From b090363af28c577bbf9da60d03c82056036588aa Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Mon, 1 Nov 2021 21:29:02 +0100
Subject: Expand knowledge base
---
content/know/category/complex-analysis.md | 9 ++
.../know/concept/cauchy-principal-value/index.pdc | 58 +++++++
content/know/concept/dyson-equation/index.pdc | 174 +++++++++++++++++++++
.../know/concept/holomorphic-function/index.pdc | 1 +
.../concept/kramers-kronig-relations/index.pdc | 1 +
content/know/concept/martingale/index.pdc | 61 ++++++++
content/know/concept/propagator/index.pdc | 2 +-
content/know/concept/sigma-algebra/index.pdc | 30 +++-
.../concept/sokhotski-plemelj-theorem/index.pdc | 115 ++++++++++++++
9 files changed, 449 insertions(+), 2 deletions(-)
create mode 100644 content/know/category/complex-analysis.md
create mode 100644 content/know/concept/cauchy-principal-value/index.pdc
create mode 100644 content/know/concept/dyson-equation/index.pdc
create mode 100644 content/know/concept/martingale/index.pdc
create mode 100644 content/know/concept/sokhotski-plemelj-theorem/index.pdc
(limited to 'content/know')
diff --git a/content/know/category/complex-analysis.md b/content/know/category/complex-analysis.md
new file mode 100644
index 0000000..fb7cbef
--- /dev/null
+++ b/content/know/category/complex-analysis.md
@@ -0,0 +1,9 @@
+---
+title: "Complex analysis"
+firstLetter: "C"
+date: 2021-11-01T12:56:57+01:00
+draft: false
+layout: "category"
+---
+
+This page will fill itself.
diff --git a/content/know/concept/cauchy-principal-value/index.pdc b/content/know/concept/cauchy-principal-value/index.pdc
new file mode 100644
index 0000000..3b59e2b
--- /dev/null
+++ b/content/know/concept/cauchy-principal-value/index.pdc
@@ -0,0 +1,58 @@
+---
+title: "Cauchy principal value"
+firstLetter: "C"
+publishDate: 2021-11-01
+categories:
+- Mathematics
+
+date: 2021-11-01T12:54:50+01:00
+draft: false
+markup: pandoc
+---
+
+# Cauchy principal value
+
+The **Cauchy principal value** $\mathcal{P}$,
+or just **principal value**,
+is a method for integrating problematic functions,
+i.e. functions with singularities,
+whose integrals would otherwise diverge.
+
+Consider a function $f(x)$ with a singularity at some finite $x = b$,
+which is hampering attempts at integrating it.
+To resolve this, we define the Cauchy principal value $\mathcal{P}$ as follows:
+
+$$\begin{aligned}
+ \boxed{
+ \mathcal{P} \int_a^c f(x) \dd{x}
+ = \lim_{\varepsilon \to 0^{+}} \!\bigg( \int_a^{b - \varepsilon} f(x) \dd{x} + \int_{b + \varepsilon}^c f(x) \dd{x} \bigg)
+ }
+\end{aligned}$$
+
+If $f(x)$ instead has a singularity at postive infinity $+\infty$,
+then we define $\mathcal{P}$ as follows:
+
+$$\begin{aligned}
+ \boxed{
+ \mathcal{P} \int_{a}^\infty f(x) \dd{x}
+ = \lim_{c \to \infty} \!\bigg( \int_{a}^c f(x) \dd{x} \bigg)
+ }
+\end{aligned}$$
+
+And analogously for $-\infty$.
+If $f(x)$ has singularities both at $+\infty$ and at $b$,
+then we simply combine the two previous cases,
+such that $\mathcal{P}$ is given by:
+
+$$\begin{aligned}
+ \mathcal{P} \int_{a}^\infty f(x) \:dx
+ = \lim_{c \to \infty} \lim_{\varepsilon \to 0^{+}}
+ \!\bigg( \int_{a}^{b - \varepsilon} f(x) \:dx + \int_{b + \varepsilon}^{c} f(x) \:dx \bigg)
+\end{aligned}$$
+
+And so on, until all problematic singularities have been dealt with.
+
+In some situations, for example involving
+the [Sokhotski-Plemelj theorem](/know/concept/sokhotski-plemelj-theorem/),
+the symbol $\mathcal{P}$ is written without an integral,
+in which case the calculations are implicitly integrated.
diff --git a/content/know/concept/dyson-equation/index.pdc b/content/know/concept/dyson-equation/index.pdc
new file mode 100644
index 0000000..7b94124
--- /dev/null
+++ b/content/know/concept/dyson-equation/index.pdc
@@ -0,0 +1,174 @@
+---
+title: "Dyson equation"
+firstLetter: "D"
+publishDate: 2021-11-01
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-11-01T14:57:54+01:00
+draft: false
+markup: pandoc
+---
+
+# Dyson equation
+
+Consider the time-dependent Schrödinger equation,
+describing a wavefunction $\Psi_0(\vb{r}, t)$:
+
+$$\begin{aligned}
+ i \hbar \pdv{t} \Psi_0(\vb{r}, t)
+ = \hat{H}_0(\vb{r}) \: \Psi_0(\vb{r}, t)
+\end{aligned}$$
+
+By definition, this equation's *fundamental solution*
+$G_0(\vb{r}, t; \vb{r}', t')$ satisfies the following:
+
+$$\begin{aligned}
+ \Big( i \hbar \pdv{t} - \hat{H}_0(\vb{r}) \Big) G_0(\vb{r}, t; \vb{r}', t')
+ = \delta(\vb{r} - \vb{r}') \: \delta(t - t')
+\end{aligned}$$
+
+From this, we define the inverse $\hat{G}{}_0^{-1}(\vb{r}, t)$
+as follows, so that $\hat{G}{}_0^{-1} G_0 = \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t')$:
+
+$$\begin{aligned}
+ \hat{G}{}_0^{-1}(\vb{r}, t)
+ &\equiv i \hbar \pdv{t} - \hat{H}_0(\vb{r})
+\end{aligned}$$
+
+Note that $\hat{G}{}_0^{-1}$ is an operator, while $G_0$ is a function.
+For the sake of consistency, we thus define
+the operator $\hat{G}_0(\vb{r}, t)$
+as a multiplication by $G_0$
+and integration over $\vb{r}'$ and $t'$:
+
+$$\begin{aligned}
+ \hat{G}_0(\vb{r}, t) \: f
+ \equiv \iint_{-\infty}^\infty G_0(\vb{r}, t; \vb{r}', t') \: f(\vb{r}', t') \: \dd{\vb{r}}' \dd{t'}
+\end{aligned}$$
+
+For an arbitrary function $f(\vb{r}, t)$,
+so that $\hat{G}{}_0^{-1} \hat{G}_0 = \hat{G}_0 \hat{G}{}_0^{-1} = 1$.
+Moving on, the Schrödinger equation can be rewritten like so,
+using $\hat{G}{}_0^{-1}$:
+
+$$\begin{aligned}
+ \hat{G}{}_0^{-1}(\vb{r}, t) \: \Psi_0(\vb{r}, t)
+ = 0
+\end{aligned}$$
+
+Let us assume that $\hat{H}_0$ is simple,
+such that $G_0$ and $\hat{G}{}_0^{-1}$ can be found without issues
+by solving the defining equation above.
+
+Suppose we now perturb this Hamiltonian with
+a possibly time-dependent operator $\hat{H}_1(\vb{r}, t)$,
+in which case the corresponding fundamental solution
+$G(\vb{r}, \vb{r}', t, t')$ satisfies:
+
+$$\begin{aligned}
+ \delta(\vb{r} - \vb{r}') \: \delta(t - t')
+ &= \Big( i \hbar \pdv{t} - \hat{H}_0(\vb{r}) - \hat{H}_1(\vb{r}, t) \Big) G(\vb{r}, t; \vb{r}', t')
+ \\
+ &= \Big( \hat{G}{}_0^{-1}(\vb{r}, t) - \hat{H}_1(\vb{r}, t) \Big) G(\vb{r}, t; \vb{r}', t')
+\end{aligned}$$
+
+This equation is typically too complicated to solve,
+so we would like an easier way to calculate this new $G$.
+The perturbed wavefunction $\Psi(\vb{r}, t)$
+satisfies the Schrödinger equation:
+
+$$\begin{aligned}
+ \Big( \hat{G}{}_0^{-1}(\vb{r}, t) - \hat{H}_1(\vb{r}, t) \Big) \Psi(\vb{r}, t)
+ = 0
+\end{aligned}$$
+
+We know that $\hat{G}{}_0^{-1} \Psi_0 = 0$,
+which we put on the right,
+and then we apply $\hat{G}_0$ in front:
+
+$$\begin{aligned}
+ \hat{G}_0^{-1} \Psi - \hat{H}_1 \Psi
+ = \hat{G}_0^{-1} \Psi_0
+ \quad \implies \quad
+ \Psi - \hat{G}_0 \hat{H}_1 \Psi
+ &= \Psi_0
+\end{aligned}$$
+
+This equation is recursive,
+so we iteratively insert it into itself.
+Note that the resulting equations are the same as those from
+[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/):
+
+$$\begin{aligned}
+ \Psi
+ &= \Psi_0 + \hat{G}_0 \hat{H}_1 \Psi
+ \\
+ &= \Psi_0 + \hat{G}_0 \hat{H}_1 \Psi_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \Psi
+ \\
+ &= \Psi_0 + \hat{G}_0 \hat{H}_1 \Psi_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \Psi_0
+ + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \Psi_0 + \: ...
+ \\
+ &= \Psi_0 + \big( \hat{G}_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \hat{G}_0 + \: ... \big) \hat{H}_1 \Psi_0
+\end{aligned}$$
+
+The parenthesized expression clearly has the same recursive pattern,
+so we denote it by $\hat{G}$ and write the so-called **Dyson equation**:
+
+$$\begin{aligned}
+ \boxed{
+ \hat{G}
+ = \hat{G}_0 + \hat{G}_0 \hat{H}_1 \hat{G}
+ }
+\end{aligned}$$
+
+Such an iterative scheme is excellent for approximating $\hat{G}(\vb{r}, t)$.
+Once a satisfactory accuracy is obtained,
+the perturbed wavefunction $\Psi$ can be calculated from:
+
+$$\begin{aligned}
+ \boxed{
+ \Psi
+ = \Psi_0 + \hat{G} \hat{H}_1 \Psi_0
+ }
+\end{aligned}$$
+
+This relation is equivalent to the Schrödinger equation.
+So now we have the operator $\hat{G}(\vb{r}, t)$,
+but what about the fundamental solution function $G(\vb{r}, t; \vb{r}', t')$?
+Let us take its definition, multiply it by an arbitrary $f(\vb{r}, t)$,
+and integrate over $G$'s second argument pair:
+
+$$\begin{aligned}
+ \iint \big( \hat{G}{}_0^{-1} \!-\! \hat{H}_1 \big) G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'}
+ = \iint^\infty \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t') \: f(\vb{r}, t) \dd{\vb{r}'} \dd{t'}
+ = f
+\end{aligned}$$
+
+Where we have hidden the arguments $(\vb{r}, t)$ for brevity.
+We now apply $\hat{G}_0(\vb{r}, t)$ to this equation
+(which contains an integral over $t''$ independent of $t'$):
+
+$$\begin{aligned}
+ \hat{G}_0 f
+ &= \big( \hat{G}_0 \hat{G}{}_0^{-1} - \hat{G}_0 \hat{H}_1 \big) \iint_{-\infty}^\infty G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'}
+ \\
+ &= \big( 1 - \hat{G}_0 \hat{H}_1 \big) \iint_{-\infty}^\infty G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'}
+\end{aligned}$$
+
+Here, the shape of Dyson's equation is clearly recognizable,
+so we conclude that, as expected, the operator $\hat{G}$
+is defined as multiplication by the function $G$ followed by integration:
+
+$$\begin{aligned}
+ \hat{G}(\vb{r}, t) \: f(\vb{r}, t)
+ \equiv \iint_{-\infty}^\infty G(\vb{r}, t; \vb{r}', t') \: f(\vb{r}', t') \: \dd{\vb{r}}' \dd{t'}
+\end{aligned}$$
+
+
+
+## References
+1. H. Bruus, K. Flensberg,
+ *Many-body quantum theory in condensed matter physics*,
+ 2016, Oxford.
diff --git a/content/know/concept/holomorphic-function/index.pdc b/content/know/concept/holomorphic-function/index.pdc
index 1c2f092..4b7221c 100644
--- a/content/know/concept/holomorphic-function/index.pdc
+++ b/content/know/concept/holomorphic-function/index.pdc
@@ -4,6 +4,7 @@ firstLetter: "H"
publishDate: 2021-02-25
categories:
- Mathematics
+- Complex analysis
date: 2021-02-25T14:40:45+01:00
draft: false
diff --git a/content/know/concept/kramers-kronig-relations/index.pdc b/content/know/concept/kramers-kronig-relations/index.pdc
index 9b67d60..01e5a3a 100644
--- a/content/know/concept/kramers-kronig-relations/index.pdc
+++ b/content/know/concept/kramers-kronig-relations/index.pdc
@@ -4,6 +4,7 @@ firstLetter: "K"
publishDate: 2021-02-25
categories:
- Mathematics
+- Complex analysis
- Physics
- Optics
diff --git a/content/know/concept/martingale/index.pdc b/content/know/concept/martingale/index.pdc
new file mode 100644
index 0000000..ffc286b
--- /dev/null
+++ b/content/know/concept/martingale/index.pdc
@@ -0,0 +1,61 @@
+---
+title: "Martingale"
+firstLetter: "M"
+publishDate: 2021-10-31
+categories:
+- Mathematics
+
+date: 2021-10-18T10:01:46+02:00
+draft: false
+markup: pandoc
+---
+
+# Martingale
+
+A **martingale** is a type of stochastic process
+(i.e. a time-indexed [random variable](/know/concept/random-variable/))
+with important and useful properties,
+especially for stochastic calculus.
+
+For a stochastic process $\{ M_t : t \ge 0 \}$
+on a probability space $(\Omega, \mathcal{F}, P)$ with filtration $\{ \mathcal{F}_t \}$
+(see [$\sigma$-algebra](/know/concept/sigma-algebra/)),
+then $\{ M_t \}$ is a martingale if it satisfies all of the following:
+
+1. $M_t$ is $\mathcal{F}_t$-adapted, meaning
+ the filtration $\mathcal{F}_t$ contains enough information
+ to reconstruct the current and all past values of $M_t$.
+2. For all times $t \ge 0$, the expectation value exists $\mathbf{E}(M_t) < \infty$.
+3. For all $s, t$ satisfying $0 \le s \le t$,
+ the [conditional expectation](/know/concept/conditional-expectation/)
+ $\mathbf{E}(M_t | \mathcal{F}_s) = M_s$,
+ meaning the increment $M_t \!-\! M_s$ is always expected
+ to be zero $\mathbf{E}(M_t \!-\! M_s | \mathcal{F}_s) = 0$.
+
+The last condition is called the **martingale property**,
+and essentially means that a martingale is an unbiased random walk.
+Accordingly, the [Wiener process](/know/concept/wiener-process/) $\{ B_t \}$
+(Brownian motion) is a prime example of a martingale
+(with respect to its own filtration),
+since each of its increments $B_t \!-\! B_s$ has mean $0$ by definition.
+
+Modifying property (3) leads to two common generalizations.
+The stochastic process $\{ M_t \}$ above is a **submartingale**
+if the current value is a lower bound for the expectation:
+
+3. For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s$.
+
+Analogouly, $\{ M_t \}$ is a **supermartingale**
+if the current value is an upper bound instead:
+
+3. For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$.
+
+Clearly, submartingales and supermartingales are *biased* random walks,
+since they will tend to increase and decrease with time, respectively.
+
+
+
+## References
+1. U.F. Thygesen,
+ *Lecture notes on diffusions and stochastic differential equations*,
+ 2021, Polyteknisk Kompendie.
diff --git a/content/know/concept/propagator/index.pdc b/content/know/concept/propagator/index.pdc
index 2f18c4d..517376c 100644
--- a/content/know/concept/propagator/index.pdc
+++ b/content/know/concept/propagator/index.pdc
@@ -61,7 +61,7 @@ $$\begin{aligned}
\end{aligned}$$
Sometimes the name "propagator" is also used to refer to
-the so-called *fundamental solution* or *Green's function* $G$
+the so-called *fundamental solution* $G$
of the time-dependent Schrödinger equation,
which is related to $K$ by:
diff --git a/content/know/concept/sigma-algebra/index.pdc b/content/know/concept/sigma-algebra/index.pdc
index 690c4cc..1a459ea 100644
--- a/content/know/concept/sigma-algebra/index.pdc
+++ b/content/know/concept/sigma-algebra/index.pdc
@@ -43,7 +43,7 @@ is a sub-family of a certain $\mathcal{F}$,
which is a valid $\sigma$-algebra in its own right.
-## Notable examples
+## Notable applications
A notable $\sigma$-algebra is the **Borel algebra** $\mathcal{B}(\Omega)$,
which is defined when $\Omega$ is a metric space,
@@ -54,6 +54,8 @@ and all the subsets of $\mathbb{R}$ obtained by countable sequences
of unions and intersections of those intervals.
The elements of $\mathcal{B}$ are **Borel sets**.
+
+
Another example of a $\sigma$-algebra is the **information**
obtained by observing a [random variable](/know/concept/random-variable/) $X$.
Let $\sigma(X)$ be the information generated by observing $X$,
@@ -84,6 +86,32 @@ if $Y$ can always be computed from $X$,
i.e. there exists a function $f$ such that
$Y(\omega) = f(X(\omega))$ for all $\omega \in \Omega$.
+
+
+The concept of information can be extended for
+stochastic processes (i.e. time-indexed random variables):
+if $\{ X_t : t \ge 0 \}$ is a stochastic process,
+its **filtration** $\mathcal{F}_t$ contains all
+the information generated by $X_t$ up to the current time $t$:
+
+$$\begin{aligned}
+ \mathcal{F}_t
+ = \sigma(X_s : 0 \le s \le t)
+\end{aligned}$$
+
+In other words, $\mathcal{F}_t$ is the "accumulated" $\sigma$-algebra
+of all information extractable from $X_t$,
+and hence grows with time: $\mathcal{F}_s \subset \mathcal{F}_t$ for $s < t$.
+Given $\mathcal{F}_t$, all values $X_s$ for $s \le t$ can be computed,
+i.e. if you know $\mathcal{F}_t$, then the present and past of $X_t$ can be reconstructed.
+
+Given some filtration $\mathcal{H}_t$, a stochastic process $X_t$
+is said to be *"$\mathcal{H}_t$-adapted"*
+if $X_t$'s own filtration $\sigma(X_s : 0 \le s \le t) \subseteq \mathcal{H}_t$,
+meaning $\mathcal{H}_t$ contains enough information
+to determine the current and past values of $X_t$.
+Clearly, $X_t$ is always adapted to its own filtration.
+
## References
diff --git a/content/know/concept/sokhotski-plemelj-theorem/index.pdc b/content/know/concept/sokhotski-plemelj-theorem/index.pdc
new file mode 100644
index 0000000..a482618
--- /dev/null
+++ b/content/know/concept/sokhotski-plemelj-theorem/index.pdc
@@ -0,0 +1,115 @@
+---
+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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