd: $a_n = 0$.

$b_n = \int_{-1}^{1} x\sin(n\pi x)\,dx$

Integration by parts: $b_n = \frac{2(-1)^{n+1}}{n\pi}$

Result:

$x = \frac{2}{\pi}\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\sin(n\pi x)$

$= \frac{2}{\pi}\left(\sin(\pi x) - \frac{1}{2}\sin(2\pi x) + \frac{1}{3}\sin(3\pi x) - \cdots\right)$

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Complex Form

Using Euler’s formula, Fourier series can be written:

$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{in\pi x/L}$

where:

$c_n = \frac{1}{2L}\int_{-L}^{L} f(x)e^{-in\pi x/L}\,dx$

Relations: $c_0 = a_0/2$, $c_n = (a_n - ib_n)/2$ for $n > 0$.