The goal is to evaluate integrals of the following form,
where is assumed to be continuous in the integration interval :
To do so, we start by splitting the integrand
into its real and imaginary parts (limit hidden):
In the real part, notice that the integrand diverges
for when ;
more specifically, there is a singularity at zero.
We therefore split the integral as follows:
This is simply the definition of the
Cauchy principal value ,
so the real part is given by:
Meanwhile, in the imaginary part,
we substitute for , and introduce :
The expression is a so-called nascent delta function,
meaning that in the limit it converges to
the Dirac delta function :
By combining the real and imaginary parts,
we thus arrive at the (real version of the)
Sokhotski-Plemelj theorem of complex analysis,
which is important in quantum mechanics:
However, this theorem is often written in the following sloppy way,
where is defined up front to be small,
the integral is hidden, and is set to .
This awkwardly leaves behind:
That was the real version of the theorem,
which is a special case of a general result in complex analysis.
Consider the following function:
Where must be holomorphic.
For all not on , this exists,
but not for , since the integral diverges then.
However, in the limit when approaching , we can still obtain a value for ,
with a caveat: the value depends on the direction we approach from!
The full Sokhotski-Plemelj theorem then states, for all on the closed contour :
Where is if is approached from the inside, and if from outside.
The above real version follows by making an infinitely large semicircle
with its flat side on the real line:
the integrand vanishes away from the real axis,
because for .