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Diffstat (limited to 'pages/complex-analysis')
10 files changed, 115 insertions, 157 deletions
diff --git a/pages/complex-analysis/index.md b/pages/complex-analysis/index.md index d07109e..60c8ed2 100644 --- a/pages/complex-analysis/index.md +++ b/pages/complex-analysis/index.md @@ -1,8 +1,7 @@ --- title: Complex Analysis -nav_order: 2 +nav_order: 3 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 index b30d18c..bbbbd30 100644 --- a/pages/complex-analysis/one-complex-variable/basics.md +++ b/pages/complex-analysis/one-complex-variable/basics.md @@ -3,21 +3,19 @@ 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. +{% theorem %} +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. +{% endtheorem %} {% 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 index ccdd0ea..3cf81f7 100644 --- a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md +++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md @@ -3,39 +3,33 @@ 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 } +{% theorem * Cauchy's Integral Formula %} +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 $\gamma$, + +$$ +f(a) = \frac{1}{2 \pi i} \int_{\gamma} \frac{f(z)}{z-a} \, dz. +$$ +{% endtheorem %} {% 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 } +{% theorem * Cauchy's Integral Formula (Generalization) %} +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 $\gamma$, + +$$ +f^{(n)} (a) = \frac{n!}{2 \pi i} \int_{\gamma} \frac{f(z)}{(z-a)^{n+1}} \, dz. +$$ +{% endtheorem %} {% proof %} {% endproof %} @@ -50,20 +44,17 @@ 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 } +{% theorem * Cauchy's Estimate %} +Let $f$ be holomorphic on an open set containing the disc with center $a$ and radius $r>0$. +Then + +$$ +\norm{f^{(n)}(a)} \le \frac{n!}{r^n} \sup_{\abs{z-a} = r} \norm{f(z)} \qquad \forall n \in \NN. +$$ +{% endtheorem %} {% proof %} -From [{{ page.title }}](#cauchys-integral-formula-generalized) +From [{{ page.title }}](#cauchy-s-integral-formula-generalization) for the circular contour around $a$ with radius $r$ we obtain $$ @@ -82,16 +73,14 @@ and the right hand side of the inequality reduces to the desired expression. Recall that an *entire* function is a holomorphic function that is defined everywhere in the complex plane. -{: .theorem-title } -> Liouville's Theorem -> {: #liouvilles-theorem } -> -> Every bounded entire function is constant. +{% theorem * Liouville's Theorem %} +Every bounded entire function is constant. +{% endtheorem %} {% proof %} Consider an entire function $f$ and assume that $\norm{f(z)} \le M$ for all $z \in \CC$ and some $M > 0$. Since $f$ is holomorphic on the whole plane, we may make -[Cauchy's Estimate](#cauchys-estimate) +[Cauchy's Estimate](#cauchy-s-estimate) for all disks centered at any point $a \in \CC$ and with any radius $r>0$. For the first derivative, we have $\norm{f'(a)} \le M/r$, which tends to $0$ for $r \to \infty$. Hence $f' = 0$ in the whole plane. This diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md index 15412bc..2445b8b 100644 --- a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md +++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md @@ -3,37 +3,34 @@ title: Cauchy's Theorem parent: One Complex Variable grand_parent: Complex Analysis nav_order: 2 -# cspell:words --- # {{ page.title }} -{: .theorem-title } -> {{ page.title }} (Homotopy Version) -> -> Let $G$ be a connected open subset of the complex plane. -> Let $f : G \to \CC$ be a holomorphic function. -> If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then -> -> $$ -> \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 -> $$ +{% theorem Cauchy's Theorem (Homotopy Version) %} +Let $G$ be a connected open subset of the complex plane. +Let $f : G \to \CC$ be a holomorphic function. +If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then + +$$ +\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 +$$ +{% endtheorem %} {% proof %} {% endproof %} {{ page.title }} has a converse: -{: .theorem-title } -> Morera's Theorem -> -> Let $G \subset \CC$ be open and let $f : G \to \CC$ be a continuous function. -> If $\int_{\gamma} f(z) \, dz = 0$ for every contour $\gamma$ contained in $G$, -> then $f$ is holomorphic in $G$. +{% theorem * Morera's Theorem %} +Let $G \subset \CC$ be open and let $f : G \to \CC$ be a continuous function. +If $\int_{\gamma} f(z) \, dz = 0$ for every contour $\gamma$ contained in $G$, +then $f$ is holomorphic in $G$. +{% endtheorem %} diff --git a/pages/complex-analysis/one-complex-variable/index.md b/pages/complex-analysis/one-complex-variable/index.md index 4942ff8..5830a81 100644 --- a/pages/complex-analysis/one-complex-variable/index.md +++ b/pages/complex-analysis/one-complex-variable/index.md @@ -3,7 +3,6 @@ title: One Complex Variable parent: Complex Analysis nav_order: 1 has_children: true -# cspell:words --- # {{ page.title }} diff --git a/pages/complex-analysis/one-complex-variable/power-series.md b/pages/complex-analysis/one-complex-variable/power-series.md index 0147f31..31793ab 100644 --- a/pages/complex-analysis/one-complex-variable/power-series.md +++ b/pages/complex-analysis/one-complex-variable/power-series.md @@ -3,59 +3,57 @@ title: Power Series parent: One Complex Variable grand_parent: Complex Analysis nav_order: 1 -# cspell:words --- # {{ page.title }} -{: .definition-title } -> Definition ({{ page.title }}) -> -> Let $X$ be a complex Banach space. -> A *power series* (with values in $X$) is an infinite series of the form -> -> -> $$ -> \sum_{n=0}^{\infty} x_n (z - a)^n = x_0 + x_1 (z-a) + x_2 (z-a)^2 + \cdots, -> $$ -> -> where $x_n \in X$ is the *$n$th coefficient*, -> $z$ is a complex variable and -> $a$ is the *center* of the series. - -{: .lemma } -> Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$. -> Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$. +{% definition Power Series %} +Let $X$ be a complex Banach space. +A *power series* (with values in $X$) is an infinite series of the form + +$$ +\sum_{n=0}^{\infty} x_n (z - a)^n = x_0 + x_1 (z-a) + x_2 (z-a)^2 + \cdots, +$$ + +where $x_n \in X$ is the *$n$th coefficient*, +$z$ is a complex variable and +$a$ is the *center* of the series. +{% enddefinition %} + +{% lemma %} +Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$. +Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$. +{% endlemma %} {% proof %} TODO {% endproof %} -{: .theorem } -> Suppose $\sum_{n=0}^{\infty} x_n (z - a)^n$ is a Banach space valued power series. -> Then either -> -> - the series converges only for $z=a$ (formally $R=0$), or -> - there exists a number $0<R<\infty$ such that -> the series converges absolutely whenever $\abs{z-a} < R$ -> and diverges whenever $\abs{z-a} > R$, or -> - the series converges absolutely for all $z \in \CC$ (formally $R=\infty$). -> -> The number $R \in [0,\infty]$ is called the *radius of convergence* of the power series. +{% theorem %} +Suppose $\sum_{n=0}^{\infty} x_n (z - a)^n$ is a Banach space valued power series. +Then either + +- the series converges only for $z=a$ (formally $R=0$), or +- there exists a number $0<R<\infty$ such that + the series converges absolutely whenever $\abs{z-a} < R$ + and diverges whenever $\abs{z-a} > R$, or +- the series converges absolutely for all $z \in \CC$ (formally $R=\infty$). + +The number $R \in [0,\infty]$ is called the *radius of convergence* of the power series. +{% endtheorem %} {% proof %} TODO {% endproof %} -{: .theorem-title } -> Cauchy–Hadamard Formula -> -> Let $\sum_{n=0}^{\infty} x_n (z - a)^n$ be a Banach space valued power series -> with radius of convergence $R$. Then -> -> $$ -> \frac{1}{R} = \limsup_{n \to \infty} \norm{x_n}^{1/n}. -> $$ +{% theorem * Cauchy–Hadamard Formula %} +Let $\sum_{n=0}^{\infty} x_n (z - a)^n$ be a Banach space valued power series +with radius of convergence $R$. Then + +$$ +\frac{1}{R} = \limsup_{n \to \infty} \norm{x_n}^{1/n}. +$$ +{% endtheorem %} {% proof %} TODO diff --git a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md index b49cdf4..a2fa53d 100644 --- a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md +++ b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md @@ -3,16 +3,13 @@ title: The Calculus of Residues parent: One Complex Variable grand_parent: Complex Analysis nav_order: 4 -# cspell:words -#published: false --- # {{ page.title }} -{: .definition-title } -> Definition (Residue) -> -> TODO +{% definition Residue %} +TODO +{% enddefinition %} Calculation of Residues @@ -32,24 +29,17 @@ $$ \Res(f,c) = \frac{g(c)}{h'(c)} $$ +{% theorem * Residue Theorem (Basic Version) %} +Let $f$ be a function meromorphic in an open subset $G \subset \CC$. +Let $\gamma$ be a contour in $G$ such that +the interior of $\gamma$ is contained in $G$ +and contains finitely many poles $c_1, \ldots, c_n$ of $f$. +Then - - -{: .theorem-title } -> Residue Theorem (Basic Version) -> {: #residue-theorem-basic-version } -> -> Let $f$ be a function meromorphic in an open subset $G \subset \CC$. -> Let $\gamma$ be a contour in $G$ such that -> the interior of $\gamma$ is contained in $G$ -> and contains finitely many poles $c_1, \ldots, c_n$ of $f$. -> Then -> -> -> $$ -> \int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k) -> $$ -> {: .katex-display .mb-0 } +$$ +\int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k) +$$ +{% endtheorem %} {% proof %} {% endproof %} 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 index 5adc3f6..4e7666c 100644 --- a/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md +++ b/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md @@ -3,16 +3,12 @@ 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 }} } -> -> ... +{% theorem %} +{% endtheorem %} {% proof %} {% endproof %} diff --git a/pages/complex-analysis/several-complex-variables/index.md b/pages/complex-analysis/several-complex-variables/index.md index 49763d5..803eea4 100644 --- a/pages/complex-analysis/several-complex-variables/index.md +++ b/pages/complex-analysis/several-complex-variables/index.md @@ -3,7 +3,6 @@ title: Several Complex Variables parent: Complex Analysis nav_order: 2 has_children: true -# cspell:words --- # {{ page.title }} diff --git a/pages/complex-analysis/weak-and-strong-analyticity.md b/pages/complex-analysis/weak-and-strong-analyticity.md index 7db1dbf..c7ffb85 100644 --- a/pages/complex-analysis/weak-and-strong-analyticity.md +++ b/pages/complex-analysis/weak-and-strong-analyticity.md @@ -3,16 +3,9 @@ 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 %} |