Is this Fourier Transform for the Dirac Delta correct?

Question

I'm reading "Schaum's Outlines: Quantum Mechanics" and in chapter 2 they define the Fourier transform of the Dirac Delta as:

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\delta(x-y)e^{-ikx}dx=\frac{1}{\sqrt{2\pi}}e^{-ikx} $$

But shouldn't it be $(1/\sqrt{2\pi})e^{-iky}$, using the fundamental property of the $\delta$-function?

Answer 1

You are right. Take a look at the third property in this link. Then take the limit as $\epsilon\rightarrow\infty$.