It is well known that the Poisson kernel
$$\eta_y(x)=\frac{1}{\pi}\frac{y}{x^2+y^2}$$
Provides a limit representation of the Dirac delta function $\lim_{y\to0}\eta_y(x)=\delta(x)$. In an attempt to verify this fact in Mathematica, I take:
Limit[y/(x^2+y^2) , y->0]
0
which gives the proper result for all x except for x=0. In fact, I would have preferred the Limit function to actually return Pi DiracDelta[x], similarly to what happens if we evaluate FourierTransform[1,x,w]. Nevertheless, we can also plot, e.g.:
Plot[y/(x^2 + y^2) /. y -> 10^-1, {x, -1, 1}, PlotRange -> All]
and convince ourselves that for ever-smaller y the function indeed approaches a delta function.
Now, consider changing the sign of x^2 in the above expression:
Plot[y/(-x^2 + y^2) /. y -> 5^-1, {x, -1, 1}, PlotRange -> All]
In this case, apart from the non-zero region around x=0 there also appear negative and positive infinities symmetrically on both sides. Considering that the normalization changes a bit:
Integrate[y/(-x^2 + y^2), {x, -Infinity, Infinity}]
one might think that
$$\lim_{y^2\to 0+i0}\eta_y(x)=\lim_{y^2\to 0+i0}\frac{i}{\pi}\frac{y}{-x^2+y^2}$$
is also a limit representation of the delta function, if we take y to zero along a path such that y^2 is slightly complex. However, so far this is just an assumption, and the Limit routine cannot be used to verify this as we have seen above. Is it possible to use Mathematica to verify this claim?