---
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 $a < 0 < b$,
and $f(x)$ 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):

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

To evaluate the real part,
we notice that for $\eta \to 0^+$ the integrand diverges for $x \to 0$,
and thus 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/):

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

By combining the real and imaginary parts,
we thus arrive at the (real version of the)
so-called **Sokhotski-Plemelj theorem** of complex analysis:

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

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:

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

The full, complex version of the Sokhotski-Plemelj theorem
evaluates integrals of the following form
over a contour $C$ in the complex plane:

$$\begin{aligned}
    \phi(z) = \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta}
\end{aligned}$$

Where $f(z)$ must be [holomorphic](/know/concept/holomorphic-function/).
The Sokhotski-Plemelj theorem then states:

$$\begin{aligned}
    \boxed{
        \lim_{w \to z} \phi(w)
        = \frac{1}{2 \pi i} \mathcal{P} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} \pm \frac{f(z)}{2}
    }
\end{aligned}$$

Where the sign is positive if $z$ is inside $C$, and negative if it is outside.
The real version follows by letting $C$ follow the whole real axis,
making $C$ an infinitely large semicircle,
so that the integrand vanishes away from the real axis,
because $1 / (\zeta \!-\! z) \to 0$ for $|\zeta| \to \infty$.