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### Messages - jfarrellhfx

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##### Chapter 5 / Section 5.1, 5.2 - ok to consider delta distribution?
« on: March 09, 2020, 10:49:21 PM »
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

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##### Chapter 2 / Re: S2.2P Problem 2 (6)
« on: January 20, 2020, 09:38:12 PM »
Hmm... Here's how I think I would start:
Begin with the equation for the integral curves:

$$\frac{dt}{1} = \frac{dx}{3} = \frac{dy}{-2} = \frac{du}{x}$$

I would try to solve for $u$ first:
$$du = x \ dt$$

But the integral is in terms of the variable $t$, but you have an $x$ in there... is there a way to get $x$ in terms of $t$?

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