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What is the meaning of $N(t)=\int_\mathbb R\int_\mathbb R\frac{e^{ix+iy}}{t-x+i}dxdy$?

What is the meaning of $N(t)=\int_\mathbb R\int_\mathbb R\frac{e^{ix+iy}}{t-x+i}dxdy$?
It comes from the fact that $f(z)=\frac{e^{iz}}{z-i}$ is the Fourier transform of $\frac{1}{z+1}$, hence
$f(z)=-\frac{1}{2}\log(1-iz)$ is the residue of $f(z)$ at $-i$ with respect to the straight line $Im(z)=-\frac{1}{2}$. But I have no idea how this came to be.

The meaning is the following.
Consider
$$I(t)=\int_\mathbb R\frac{e^{it(x+iy)}}{t-x-iy}\,dx\,dy$$
and compare it with $1/(t-x-iy)$.
Note that this is a smooth function of $t$ and you can safely compute the Cauchy integral.
Now define $N(t)=\frac{d}{dt} I(t)$ and observe that $N$ is a function of $t$ and absolutely integrable.
Therefore $N$ must be equal to some $A(t)$, and so
$$\frac{d}{dt} I(t)=A(
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By, darckamm