APM346--2020S > Chapter 5

Section 5.1, 5.2 - ok to consider delta distribution?

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jfarrellhfx:
From a physics class, I remember the statement "the Fourier Transform of a plane wave is the delta distribution", e.g. for the unitary transform, $\widehat{e^{ibx}} = \sqrt{2 \pi}\ \delta (k - b)$.  I understand that these plane waves are not "square integrable", so we do not formally consider them. But I wonder if it is acceptable to use this on homework / quiz / test?

For example, when trying to calculate the transform of a function like $f(x) = e^{-\alpha \left|x\right|}\cos \beta x$, I want to consider the exponential and the $\cos$ functions separately, find their Fourier Transforms, and then convolute.  The $\cos$ will give these delta distributions by Euler's Identity, and the convolution will be easy.
Cheers,
Jack

Victor Ivrii:
Then you need to explain more or less rigorously, what is $\delta$-distribution and how Fourier transform is defined for it. Yes, online textbook covers this topic and much more than your physics class but in more advanced chapters, we do not cover

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