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Given $t \in \mathbb{R}$ and $z = x + iy$ and $y>0$. $\lim_{y\to0^+} \frac{1}{t - z} = \frac{1}{t-x} + \pi i \delta(t-x)$

This limit is given in the book Integral Transforms and Their Applications - Debnath 2nd ed. (pg 379)

I don't understand how this limit was evaluated. Please help out.

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    I$f$ I understand you correctly, the "original limit" $\displaystyle\lim\limits_{y\to0+}\$f$rac{1}{t-z}$ is not to be $u$nderstood the way limits of functions are usually understood, but rather its meaning concerns what happens when you multiply the exression by a test function and then integrate and then take the limit.2011-10-24

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