Section 12.6 Examples of Fourier Transforms
This section asks you to find the Fourier transform of a cosine function and a Gaussian. Hints and answers are provided, but the details are left for the reader.
Activity 12.6.1. The Fourier Transform of the Cosine.
Find the Fourier transform of the cosine function \(f(x)=\cos kx.\)
Hint.
Answer.
Hint: You must integrate a combination of an exponential and a cosine. It is ALWAYS easier to integrate exponentials, so use the exponential form of the cosine function, (2.5.4).
\begin{align}
{\cal{F}}(\cos kx)\amp =\tilde{f}(k^{\prime})\tag{12.6.1}\\
\amp = \pi\, \left(\delta(k-k^{\prime})+\delta(k+k^{\prime})\right)\tag{12.6.2}
\end{align}
Activity 12.6.2. The Fourier Transform of a Gaussian.
Find the Fourier transform of the normalized Gaussian (12.2.10)
\begin{equation}
f(x)=\frac{1}{\sqrt{2\pi}\,\sigma}\,
e^{-\frac{(x-x_0)^2}{2\sigma^2}}\tag{12.6.3}
\end{equation}
Hint.
Answer.
Complete the square in the exponential, see Section A.1, and use the formula for the integral of a Gaussian, see (12.2.1).
\begin{align}
{\cal{F}}\left(\frac{1}{\sqrt{2\pi}\,\sigma}\,
e^{-\frac{(x-x_0)^2}{2\sigma^2}}\right)\amp =\tilde{f}(k)\tag{12.6.4}\\
\amp = \frac{1}{\sqrt{2\pi}}\, e^{-\frac{k^2\sigma^2}{2}}\, e^{-ikx_0}\tag{12.6.5}
\end{align}
Notice that the Fourier transform of a Gaussian is also a Gaussian, but now the factor of \(\sigma^2\) is in the numerator of the exponential instead of the denominator. How are the shapes of the two Gaussian's related to each other?