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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: 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}

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?