From 7c412050570ef229dd78cbcffbf80f23728a630d Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 13 May 2023 15:42:47 +0200
Subject: Improve knowledge base

---
 .../concept/sokhotski-plemelj-theorem/index.md     | 47 +++++++++++++---------
 1 file changed, 27 insertions(+), 20 deletions(-)

(limited to 'source/know/concept/sokhotski-plemelj-theorem')

diff --git a/source/know/concept/sokhotski-plemelj-theorem/index.md b/source/know/concept/sokhotski-plemelj-theorem/index.md
index 66e89bc..445b029 100644
--- a/source/know/concept/sokhotski-plemelj-theorem/index.md
+++ b/source/know/concept/sokhotski-plemelj-theorem/index.md
@@ -9,8 +9,8 @@ categories:
 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 $$[a, b]$$:
+The goal is to evaluate integrals of the following form,
+where $$f(x)$$ is assumed to be continuous in the integration interval $$[a, b]$$:
 
 $$\begin{aligned}
     \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x}
@@ -22,12 +22,14 @@ into its real and imaginary parts (limit hidden):
 $$\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}
+    \\
+    &= \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}$$
 
-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:
+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:
 
 $$\begin{aligned}
     \lim_{\eta \to 0^+} \int_a^b \frac{x f(x)}{x^2 + \eta^2} \dd{x}
@@ -56,7 +58,7 @@ $$\begin{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/):
+the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(x)$$:
 
 $$\begin{aligned}
     \lim_{\eta \to 0^+} \int_a^b \frac{\eta \: f(x)}{x^2 + \eta^2} \dd{x}
@@ -66,7 +68,8 @@ $$\begin{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:
+**Sokhotski-Plemelj theorem** of complex analysis,
+which is important in quantum mechanics:
 
 $$\begin{aligned}
     \boxed{
@@ -82,29 +85,33 @@ This awkwardly leaves $$\mathcal{P}$$ behind:
 
 $$\begin{aligned}
     \frac{1}{x + i \eta}
-    = \mathcal{P} \Big( \frac{1}{x} \Big) - i \pi \delta(x)
+    = \mathcal{P} \frac{1}{x} - i \pi \delta(x)
 \end{aligned}$$
 
-The full, complex version of the Sokhotski-Plemelj theorem
-evaluates integrals of the following form
-over a contour $$C$$ in the complex plane:
+That was the real version of the theorem,
+which is a special case of a general result in complex analysis.
+Consider the following function:
 
 $$\begin{aligned}
-    \phi(z) = \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta}
+    \phi(z) = \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta}
 \end{aligned}$$
 
 Where $$f(z)$$ must be [holomorphic](/know/concept/holomorphic-function/).
-The Sokhotski-Plemelj theorem then states:
+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 $$C$$:
 
 $$\begin{aligned}
     \boxed{
-        \lim_{w \to z} \phi(w)
-        = \frac{1}{2 \pi i} \mathcal{P} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} \pm \frac{f(z)}{2}
+        \lim_{y \to z} \phi(y)
+        = \mathcal{P} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} \pm \: i \pi f(z)
     }
 \end{aligned}$$
 
-Where 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,
+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 $$|\zeta| \to \infty$$.
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