Fourier Series

Level 1 - Calculus topic page in Series And Approximations.

Principle

A Fourier series represents a periodic function as a sum of sines and cosines. It decomposes a signal into frequency components.

Notation

\(L\)

half-period for an interval [-L,L]

\(a_n,b_n\)

Fourier coefficients

\(\sin,\cos\)

basic periodic waves

The Core Method

For a function with period \(2L\), the real Fourier series has the form

Fourier series

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

Even functions have only cosine terms. Odd functions have only sine terms. Symmetry is often the fastest simplification.

Worked Cases

Question

If \(f\) is odd on \([-L,L]\), which Fourier coefficients are zero?

Answer

For an odd function, \(f(x)\cos(n\pi x/L)\) is odd because cosine is even, so its symmetric integral is zero. Also \(a_0=0\). Therefore all cosine coefficients \(a_n\) and \(a_0\) are zero; only sine coefficients \(b_n\) can remain.

Examples

Question

What is the period of \(\sin(3x)\)?

Answer

The sine function repeats when its input increases by \(2\pi\). For \(\sin(3x)\), require \(3(x+T)=3x+2\pi\). Thus \(3T=2\pi\), so \(T=2\pi/3\).

Taylor Series

Multivariable Functions