From b090363af28c577bbf9da60d03c82056036588aa Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Mon, 1 Nov 2021 21:29:02 +0100
Subject: Expand knowledge base

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+---
+title: "Sokhotski-Plemelj theorem"
+firstLetter: "S"
+publishDate: 2021-11-01
+categories:
+- Mathematics
+- Complex analysis
+- Quantum mechanics
+
+date: 2021-11-01T12:54:37+01:00
+draft: false
+markup: pandoc
+---
+
+# Sokhotski-Plemelj theorem
+
+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 $|\zeta| \to \infty$.
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