Fourier Series

(Lecture 25 of Mathematical Methods II.)

We have seen several ways to rewrite functions to put forward some of their attributes. This can be combined in countless ways. For instance, a function can always be decomposed as and odd and an even function. Indeed:

\begin{equation} f(z)=\frac{f(z)+f(-z)}{2}+\frac{f(z)-f(-z)}{2} \end{equation}

where the first term $g(z)=(f(z)+f(-z))/2$ is even ($g(z)=g(-z)$) and the second $h(z)=(f(z)-f(-z))/2$ is odd ($h(z)=-h(-z)$). This is a basic decomposition (involving a process of (anti)symmetrization). More elaborate ones typically involve an infinite number of terms. We have studied in some detail various types of such series expansions, like the Taylor expansion and its generalization to include singularities, the Laurent expansion. For a function $U(t)$ defined on a finite compact support, say on $[-\pi,\pi]$, which has a derivative (the general condition is for piecewise continuous functions which have piecewise derivatives), one can define its Fourier Series $S(t)$ as:

\begin{equation} \tag{1} S(t)=\frac{a_0}{2}+\sum_{j=1}^\infty(a_j\cos jt+b_j\sin jt)\,, \end{equation}

where the coefficients $a_j$ and $b_j$ read:

\begin{equation} \tag{2} a_j=\frac{1}{\pi}\int_{-\pi}^\pi U(t)\cos(jt)\,dt\quad\mathrm{and}\quad b_j=\frac{1}{\pi}\int_{-\pi}^\pi U(t)\sin(jt)\,dt\,. \end{equation}

This is a decomposition that describes periodic functions, as equivalently as considering $[-\pi,\pi]$ only, one can make copies of the functions over the entire line.

There is a huge amount of literature on Fourier Series, that would require a full course in itself. Some properties of power series that also apply here are: the coefficients of two Fourier Series add up to form the Fourier expansion of the sum of two functions, and Fourier series can be integrated and differentiated termwise. We go directly to the complex version of the idea. Of course cos and sine are really much related to the complex variable. The complex Fourier Series is defined as:

\begin{equation} \tag{3} \tilde S(t)=\sum_{n=-\infty}^{\infty}c_ne^{int}\,, \end{equation}

with, this time:

\begin{equation} \tag{4} c_n=\frac{1}{2\pi}\int_{-\pi}^\pi U(t)e^{-int}\,dt\,, \end{equation}

for all $n\in\mathbf{N}$. To relate Eqs. (1) and (3) together, we expand the latter in trigonometric form through the Moivre formula:

\begin{align} \tag{5} \tilde S(t)&=c_0+\sum_{n=1}^{\infty}c_{-n}e^{-int}+\sum_{n=1}^{\infty}c_ne^{int}\,,\\ &=c_0+\sum_{n=1}^{\infty}c_{-n}[\cos(-nt)+i\sin(-nt)]+\sum_{n=1}^{\infty}c_n[\cos(nt)+i\sin(nt)]\,,\\ &=c_0+\sum_{n=1}^{\infty}[c_n+c_{-n}]\cos(nt)+i[c_n-c_{-n}]\sin(nt) \end{align}

which shows that:

\begin{equation} \tag{6} c_0=a_0/2\,,\quad a_n=c_n+c_{-n}\quad\mathrm{and}\quad b_n=i(c_{n}-c_{-n})\,. \end{equation}

By rescaling $-\pi\le t\le\pi$ by $L/\pi$, we can stretch the function over the interval $[-L,L]$ for the variable $\tau=tL/\pi$, providing:

\begin{equation} \tilde S(t)=\sum_{n=-\infty}^{\infty}c_n\exp\big(i\frac{\pi nt}{L}\big)\quad\mathrm{with}\quad c_n=\frac{1}{2L}\int_{-L}^L U(\tau)e^{-i\displaystyle{\frac{\pi n\tau}{L}}}\,d\tau\,, \end{equation}

The terms $\omega_n=\frac{\pi n}{L}$ are called the frequencies and the set $\{\omega_n\}$ the frequency spectrum. As $L\rightarrow\infty$, it is clear that the frequency spectrum forms a continuum, leading eventually to a continuous function also in the frequency space. This function is called the Fourier transform of the original function $\mathcal{F}[f(t)](\omega)$. One usually uses lighter notations than this. The way to compute the coefficient is now given in a beautiful symmetrical form of the transform itself:

\begin{equation} \tag{7} F(\omega)=\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt\,,\qquad f(t)=\int_{-\infty}^\infty F(\omega)e^{i\omega t}\, d\omega\,. \end{equation}

One speaks of the Fourier and the Inverse Fourier transforms. They are linked through the symmetry property: if $F(\omega)=\mathcal{F}(U(t))$, then $\mathcal{F}(F(t))=U(-\omega)$.

For instance, $\mathcal{F}[\exp(-|t|)](\omega)=\displaystyle\frac{1}{1+\omega^2}$. This is readily established by direct computation:

\begin{align} \tag{8} \mathcal{F}(\omega)&=\int_{-\infty}^\infty e^{-|t|}e^{i\omega t}\,dt\\ &=\int_{-\infty}^0 e^{(1+i\omega)t}\,dt+\int_{0}^\infty e^{(-1+i\omega)t}\,dt\\ &=\frac{1}{1+i\omega}e^{(1+i\omega)t}\Big|_{-\infty}^0 +\frac{1}{-1+i\omega}e^{(-1+i\omega)t}\Big|_0^{\infty}\\ &=\frac{1}{1+i\omega}-\frac{1}{-1+i\omega}=\frac{1}{1+i\omega}+\frac{1}{1-i\omega}=\frac{1}{1+\omega^2}\,. \end{align}

Important properties of the Fourier transform that are useful for calculations are:

  1. Linearity: $\mathcal{F}(aU+bV)=a\mathcal{F}(U)+b\mathcal{F}(V)$.
  2. Scaling: $\mathcal{F}(U(at))=\frac{1}{|a|}\mathcal{F}(\omega/a)$.
  3. Derivatives become products: $\mathcal{F}(U'(t))=i\omega F(\omega)$.

Fourier transforms are useful in many applications. For intance, Schrödinger's equation $i\hbar\partial_t\psi(x,t)=-\frac{\hbar^2\nabla^2}{2m}\psi(x,t)$ is readily solved through a Fourier transform: $i\hbar\partial_t\psi(k,t)=\frac{(\hbar k)^2}{2m}\psi(k,t)$ and therefore $\psi(k,t)=\exp(-i\frac{\hbar k^2}{2m})\psi(k,0)$.