---
title: "Sokhotski-Plemelj theorem"
sort_title: "Sokhotski-Plemelj theorem"
date: 2021-11-01
categories:
- Mathematics
- Complex analysis
- Quantum mechanics
layout: "concept"
---

The goal is to evaluate integrals of the following form,
where $$f(x)$$ is assumed to be continuous in the integration interval $$[a, b]$$:

$$\begin{aligned}
    \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x}
\end{aligned}$$

To do so, we start by splitting the integrand
into its real and imaginary parts (limit hidden):

$$\begin{aligned}
    \int_a^b \frac{f(x)}{x + i \eta} \dd{x}
    &= \int_a^b \frac{x - i \eta}{x^2 + \eta^2} f(x) \dd{x}
    \\
    &= \int_a^b \frac{x}{x^2 + \eta^2} f(x) \dd{x} - i \int_a^b \frac{\eta}{x^2 + \eta^2} f(x) \dd{x}
\end{aligned}$$

In the real part, notice that the integrand diverges
for $$x \to 0$$ when $$\eta \to 0^+$$;
more specifically, there is a singularity at zero.
We therefore split the integral as follows:

$$\begin{aligned}
    \lim_{\eta \to 0^+} \int_a^b \frac{x f(x)}{x^2 + \eta^2} \dd{x}
    &= \lim_{\eta \to 0^+} \bigg( \int_a^{-\eta} \frac{x f(x)}{x^2 + \eta^2} \dd{x} + \int_\eta^b \frac{x f(x)}{x^2 + \eta^2} \dd{x} \bigg)
\end{aligned}$$

This is simply the definition of the
[Cauchy principal value](/know/concept/cauchy-principal-value/) $$\mathcal{P}$$,
so the real part is given by:

$$\begin{aligned}
    \lim_{\eta \to 0^+} \int_a^b \frac{x f(x)}{x^2 + \eta^2} \dd{x}
    &= \mathcal{P} \int_a^b \frac{x f(x)}{x^2} \dd{x}
    = \mathcal{P} \int_a^b \frac{f(x)}{x} \dd{x}
\end{aligned}$$

Meanwhile, in the imaginary part,
we substitute $$\eta$$ for $$1 / m$$, and introduce $$\pi$$:

$$\begin{aligned}
    \lim_{\eta \to 0^+} \int_a^b \frac{\eta \: f(x)}{x^2 + \eta^2} \dd{x}
    &= \lim_{m \to +\infty} \frac{\pi}{\pi} \int_a^b \frac{1/m}{x^2 + 1/m^2} f(x) \dd{x}
    \\
    &= \lim_{m \to +\infty} \frac{\pi}{\pi} \int_a^b \frac{m}{1 + m^2 x^2} f(x) \dd{x}
\end{aligned}$$

The expression $$m / \pi (1 + m^2 x^2)$$ is a so-called *nascent delta function*,
meaning that in the limit $$m \to +\infty$$ it converges to
the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(x)$$:

$$\begin{aligned}
    \lim_{\eta \to 0^+} \int_a^b \frac{\eta \: f(x)}{x^2 + \eta^2} \dd{x}
    &= \pi \int_a^b \delta(x) \: f(x) \dd{x}
    = \pi f(0)
\end{aligned}$$

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:

$$\begin{aligned}
    \boxed{
        \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x}
        = \mathcal{P} \int_a^b \frac{f(x)}{x} \dd{x} - i \pi f(0)
    }
\end{aligned}$$

However, this theorem is often written in the following sloppy way,
where $$\eta$$ is defined up front to be small,
the integral is hidden, and $$f(x)$$ is set to $$1$$.
This awkwardly leaves $$\mathcal{P}$$ behind:

$$\begin{aligned}
    \frac{1}{x + i \eta}
    = \mathcal{P} \frac{1}{x} - i \pi \delta(x)
\end{aligned}$$

That was the real version of the theorem,
which is a special case of a general result in complex analysis.
Consider the following function:

$$\begin{aligned}
    \phi(z) = \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta}
\end{aligned}$$

Where $$f(z)$$ must be [holomorphic](/know/concept/holomorphic-function/).
For all $$z$$ not on $$C$$, this $$\phi(z)$$ exists,
but not for $$z \in C$$, since the integral diverges then.
However, in the limit when approaching $$C$$, we can still obtain a value for $$\phi$$,
with a caveat: the value depends on the direction we approach $$C$$ from!
The full Sokhotski-Plemelj theorem then states, for all $$z$$ on the closed contour $$C$$:

$$\begin{aligned}
    \boxed{
        \lim_{y \to z} \phi(y)
        = \mathcal{P} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} \pm \: i \pi f(z)
    }
\end{aligned}$$

Where $$\pm$$ is $$+$$ if $$C$$ is approached from the inside, and $$-$$ if from outside.
The above real version follows by making $$C$$ an infinitely large semicircle
with its flat side on the real line:
the integrand vanishes away from the real axis,
because $$1 / (\zeta \!-\! z) \to 0$$ for $$|\zeta| \to \infty$$.