From 777f9d3fd8caf56e6bc6999a4b05379307d0733f Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Tue, 12 Sep 2023 07:36:33 +0200
Subject: Initial commit

---
 pages/complex-analysis/index.md                    |   8 ++
 .../one-complex-variable/basics.md                 |  23 +++++
 .../cauchys-integral-formula.md                    | 102 +++++++++++++++++++++
 .../one-complex-variable/cauchys-theorem.md        |  39 ++++++++
 .../complex-analysis/one-complex-variable/index.md |   9 ++
 .../one-complex-variable/power-series.md           |  62 +++++++++++++
 .../the-calculus-of-residues.md                    |  60 ++++++++++++
 .../several-complex-variables/edge-of-the-wedge.md |  18 ++++
 .../several-complex-variables/index.md             |  12 +++
 .../weak-and-strong-analyticity.md                 |  18 ++++
 10 files changed, 351 insertions(+)
 create mode 100644 pages/complex-analysis/index.md
 create mode 100644 pages/complex-analysis/one-complex-variable/basics.md
 create mode 100644 pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
 create mode 100644 pages/complex-analysis/one-complex-variable/cauchys-theorem.md
 create mode 100644 pages/complex-analysis/one-complex-variable/index.md
 create mode 100644 pages/complex-analysis/one-complex-variable/power-series.md
 create mode 100644 pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md
 create mode 100644 pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md
 create mode 100644 pages/complex-analysis/several-complex-variables/index.md
 create mode 100644 pages/complex-analysis/weak-and-strong-analyticity.md

(limited to 'pages/complex-analysis')

diff --git a/pages/complex-analysis/index.md b/pages/complex-analysis/index.md
new file mode 100644
index 0000000..d07109e
--- /dev/null
+++ b/pages/complex-analysis/index.md
@@ -0,0 +1,8 @@
+---
+title: Complex Analysis
+nav_order: 2
+has_children: true
+# cspell:words
+---
+
+# {{ page.title }}
diff --git a/pages/complex-analysis/one-complex-variable/basics.md b/pages/complex-analysis/one-complex-variable/basics.md
new file mode 100644
index 0000000..b30d18c
--- /dev/null
+++ b/pages/complex-analysis/one-complex-variable/basics.md
@@ -0,0 +1,23 @@
+---
+title: Basics
+parent: One Complex Variable
+grand_parent: Complex Analysis
+nav_order: 1
+# cspell:words
+---
+
+# {{ page.title }}
+
+{: .theorem }
+> {: #holomorphic-function-is-constant-if-derivative-vanishes }
+>
+> If the derivative of a holomorphic function vanishes
+> throughout a connected open subset of the complex plane,
+> then it must be constant on that set.
+>
+> More generally, if the derivative of a holomorphic function vanishes
+> throughout an open subset of the complex plane,
+> then it must be constant on any connected component of that set.
+
+{% proof %}
+{% endproof %}
diff --git a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
new file mode 100644
index 0000000..ccdd0ea
--- /dev/null
+++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
@@ -0,0 +1,102 @@
+---
+title: Cauchy's Integral Formula
+parent: One Complex Variable
+grand_parent: Complex Analysis
+nav_order: 3
+# cspell:words
+---
+
+# {{ page.title }}
+
+{: .theorem-title }
+> {{ page.title }}
+>
+> Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$.
+> Let $\gamma$ be a contour in $G$ such that the interior of $\gamma$ is contained in $G$.
+> Then for any point $a$ in the interior of $\gamma$,
+>
+> $$
+> f(a) = \frac{1}{2 \pi i} \int_{\gamma} \frac{f(z)}{z-a} \, dz.
+> $$
+> {: .katex-display .mb-0 }
+
+{% proof %}
+{% endproof %}
+
+{: .theorem-title }
+> {{ page.title }} (Generalization)
+> {: #cauchys-integral-formula-generalized }
+>
+> Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$.
+> Then the $n$th derivative $f^{(n)}$ exists for every $n \in \NN$.
+> If $\gamma$ is a contour in $G$ such that the interior of $\gamma$ is contained in $G$,
+> then for any point $a$ in the interior of $\gamma$,
+>
+> $$
+> f^{(n)} (a) = \frac{n!}{2 \pi i} \int_{\gamma} \frac{f(z)}{(z-a)^{n+1}} \, dz.
+> $$
+> {: .katex-display .mb-0 }
+
+{% proof %}
+{% endproof %}
+
+The last formula may be rewritten as
+
+$$
+\int_{\gamma} \frac{f(z)}{(z-a)^n} \, dz = \frac{2 \pi i}{(n-1)!} f^{(n-1)}(a)
+$$
+
+and is often used to compute the integral.
+
+## Many Consequences
+
+{: .theorem-title }
+> Cauchy's Estimate 
+> {: #cauchys-estimate }
+>
+> Let $f$ be holomorphic on an open set containing the disc with center $a$ and radius $r>0$.
+> Then
+>
+> $$
+> \norm{f^{(n)}(a)} \le \frac{n!}{r^n} \sup_{\abs{z-a} = r} \norm{f(z)} \qquad \forall n \in \NN.
+> $$
+> {: .katex-display .mb-0 }
+
+{% proof %}
+From [{{ page.title }}](#cauchys-integral-formula-generalized)
+for the circular contour around $a$ with radius $r$ we obtain
+
+$$
+\begin{aligned}
+\norm{f^{(n)}(a)} &\le \frac{n!}{2\pi} \sup_{\abs{z-a} = r} \norm{f(z)} \, \int_{\abs{z-a} = r} \frac{dz}{\abs{z-a}^{n+1}}.
+\end{aligned}
+$$
+
+Note that the supremum is finite (and is attained),
+because $f$ is continuous and the circle is compact.
+Clearly, the integral evaluates to $2 \pi r / r^{n+1}$
+and the right hand side of the inequality reduces to the desired expression.
+{% endproof %}
+
+---
+
+Recall that an *entire* function is a holomorphic function that is defined everywhere in the complex plane.
+
+{: .theorem-title }
+> Liouville's Theorem
+> {: #liouvilles-theorem }
+>
+> Every bounded entire function is constant.
+
+{% proof %}
+Consider an entire function $f$ and assume that $\norm{f(z)} \le M$ for all $z \in \CC$ and some $M > 0$.
+Since $f$ is holomorphic on the whole plane, we may make
+[Cauchy's Estimate](#cauchys-estimate)
+for all disks centered at any point $a \in \CC$ and with any radius $r>0$.
+For the first derivative, we have $\norm{f'(a)} \le M/r$, which tends to $0$ for $r \to \infty$.
+Hence $f' = 0$ in the whole plane. This
+[implies](/pages/complex-analysis/one-complex-variable/basics.html#holomorphic-function-is-constant-if-derivative-vanishes)
+that $f$ is constant.
+{% endproof %}
+
+---
diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
new file mode 100644
index 0000000..15412bc
--- /dev/null
+++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
@@ -0,0 +1,39 @@
+---
+title: Cauchy's Theorem
+parent: One Complex Variable
+grand_parent: Complex Analysis
+nav_order: 2
+# cspell:words
+---
+
+# {{ page.title }}
+
+{: .theorem-title }
+> {{ page.title }} (Homotopy Version)
+>
+> Let $G$ be a connected open subset of the complex plane.
+> Let $f : G \to \CC$ be a holomorphic function.
+> If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then
+>
+> $$
+> \int_{\gamma_0} \! f(z) \, dz =
+> \int_{\gamma_1} \! f(z) \, dz 
+> $$
+>
+> If $\gamma$ is a null-homotopic closed curve in $G$, then
+>
+> $$
+> \int_{\gamma} f(z) \, dz = 0
+> $$
+
+{% proof %}
+{% endproof %}
+
+{{ page.title }} has a converse:
+
+{: .theorem-title }
+> Morera's Theorem
+>
+> Let $G \subset \CC$ be open and let $f : G \to \CC$ be a continuous function.
+> If $\int_{\gamma} f(z) \, dz = 0$ for every contour $\gamma$ contained in $G$,
+> then $f$ is holomorphic in $G$.
diff --git a/pages/complex-analysis/one-complex-variable/index.md b/pages/complex-analysis/one-complex-variable/index.md
new file mode 100644
index 0000000..4942ff8
--- /dev/null
+++ b/pages/complex-analysis/one-complex-variable/index.md
@@ -0,0 +1,9 @@
+---
+title: One Complex Variable
+parent: Complex Analysis
+nav_order: 1
+has_children: true
+# cspell:words
+---
+
+# {{ page.title }}
diff --git a/pages/complex-analysis/one-complex-variable/power-series.md b/pages/complex-analysis/one-complex-variable/power-series.md
new file mode 100644
index 0000000..0147f31
--- /dev/null
+++ b/pages/complex-analysis/one-complex-variable/power-series.md
@@ -0,0 +1,62 @@
+---
+title: Power Series
+parent: One Complex Variable
+grand_parent: Complex Analysis
+nav_order: 1
+# cspell:words
+---
+
+# {{ page.title }}
+
+{: .definition-title }
+> Definition ({{ page.title }})
+>
+> Let $X$ be a complex Banach space.
+> A *power series* (with values in $X$) is an infinite series of the form
+>
+> 
+> $$
+> \sum_{n=0}^{\infty} x_n (z - a)^n = x_0 + x_1 (z-a) + x_2 (z-a)^2 + \cdots,
+> $$
+>
+> where $x_n \in X$ is the *$n$th coefficient*,
+> $z$ is a complex variable and
+> $a$ is the *center* of the series.
+
+{: .lemma }
+> Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$.
+> Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$.
+
+{% proof %}
+TODO
+{% endproof %}
+
+{: .theorem }
+> Suppose $\sum_{n=0}^{\infty} x_n (z - a)^n$ is a Banach space valued power series.
+> Then either
+>
+> - the series converges only for $z=a$ (formally $R=0$), or
+> - there exists a number $0<R<\infty$ such that
+>   the series converges absolutely whenever $\abs{z-a} < R$ 
+>   and diverges whenever $\abs{z-a} > R$, or
+> - the series converges absolutely for all $z \in \CC$ (formally $R=\infty$).
+>
+> The number $R \in [0,\infty]$ is called the *radius of convergence* of the power series.
+
+{% proof %}
+TODO
+{% endproof %}
+
+{: .theorem-title }
+> Cauchy–Hadamard Formula
+>
+> Let $\sum_{n=0}^{\infty} x_n (z - a)^n$ be a Banach space valued power series
+> with radius of convergence $R$. Then
+> 
+> $$
+> \frac{1}{R} = \limsup_{n \to \infty} \norm{x_n}^{1/n}.
+> $$
+
+{% proof %}
+TODO
+{% endproof %}
diff --git a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md
new file mode 100644
index 0000000..b49cdf4
--- /dev/null
+++ b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md
@@ -0,0 +1,60 @@
+---
+title: The Calculus of Residues
+parent: One Complex Variable
+grand_parent: Complex Analysis
+nav_order: 4
+# cspell:words
+#published: false
+---
+
+# {{ page.title }}
+
+{: .definition-title }
+> Definition (Residue)
+>
+> TODO
+
+Calculation of Residues
+
+If $f$ has a simple pole at $c$, then
+$\Res(f,c) = \lim_{z \to c} (z-c) f(z)$.
+
+If $f$ has a pole of order $k$ at $c$, then
+
+$$
+\Res(f,c) = \frac{1}{(k-1)!} g^{(k-1)}(c), \quad \text{where} \ g(z) = (z-c)^k f(z).
+$$
+
+If $g$ and $h$ are holomorphic near $c$ and $h$ has a simple zero at $c$,
+then $f = g/h$ has a simple pole at $c$ and
+
+$$
+\Res(f,c) = \frac{g(c)}{h'(c)}
+$$
+
+
+
+
+{: .theorem-title }
+> Residue Theorem (Basic Version)
+> {: #residue-theorem-basic-version }
+>
+> Let $f$ be a function meromorphic in an open subset $G \subset \CC$.
+> Let $\gamma$ be a contour in $G$ such that
+> the interior of $\gamma$ is contained in $G$
+> and contains finitely many poles $c_1, \ldots, c_n$ of $f$.
+> Then
+>
+> 
+> $$
+> \int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k)
+> $$
+> {: .katex-display .mb-0 }
+
+{% proof %}
+{% endproof %}
+
+TODO
+- argument principle
+- Rouché's theorem
+- winding number
diff --git a/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md b/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md
new file mode 100644
index 0000000..5adc3f6
--- /dev/null
+++ b/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md
@@ -0,0 +1,18 @@
+---
+title: Edge of the Wedge
+parent: Several Complex Variables
+grand_parent: Complex Analysis
+nav_order: 1
+# cspell:words
+---
+
+# {{ page.title }}
+
+{: .theorem-title }
+> {{ page.title }}
+> {: #{{ page.title | slugify }} }
+>
+> ...
+
+{% proof %}
+{% endproof %}
diff --git a/pages/complex-analysis/several-complex-variables/index.md b/pages/complex-analysis/several-complex-variables/index.md
new file mode 100644
index 0000000..49763d5
--- /dev/null
+++ b/pages/complex-analysis/several-complex-variables/index.md
@@ -0,0 +1,12 @@
+---
+title: Several Complex Variables
+parent: Complex Analysis
+nav_order: 2
+has_children: true
+# cspell:words
+---
+
+# {{ page.title }}
+
+TODO
+- Edge of the Wedge Theorem
diff --git a/pages/complex-analysis/weak-and-strong-analyticity.md b/pages/complex-analysis/weak-and-strong-analyticity.md
new file mode 100644
index 0000000..7db1dbf
--- /dev/null
+++ b/pages/complex-analysis/weak-and-strong-analyticity.md
@@ -0,0 +1,18 @@
+---
+title: Weak and Strong Analyticity
+parent: Complex Analysis
+nav_order: 3
+published: false
+# cspell:words
+---
+
+# {{ page.title }}
+
+{: .definition-title }
+> {{ page.title }}
+> {: #{{ page.title | slugify }} }
+>
+> ...
+
+{% proof %}
+{% endproof %}
-- 
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