The Fourier transform of sinx/x and (sinx/x)^2 and more

In this post

We are going to evaluate the Fourier transform of $\frac{\sin{x}}{x}$ and $\left(\frac{\sin{x}}{x}\right)^2$. And it turns out to be a comprehensive application of many elementary theorems in real and complex analysis. It is advised to make sure that you can compute and understand all the identities in this post by yourself in the end. Also, you are expected to be able to recall what all words in italics mean.

To be clear, by the Fourier transform of $f$ we actually mean

$$\hat{f}(t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-itx}dx.$$

There is a matter of convenience. Indeed, the coefficient $\frac{1}{\sqrt{2\pi}}$ is superfluous, but without this coefficient, when computing the Fourier inverse, one has to write $\frac{1}{2\pi}$ on the other side. Instead of making the transform-inverse unbalanced, we write $\frac{1}{\sqrt{2\pi}}$ all the time and pretend it is not here.

We say a function $f$ is in $L^1$ if $\int_{-\infty}^{+\infty}|f(x)|dx<+\infty$. As a classic exercise in elementary calculus, one can show that $\frac{\sin{x}}{x} \not\in L^1$ but $\left(\frac{\sin{x}}{x}\right)^2 \in L^1$.

Problem 1

For real $t$, find the following limit:

$$\lim_{A \to \infty}\int_{-A}^{A}\frac{\sin{x}}{x}e^{itx}dx.$$

Since $\frac{\sin{x}}{x}e^{itx}\not\in L^1$, we cannot evaluate the integral over $\mathbb{R}$ in the ordinary sense (the reader can safely ignore this if he or she has no background in Lebesgue integration at this moment, but do keep in mind that being in $L^1$ is a big deal). However, for given $A>0$, the integral over $[-A,A]$ is defined, and we evaluate this limit as $A \to \infty$ to get what we want (by abuse of language). The reader is highly encouraged to write down calculation and supply images that should’ve been here.

We will do this using contour integration. Since the complex function $f(z)=\frac{\sin{z}}{z}e^{itz}$ is entire, by Cauchy’s theorem, its integral over $[-A,A]$ is equal to the integration over the path $\Gamma_A$ which consists of three parts: $[-A,-1]$, lower half of the unit circle at the origin and $[1,A]$.

Since the path $\Gamma_A$ avoids the origin, we are safe to use the identity

$$2i\sin{z}=e^{iz}-e^{-iz}.$$

Replacing $\sin{z}$ with $\frac{1}{2i}(e^{itz}-e^{-itz})$, we get

$$I_A(t)=\int_{\Gamma_A}f(z)dz=\int_{\Gamma_A}\frac{1}{2iz}(e^{i(t+1)z}-e^{i(t-1)z})dz.$$

If we put $\varphi_A(t)=\int_{\Gamma_A}\frac{1}{2iz}e^{itz}dz$, we see $I_A(t)=\varphi_A(t+1)-\varphi_A(t-1)$. It is convenient to divide $\varphi_A$ by $\pi$ since we therefore get

$$\frac{1}{\pi}\varphi_A(t)=\frac{1}{2\pi i}\int_{\Gamma_A}\frac{e^{itz}}{z}dz$$

and we are cool with the denominator $2\pi i$.

Now, close the path $\Gamma_A$ in two ways. First, by the semicircle from $A$ to $-Ai$ to $-A$ (that is, lower semicircle at the origin with radius $A$; clockwise); second, by the semicircle from $A$ to $Ai$ to $-A$ (lower semicircle and anticlockwise). For simplicity we denote the two closed paths by $\Gamma_L$ and $\Gamma_U$. Again by the Cauchy theorem, the first path $\Gamma_L$ gives us an integral with value $0$, thus by Cauchy’s theorem,

$$\frac{1}{\pi}\varphi_A(t)=\frac{1}{2\pi i}\int_{-\pi}^{0}\frac{\exp{(itAe^{i\theta})}}{Ae^{i\theta}}dAe^{i\theta}=\frac{1}{2\pi}\int_{-\pi}^{0}\exp{(itAe^{i\theta})}d\theta.$$

Notice that

$$\begin{aligned} |\exp(itAe^{i\theta})|&=|\exp(itA(\cos\theta+i\sin\theta))| \\ &=|\exp(itA\cos\theta)|\cdot|\exp(-At\sin\theta)| \\ &=\exp(-At\sin\theta) \end{aligned}$$

hence if $t\sin\theta>0$, we have $|\exp(iAte^{i\theta})| \to 0$ as $A \to \infty$. When $-\pi < \theta <0$ we have $\sin\theta<0$. This allows us to compute the case when $t<0$:

$$\frac{1}{\pi}\varphi_{A}(t)=\frac{1}{2\pi}\int_{-\pi}^{0}\exp(itAe^{i\theta})d\theta \to 0\quad (A \to \infty,t<0).$$

(The reader should be able to prove the convergence above.) Also trivially

$$\varphi_A(0)=\frac{1}{2}\int_{-\pi}^{0}1d\theta=\frac{\pi}{2}.$$

But what if $t>0$? Indeed, it would be difficult to obtain the limit using the integral over $[-\pi,0]$. Thus we will seek help from another contour $\Gamma_U$.

Note that $\frac{e^{itz}}{z}$ is a meromorphic function in $\mathbb{C}$ with a simple pole at $0$ with residue $1$ since

$$\frac{e^{itz}}{z}=\frac{1}{z}\left(1+itz+\frac{(itz)^2}{2!}+\cdots\right)=\frac{1}{z}+it+\frac{(it)^2z}{2!}+\cdots.$$

By the residue theorem,

$$\begin{aligned} \frac{1}{2\pi{i}}\int_{\Gamma_U}\frac{e^{itz}}{z}dz&=\frac{1}{2\pi{i}}\int_{\Gamma_A}\frac{e^{itz}}{z}dz+\frac{1}{2\pi}\int_{0}^{\pi}\exp(itAe^{i\theta})d\theta \\ &=1\cdot\operatorname{Ind}_{\Gamma_U}(0)=1. \end{aligned}$$

Note that we have used the change-of-variable formula as we did for the upper one. $\operatorname{Ind}_{\Gamma_L}(0)$ denotes the winding number of $\Gamma_U$ around $0$, which is $1$ of course. The identity above implies

$$\frac{1}{\pi}\varphi_A(t)=1-\frac{1}{2\pi}\int_{0}^{\pi}\exp{(itAe^{i\theta})}d\theta.$$

Now we are ready to find the limit of $\frac{1}{\pi}\varphi_A(t)$ as $A \to \infty$ and when $t>0$. Notice that

$$\left|\frac{1}{2\pi}\int_{0}^{\pi}\exp{(itAe^{i\theta})}d\theta\right| \le \frac{1}{2\pi}\int_0^{2\pi}|\exp(itAe^{i\theta})|d\theta=\frac{1}{2\pi}\int_0^{2\pi}\exp(-At\sin\theta)d\theta.$$

Therefore, when $t>0$, since $\sin\theta>0$ when $0<\theta<\pi$, we get

$$\frac{1}{\pi}\varphi_A(t)\to 1 \quad(A \to \infty,t>0).$$

But as is already shown, $I_A(t)=\varphi_A(t+1)-\varphi_A(t-1)$. To conclude,

$$\lim_{A\to\infty}I_A(t)= \begin{cases} \pi\quad &|t|<1, \\ 0 \quad &|t|>1 ,\\ \frac{1}{2\pi} \quad &|t|=1. \end{cases}$$

What we can learn from this integral

Since $\psi(x)=\left(\frac{\sin{x}}{x}\right)$ is even, dividing $I_A$ by $\sqrt{\frac{1}{2\pi}}$, we actually obtain the Fourier transform of $\psi$ by abuse of language. We also get

$$\hat\psi(t)= \begin{cases} \sqrt{\frac{\pi}{2}}\quad & |t|<1, \\ 0 \quad & |t|>1, \\ \frac{1}{2\pi\sqrt{2\pi}} & |t|=1. \end{cases}$$

Note that $\hat\psi(t)$ is not continuous, let alone being uniformly continuous. Therefore, $\psi(x) \notin L^1$. The reason is, if $f \in L^1$, then $\hat{f}$ is uniformly continuous (proof). Another interesting fact is, this also gives us the value of the Dirichlet integral since

$$\begin{aligned} \int_{-\infty}^{\infty}\left(\frac{\sin{x}}{x}\right)dx&=\int_{-\infty}^{\infty}\left(\frac{\sin{x}}{x}\right)e^{0\cdot ix}dx \\ &=\sqrt{2\pi}\hat\psi(0) \\ &=\pi. \end{aligned}$$

We end this section by evaluating the inverse of $\hat\psi(t)$. The calculation is not that difficult. Now you can see why we put $\sqrt\frac{1}{2\pi}$.

$$\begin{aligned} \sqrt{\frac{1}{2\pi}}\int_{-\infty}^{\infty}\hat\psi(t)e^{itx}dt &= \sqrt{\frac{1}{2\pi}}\int_{-1}^{1}\sqrt{\frac{\pi}{2}}e^{itx}dt \\ &=\frac{1}{2}\cdot\frac{1}{ix}(e^{ix}-e^{-ix}) \\ &=\frac{\sin{x}}{x}. \end{aligned}$$

Problem 2

For real $t$, compute

$$J=\int_{-\infty}^{\infty}\left(\frac{\sin{x}}{x}\right)^2e^{itx}dx.$$

Now since $h(x)=\frac{\sin^2{x}}{x^2} \in L^1$, we are able to say with ease that the integral above is the Fourier transform of $h(x)$ (multiplied by $\sqrt{2\pi}$). But still we use the limit form

$$J(t)=\lim_{A \to \infty}J_A(t)$$

where

$$J_A(t)=\int_{-A}^{A}\left(\frac{\sin{x}}{x}\right)^2e^{itx}dx.$$

And we are still using the contour integration as in problem 1(keep $\Gamma_A$, $\Gamma_U$ and $\Gamma_L$ in mind!). For this we get

$$\left(\frac{\sin z}{z}\right)^2e^{itz}=\frac{e^{i(t+2)z}+e^{i(t-2)z}-2e^{itz}}{-4z^2}.$$

Therefore it suffices to discuss the function

$$\mu_A(t)=\int_{\Gamma_A}\frac{e^{itz}}{2z^2}dz$$

since we have

$$J_A(t)=\mu_A(t)-\frac{1}{2}(\mu_A(t+2)-\mu_A(t-2)).$$

Dividing $\mu_A(t)$ by $\frac{1}{\pi i}$, we see

$$\frac{1}{\pi i}\mu_A(t)=\frac{1}{2\pi i}\int_{\Gamma_A}\frac{e^{itz}}{z^2}dz.$$

An integration of $\frac{e^{itz}}{z^2}$ over $\Gamma_L$ yields

$$\begin{aligned} \frac{1}{\pi i}\mu_A(t)&=\frac{1}{2\pi i}\int_{-\pi}^{0}\frac{\exp(itAe^{i\theta})}{A^2e^{2i\theta}}dAe^{i\theta} \\ &=\frac{1}{2\pi}\int_{-\pi}^{0}\frac{\exp(itAe^{i\theta})}{Ae^{i\theta}}d\theta. \end{aligned}$$

Since we still have

$$\left|\frac{\exp(itAe^{i\theta})}{Ae^{i\theta}}\right|=\frac{1}{A}\exp(-At\sin\theta),$$

if $t<0$ in this case, $\frac{1}{\pi i}\mu_A(z) \to 0$ as $A \to \infty$. For $t>0$, integrating along $\Gamma_U$, we have

$$\frac{1}{\pi i}\mu_A(t)=it-\frac{1}{2\pi}\int_{0}^{\pi}\frac{\exp(itAe^{i\theta})}{Ae^{i\theta}}d\theta \to it \quad (A \to \infty)$$

We can also evaluate $\mu_A(0)$ by computing the integral but we are not doing that. To conclude,

$$\lim_{A \to\infty}\mu_A(t)=\begin{cases} 0 \quad &t>0, \\ -\pi t \quad &t<0. \end{cases}$$

Therefore for $J_A$ we have

$$J(t)=\lim_{A \to\infty}J_A(t)=\begin{cases} 0 \quad &|t| \geq 2, \\ \pi(1+\frac{t}{2}) \quad &-2<t \leq 0, \\ \pi(1-\frac{t}{2}) \quad & 0<t <2. \end{cases}$$

Now you may ask, how to find the value of $J(t)$ at $0$, $2$ or $-2$? $\mu_A(0)$ is not even evaluated. But $h(t) \in L^1$, $\hat{h}(t)=\sqrt{\frac{1}{2\pi}}J(t)$ is uniformly continuous (!), thus continuous, and the values at these points follows from continuity.

What we can learn from this integral

Again, we get the value of a classic improper integral by

$$\int_{-\infty}^{\infty}\left(\frac{\sin{x}}{x}\right)^2dx = J(0)=\pi.$$

And this time it’s not hard to find the Fourier inverse:

$$\begin{aligned} \sqrt{\frac{1}{2\pi}}\int_{-\infty}^{\infty}\hat{h}(t)e^{itx}dt&=\frac{1}{2\pi}\int_{-\infty}^{\infty}J(t)e^{itx}dt \\ &=\frac{1}{2\pi}\int_{-2}^{2}\pi(1-\frac{1}{2}|t|)e^{itx}dt \\ &=\frac{e^{2ix}+e^{-2ix}-2}{-4x^2} \\ &=\frac{(e^{ix}-e^{-ix})^2}{-4x^2} \\ &=\left(\frac{\sin{x}}{x}\right)^2. \end{aligned}$$