From 6b79521290a430f4663777b71fc4264f6b982a3f Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Tue, 11 Nov 2025 15:09:36 +0100 Subject: [PATCH] Working --- .vscode/ltex.dictionary.de-DE.txt | 5 + .vscode/ltex.hiddenFalsePositives.de-DE.txt | 1 + 05-cauchy.tex | 6 +- 06-potenz.tex | 8 +- 09-singularities.tex | 117 ++++++++++++++++++++ Notizen/220603-Vorlesung.pdf | Bin 235771 -> 0 bytes Notizen/220615-Vorlesung.pdf | Bin 197884 -> 0 bytes SuggestionsForImprovement.md | 6 + 8 files changed, 135 insertions(+), 8 deletions(-) delete mode 100644 Notizen/220603-Vorlesung.pdf delete mode 100644 Notizen/220615-Vorlesung.pdf create mode 100644 SuggestionsForImprovement.md diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index 3c1796b..1d9aee1 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -48,3 +48,8 @@ Morera Novara Amandus Biholomorphie +Hebbarkeitssatz +Ostenfelde +Casorati +Casteggio +Hebbarkeitsatz diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt index 3e3598b..763371b 100644 --- a/.vscode/ltex.hiddenFalsePositives.de-DE.txt +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -26,3 +26,4 @@ {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QEs ist \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\QId.\\E$"} {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QInsbesondere gibt es offene Umgebung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, sodass auf \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q eine \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q-te Wurzelfunktion existiert, also eine Funktion \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q sodass für jedes \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q die Gleichung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q gilt.\\E$"} {"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QWegen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q gibt es eine offene Umgebung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q auf der eine \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q-te Wurzelfunktion existiert, also eine Funktion \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q sodass für jedes \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q die Gleichung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q gilt.\\E$"} +{"rule":"UNPAIRED_BRACKETS","sentence":"^\\QDie Funktionen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q und \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q stimmen demnach auf \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q überein, sind nach Korollar\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q (“Identitätssatz für holomorphe Funktionen”) also gleich!\\E$"} diff --git a/05-cauchy.tex b/05-cauchy.tex index f02aea4..ff02516 100644 --- a/05-cauchy.tex +++ b/05-cauchy.tex @@ -353,9 +353,9 @@ arbeiten. \end{proof} \begin{kor}[Hebbarkeitssatz]\label{kor:5-2-7}% - Sei $U ⊂ ℂ$ offen und sei $L ⊂ ℂ$ eine Gerade (nicht unbedingt durch den - Ursprung). Falls $f : U → ℂ$ stetig und $f|_{U ∖ L} ∈ 𝒪(U ∖ L)$ ist, dann - ist $f ∈ 𝒪(U)$. + \index{Hebbarkeitssatz}Sei $U ⊂ ℂ$ offen und sei $L ⊂ ℂ$ eine Gerade (nicht + unbedingt durch den Ursprung). Falls $f : U → ℂ$ stetig und $f|_{U ∖ L} ∈ + 𝒪(U ∖ L)$ ist, dann ist $f ∈ 𝒪(U)$. \end{kor} \begin{proof} Nach Drehung (= Multiplikation mit einer Zahl der Form $e^{iα}$ aus $ℂ^*$) und diff --git a/06-potenz.tex b/06-potenz.tex index 1366142..f534d32 100644 --- a/06-potenz.tex +++ b/06-potenz.tex @@ -64,8 +64,6 @@ Man beachte: Genau wie in der reellen Situation machen wir in \ref{il:6-0-2-4} keinerlei Aussagen über Konvergenz der Reihe für Punkte $z$, die auf dem Rand $∂ B_r(ρ)$ der Kreisscheibe liegen. -\sideremark{Vorlesung 10} - \begin{bsp}[Exponentialreihe] \index{Exponentialreihe}Die Potenzreihe $\sum_{i=0}^∞ \frac{zⁱ}{i!}$ kennen wir schon, das ist die Exponentialfunktion. Der Konvergenzradius ist $∞$. @@ -151,9 +149,9 @@ B_r(ρ)$ der Kreisscheibe liegen. \section{Holomorphe Funktionen liefern Potenzreihen} \begin{proberechnung}[Konstruktion von Potenzeihen]\label{prorech:6-1-2}% - Es sei $r > 0$ und es sei $f ∈ 𝒪(B_r(0))$. Wenn $ρ < r$ eine positive Zahl - ist, dann nach der Integralformel von Cauchy, Satz~\ref{satz:4-4-1}, für jede - Zahl $w ∈ B_ρ(0)$ die Gleichung + \sideremark{Vorlesung 10}Es sei $r > 0$ und es sei $f ∈ 𝒪(B_r(0))$. Wenn $ρ + < r$ eine positive Zahl ist, dann nach der Integralformel von Cauchy, + Satz~\ref{satz:4-4-1}, für jede Zahl $w ∈ B_ρ(0)$ die Gleichung \begin{equation}\label{eq:6-0-6-1} f(w) = \frac{1}{2πi} \int_{∂B_ρ(0)} \frac{f(z)}{z - w}\,dz. \end{equation} diff --git a/09-singularities.tex b/09-singularities.tex index 3898db3..d94de90 100644 --- a/09-singularities.tex +++ b/09-singularities.tex @@ -56,4 +56,121 @@ Wir interessieren uns für die folgende Situation: Sei $U ⊂ ℂ$ ein Gebiet, s \end{enumerate} \end{definition} +\begin{satz}[Hebbarkeitsatz von Riemann]\label{satz:9-1-hebbarkeit}% + \index{Hebbarkeitsatz von Riemann}In der Situation von + Definition~\ref{def:9-1-1} sei $ρ ∈ T$. Falls die Funktion $|f|$ in der Nähe + von $ρ$ beschränkt ist, dann hat $f$ bei $ρ$ eine hebbare Singularität. + Genauer: Falls es $ε > 0$ und $M > 0$ gibt, sodass für jedes $z ∈ B_ε(ρ) ∖ T$ + die Ungleichung $|f(z)| < M$ gilt, dann hat $f$ bei $ρ$ eine hebbare + Singularität. +\end{satz} +\begin{proof} + Nach Verkleinerung von $U$ können wir ohne Einschränkung annehmen, dass $U$ + eine Kreisscheibe um $ρ$ ist und dass $ρ$ der einzige Punkt in $U ∩ T$ ist. + Betrachte die Funktion + \[ + φ : U \longrightarrow ℂ, \quad z ↦ \begin{cases} + f(z)·(z - ρ) & \text{falls } z ≠ ρ \\ + 0 & \text{falls } z = ρ. + \end{cases} + \] + Per Annahme ist diese Funktion stetig, und auf $U ∖ \{ρ\}$ holomorph. Also + ist nach Korollar~\ref{kor:5-2-7} („Hebbarkeitssatz“) die Abbildung $φ$ auf + ganz $U$ holomorph. Weil $φ$ aber bei $ρ$ eine Nullstelle hat, finden durch + Potenzreihenentwicklung eine holomorphe Funktion $g ∈ 𝒪(U)$, sodass für jedes + $z ∈ U$ die Gleichung $φ(z) = (z - ρ)·g(z)$ gilt. Die Funktionen $g$ und $f$ + stimmen demnach auf $U ∖ \{ρ\}$ überein, sind nach Korollar~\ref{kor:7-2-2} + („Identitätssatz für holomorphe Funktionen“) also gleich! +\end{proof} + +\begin{bemerkung} + Die Aussage ist in der reellen Analysis überhaupt nicht differenzierbar! + Betrachte die Funktion $f : ℝ² ∖ \{0\} → ℝ$, $\vec{v} ↦ |\vec{v}|$. Die + Abbildung ist stetig und außerhalb von $0$ differenzierbar. Aber auf ganz + $ℝ²$ überhaupt nicht differenzierbar. +\end{bemerkung} + +\begin{frage} + Was kann ich über Beiträge von Funktionen mit Polstelle sagen? +\end{frage} + +Als Antwort eine Beispielrechnung. + +\begin{rem}[Beispielrechnung zu Funktionen mit Polstellen] + In der Situation von Definition~\ref{def:9-1-1} sei $ρ ∈ T$ eine Polstelle der + Ordnung $n > 0$ hat. Also gibt es eine Zahl $ε > 0$ und eine holomorphe + Funktion $g ∈ 𝒪(B_ε(ρ))$, sodass folgendes gilt. + \begin{enumerate} + \item Die Kreisscheibe $B_ε(ρ)$ liegt in $U$. + \item Es ist $g(ρ) ≠ 0$. + \item Für jedes $z ∈ B_ε(ρ) ∖ \{ρ\}$ gilt die Gleichung $g(z) = + (z - ρ)^n · f(z)$. + \end{enumerate} + Die Funktion $|g|$ ist stetig. Wenn ich $ε$ verkleinere, kann ich ohne + Beschränkung der Allgemeinheit zusätzlich annehmen, dass für jedes $z ∈ + B_ε(ρ)$ die Gleichung + \[ + |g(z)| > \tfrac{1}{2} |g(ρ)| + \] + gilt. Also ist für jedes $z ∈ B_ε(ρ) ∖ \{ρ\}$ + \[ + \frac{1}{2} |g(ρ)| < |g(z)| = |z - ρ|^n · |f(z)| < ε^n · |f(z)|. + \] + Die Beispielrechnung endet hier. +\end{rem} + +Zusammengefasst sagt uns die Beispielrechnung folgendes: Hat $f$ bei $ρ$ einen +Pol, dann gilt für jedes $M ∈ ℝ⁺$ und alle ausreichend kleinen $ε > 0$. +\begin{enumerate} + \item Es ist $B_ε(ρ) ⊂ U$ und $ρ$ ist die einzige Singularität von $f$ auf + $B_ε(ρ)$. + + \item Für jedes $z ∈ B_ε(ρ) ∖ \{ρ\}$ ist $|f(z)| > M$. +\end{enumerate} +Die Funktionswerte von $f$ explodieren also betragsmäßig, wenn ich mich dem +Punkt $ρ$ annähere. Auf jeden Fall sind die Funktionswerte von $0$ weg +beschränkt. \textbf{Das ist bei wesentlichen Singularitäten ganz anders!} + +\begin{satz}[Satz von Casorati\footnote{Felice Casorati (* 17.~Dezember 1835 in + Pavia; † 11.~September 1890 in Casteggio) war ein italienischer + Mathematiker.}-Weierstraß\footnote{Karl Theodor Wilhelm Weierstraß (* + 31.~Oktober 1815 in Ostenfelde bei Ennigerloh, Münsterland; † 19.~Februar 1897 + in Berlin) war ein deutscher Mathematiker, der sich um die logisch fundierte + Aufarbeitung der Analysis verdient gemacht hat. Daneben leistete er + bahnbrechende Beiträge in verschiedenen Gebieten der Mathematik, wie der + Funktionentheorie, der Variationsrechnung, der Differenzialgeometrie und der + Theorie der elliptischen Funktionen.}]\label{satz:9-1-casorati-weierstrass}% + \index{Satz von Casorati-Weierstraß}In der Situation von + Definition~\ref{def:9-1-1} sei $ρ ∈ T$. Falls $f$ bei $ρ$ eine wesentliche + Singularität hat, dann ist $f(U ∖ T) ⊂ ℂ$ dicht. +\end{satz} + +\begin{bemerkung}[Anwendungsidee] + Betrachte weiter die Situation von Definition~\ref{def:9-1-1}. Wenn $ρ ∈ T$ + eine wesentliche Singularität ist, dann gilt für jedes $ε > 0$, dass die + Bildmenge $f\bigl(B_ε(ρ) ∖ (U ∖ T)\bigr)$ in ganz $ℂ$ dicht ist. Die + Funktionswerte sind also in der Nähe von $ρ$ kein bisschen von $0$ weg + beschränkt --- ganz im Gegensatz zum Verhalten von Funktionen mit Polstellen. +\end{bemerkung} + +\begin{proof} + Wir beweisen die Kontraposition: Angenommen, $f(U ∖ T) ⊂ ℂ$ wäre nicht dicht. + Dann gibt es einen Punkt $z_0 ∈ ℂ$ und ein $ε > 0$, sodass $B_ε(z_0)$ die + Bildmenge $f(U ∖ T)$ nicht schneidet. Betrachte als Nächstes die Funktion $f + - z_0$, dann gilt für jedes $z ∈ U ∖ T$ die Ungleichung + \[ + |f(z) - z_0| > ε. + \] + Jetzt betrachte + \[ + \frac{1}{f - z_0} ∈ 𝒪\bigl(U ∖ T\bigr) + \] + und stelle fest: Die Beträge dieser Funktion sind nach oben durch + $\frac{1}{ε}$ beschränkt. Also sind nach Satz~\ref{satz:9-1-hebbarkeit} + („Hebbarkeitsatz von Riemann“) alle Singularitäten hebbar und es gibt eine + holomorphe Funktion $h ∈ 𝒪(U)$ die auf $U ∖ T$ mit $\frac{1}{f - z_0}$ + übereinstimmt. Es folgt direkt, dass $f = h^{-1} + z_0$ nur Polstellen und + deshalb keine wesentlichen Singularitäten hat. +\end{proof} + % !TEX root = Funktionentheorie diff --git a/Notizen/220603-Vorlesung.pdf b/Notizen/220603-Vorlesung.pdf deleted file mode 100644 index d3286b9252ad2bce28ec120d65ec4ad91cd6b094..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 235771 zcmbrk1yo(jvNnpldvJGHxVyVUaCdiicX!v|?(P9XfZ*=#9vmLoXP@33+%30V*E$CWHWC z17`zkJF`#T=LaC~;%x8YENSa(V(aWgs71*3*+Pj>=kK=)24*Hsge-q}e{*C_oDGaW zH4rlW&E(^QF|jrN)5#ys|LpAF;(xbRb}@AR%LsK@nZH>wCbnkI=7fxlEF7PlPi;=l zjwS{+Fg*P3_69~)CeDP0CT144yu_ECy~KnT#=OK|S!Eey?S)OuEhIc0O_V+5RE#{W zj5v*n`T1aY+_~Is>}@`cBy_j2wsqoi=Os4yw9bU<^Z5@OK>R7>Xl%-*BqH{=$S1{1 zZ0_uA&jkRuxw+B1G1J>QngJL&IXM9gOaLY(x=#r@Cl6a^19v)GCz4MMpY!=ghlq)j zk)wsZvxS{4;UC(-(9Xq~mxAJN#rB_yDgIV$>qP&T9rQ+aHh|9;07iOd0L33opSnJ2 zE(J$BV;3V6M?NPb16yZ1J8LJNKXU)!|3_5X#KX54--iwtmjq z|J_!11N%=~8UC^LQ;)&{m_ycF(q 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