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Mathematics for Physics

347 Fourier Series

A series of sine and cosine terms that is used to represent a periodic function on a finite interval.

Note: Also called the Fourier series of \(f\). Also written with Euler formulas for its coefficients.

definition [d] (Fourier Series) From Farlow: consider representing a function \(f\) by a series of the form

  • \(\displaystyle f(x) = \dfrac{a_{0}}{2} + \sum_{n=1}^{\infty}\left\{ a_{n}\cos\left(\dfrac{n\pi x}{L}\right) + b_{n}\sin\left(\dfrac{n\pi x}{L}\right) \right\}\) ,

where the coefficients are determined by Euler’s formulas

  • \(\displaystyle a_{0} = \dfrac{1}{L}\int_{-L}^{L} f(x)\, dx\) ,
  • \(\displaystyle a_{m} = \dfrac{1}{L}\int_{-L}^{L} f(x)\cos\left(\dfrac{m\pi x}{L}\right)\, dx,\quad m=1,2,\ldots\) ,
  • \(\displaystyle b_{m} = \dfrac{1}{L}\int_{-L}^{L} f(x)\sin\left(\dfrac{m\pi x}{L}\right)\, dx,\quad m=1,2,\ldots\) .

The series whose coefficients are determined by Euler’s formulas is called the Fourier series of \(f\).

where

  • \(f\) is the function being expanded.
  • \(2L\) is the period of \(f\).
  • \(a_{0}\), \(a_{m}\), \(b_{m}\) are the Fourier coefficients.
  • \(n\) and \(m\) are positive integer indices.
  • \(x\) is the independent variable.

definition [d] (Fourier Series on \([-\pi,\pi]\)) From Olver: the Fourier series of a function \(f(x)\) defined on \(-\pi\leq x\leq\pi\) is

  • \(\displaystyle f(x)\sim\dfrac{a_{0}}{2} + \sum_{k=1}^{\infty}\bigl[a_{k}\cos kx + b_{k}\sin kx\bigr]\) ,

with coefficients

  • \(\displaystyle a_{k} = \dfrac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos kx\, dx,\quad k=0,1,2,\ldots\) ,
  • \(\displaystyle b_{k} = \dfrac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin kx\, dx,\quad k=1,2,3,\ldots\) .

where

  • \(f\) is defined on \([-\pi,\pi]\).
  • \(a_{k}\) and \(b_{k}\) are the Fourier coefficients.
  • \(k\) is a nonnegative integer index.
  • \(x\) is the independent variable.

Euler formulas. The coefficients are integrals of \(f\) against cosine and sine. This principle is used to compute each term of the series from \(f\).

Trigonometric expansion. The series is a constant term plus cosine and sine harmonics. This principle is used to write a periodic function as an infinite sum of elementary oscillations.

347.1 Examples

347.1.1 Simple

Take the square wave of period \(2\pi\) with value \(-1\) on \([-\pi,0)\) and value \(1\) on \([0,\pi)\). Then \(L=\pi\), every \(a_{n}\) vanishes, and only odd sine terms remain.

\[ f(x) = \dfrac{4}{\pi}\left\{\sin x + \dfrac{1}{3}\sin 3x + \dfrac{1}{5}\sin 5x + \cdots\right\} \]

where

  • \(f\) is the square wave of period \(2\pi\).
  • \(b_{n} = \dfrac{4}{n\pi}\) for odd \(n\) and \(b_{n}=0\) for even \(n\).

347.1.2 General

For \(f(x)=x\) on \([-\pi,\pi]\), the cosine coefficients vanish and the sine coefficients are \(b_{k}=\dfrac{2}{k}(-1)^{k+1}\).

\[ x\sim 2\left(\sin x - \dfrac{\sin 2x}{2} + \dfrac{\sin 3x}{3} - \dfrac{\sin 4x}{4} + \cdots\right) \]

where

  • \(f(x)=x\) is the function on \([-\pi,\pi]\).
  • \(a_{k}=0\) for every \(k\geq 0\).
  • \(b_{k}\) is the \(k\)th sine coefficient.

347.2 References

  1. Farlow, S. J. An Introduction to Differential Equations and Their Applications. Dover, 2006. Chapter 9, §9.1, pp. 384–387 — Fourier series on \([-L,L]\); Euler formulas; square-wave example.
  2. Olver, P. J. Introduction to Partial Differential Equations. Springer, 2016. Chapter 3, §3.2, Definition 3.2, p. 74 — Fourier series on \([-\pi,\pi]\); Example 3.3 for \(f(x)=x\).