From 4fef3e94f14ea9397a199c70497e828913619f5c Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Tue, 26 Aug 2025 12:21:50 +0200 Subject: [PATCH] Next lecture --- .vscode/ltex.dictionary.de-DE.txt | 2 + .vscode/ltex.hiddenFalsePositives.de-DE.txt | 2 + 01-Wiederholung.tex => 01-komplexeZahlen.tex | 241 ++++++++++++++++++- 02-diffbarkeit.tex | 167 +++++++++++++ Funktionentheorie.tex | 3 +- Notizen/220429-Vorlesung.pdf | Bin 263380 -> 0 bytes 6 files changed, 404 insertions(+), 11 deletions(-) rename 01-Wiederholung.tex => 01-komplexeZahlen.tex (52%) create mode 100644 02-diffbarkeit.tex delete mode 100644 Notizen/220429-Vorlesung.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index 44e16a1..3a58356 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -1 +1,3 @@ Multiplikationsabbildung +Majorantenkriterium +Summenregel diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt index 45be18f..00cba17 100644 --- a/.vscode/ltex.hiddenFalsePositives.de-DE.txt +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -1,3 +1,5 @@ {"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QFür komplexe Zahlen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q bezeichnet man den Winkel zwischen der \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q-Achse und der Achse \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q als das “Argument” von \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} {"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QMan invertiert eine komplexe Zahl, indem man den Betrag invertiert und das Argument negiert.\\E$"} {"rule":"GERMAN_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QEs ist \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} +{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qist.\\E$"} +{"rule":"GERMAN_SPELLER_RULE","sentence":"^\\QEine Abbildung \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q heißt bei \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q differenzierbar mit Ableitungsmatrix \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\QMat\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, wenn \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} diff --git a/01-Wiederholung.tex b/01-komplexeZahlen.tex similarity index 52% rename from 01-Wiederholung.tex rename to 01-komplexeZahlen.tex index 927910d..9f3b1fe 100644 --- a/01-Wiederholung.tex +++ b/01-komplexeZahlen.tex @@ -5,12 +5,10 @@ \section{Konstruktion, Notation und elementare Eigenschaften} -\sideremark{Vorlesung 1} - -In der Vorlesung „Analysis I“ haben Sie Funktionen $f : ℝ → ℝ$ diskutiert. In -der Vorlesung „Funktionentheorie“ diskutieren wir Funktionen $f : ℂ → ℂ$. Damit -alle auf demselben Stand sind, wiederhole ich in aller Kürze die wesentlichen -Begriffe zu den komplexen Zahlen. +\sideremark{Vorlesung 1}In der Vorlesung „Analysis I“ haben Sie Funktionen $f : +ℝ → ℝ$ diskutiert. In der Vorlesung „Funktionentheorie“ diskutieren wir +Funktionen $f : ℂ → ℂ$. Damit alle auf demselben Stand sind, wiederhole ich in +aller Kürze die wesentlichen Begriffe zu den komplexen Zahlen. \begin{konstruktion}[Konstruktion der komplexen Zahlen] Wir definieren auf der reellen Ebene $ℝ²$ zwei Abbildungen (oder: @@ -219,9 +217,9 @@ Quadratwurzel besitzt. \end{kons} \begin{proof} Gegeben eine Zahl $z ∈ ℂ$. Falls $z = 0$ ist, dann ist $w := 0$ eine - Quadratwurzel von $z$. Ansonsten sei $λ := |z|$ und $α := \arg z$. Weiter sei - $w ∈ ℂ^*$ die Zahl mit Betrag $\sqrt{λ}$ und Argument $α/2$. Mit dieser Wahl - ist $w² = z$, die Zahl $w$ ist also eine Quadratwurzel von $z$. + Quadratwurzel von $z$. Ansonsten sei $λ := |z|$ und $α := \arg z$. Weiter + sei $w ∈ ℂ^*$ die Zahl mit Betrag $\sqrt{λ}$ und Argument $α/2$. Mit dieser + Wahl ist $w² = z$, die Zahl $w$ ist also eine Quadratwurzel von $z$. \end{proof} Quadratwurzeln sind natürlich nicht eindeutig, denn wenn $w$ eine Quadratwurzel @@ -237,7 +235,7 @@ Aber Achtung! Obwohl jede komplexe Zahl eine Quadratwurzel besitzt, gibt es keine stetige Wurzelfunktion! Das folgende Lemma zeigt, dass es nicht einmal auf der Teilmenge $ℂ^* ⊂ ℂ$ eine stetige Wurzelfunktion geben kann. -\begin{lem}[Nichtexistenz einer stetigen Wurzelfunktion] +\begin{lem}[Nichtexistenz einer stetigen Wurzelfunktion]\label{lem:1-1-23}% Es sei $w: ℂ^* → ℂ^*$ eine Funktion, sodass für alle $z ∈ ℂ^*$ gilt: $w(z)² = z$. Dann ist die Abbildung $w$ nicht stetig. \end{lem} @@ -260,5 +258,228 @@ auf der Teilmenge $ℂ^* ⊂ ℂ$ eine stetige Wurzelfunktion geben kann. \end{proof} +\section{Die Exponentialfunktion} + +\sideremark{Vorlesung 2}In der Vorlesung ``Analysis'' wurden Folgen und +Grenzwerte in $ℂ = ℝ²$ ausführlich betrachtet. + +\begin{erinnerung}[Konvergenz von Folgen und Reihen] + Eine Folge von komplexen Zahlen, $(z_n)_{n ∈ ℕ}$, mit $z_n = x_n + i · y_n$ + konvergiert genau dann (absolut), wenn die Folge $x_n$ der Realteile und die + Folge $y_n$ der Imaginärteile jeweils einzeln (absolut) konvergieren. Analoges + gilt für Reihen $\sum_{n=0}^{∞} z_n$. +\end{erinnerung} + +\begin{erinnerung}[Stetigkeit] + Der Begriff der Stetigkeit für Funktionen $f: ℂ → ℂ$ ist der Begriff der + Stetigkeit für Abbildungen $ℝ² → ℝ²$. +\end{erinnerung} + +Das wichtigste Beispiel für Reihen war die Exponentialfunktion. + +\begin{proposition}[Exponentialreihe] + Für jede komplexe Zahl $z ∈ ℂ$ konvergiert die Reihe + \[ + \sum_{k=0}^{∞} \frac{z^k}{k!} + \] + absolut. +\end{proposition} +\begin{proof} + Gegeben $z ∈ ℂ$ und $k ∈ ℕ$, so gilt + \[ + |\operatorname{Re}(z^k)| ≤ |z^k| = |z|^k + \quad\text{und}\quad + |\operatorname{Im}(z^k)| ≤ |z^k| = |z|^k. + \] + Verwende jetzt dieselbe Argumentation wie bei der Diskussion der + Exponentialreihe in Analysis I/II. Alternativ: verwende das + Majorantenkriterium mit Reihe $\sum \frac{1}{k!} |z|^k$.. +\end{proof} + +\begin{notation}[Exponentialfunktion] + Die Abbildung + \[ + \exp : ℂ → ℂ, \quad z ↦ \sum_{k=0}^{∞} \frac{z^k}{k!} + \] + wird als komplexe Exponentialfunktion\index{Exponentialfunktion} bezeichnet. +\end{notation} + + +\subsection{Die Exponentialfunktion an reellen und rein imaginären Stellen} + +\begin{beobachtung}[Komplexe Exponentialfunktion an reellen Stellen] + Für reelle Zahlen $z$ stimmt die komplexe Exponentialfunktion mit der reellen + Exponentialfunktion überein, die wir aus der Vorlesung ``Analysis'' kennen. +\end{beobachtung} + +\begin{beobachtung}[Komplexe Exponentialfunktion an rein imaginären Stellen] + Für reelle Zahlen $α$ gilt + \begin{align*} + \exp(i·α) &= \cos α + i·\sin α\\ + \exp(-i·α) &= \cos(-α) + i·\sin(-α) = \cos(α) - i·\sin(α) = \overline{\exp(i·α)} + \end{align*} +\end{beobachtung} +\begin{proof} + Vergleiche die Exponentialreihe $\sum_{k=0}^{∞} \frac{(i·α)^k}{k!} = + \sum_{k=0}^{∞} \frac{i^k · z^k}{k!}$ mit den bekannten Reihen für Sinus und + Cosinus. Beachte, dass + \[ + i^k = + \begin{cases} + 1 & \text{falls } k \equiv 0 \bmod 4\\ + i & \text{falls } k \equiv 1 \bmod 4\\ + -1 & \text{falls } k \equiv 2 \bmod 4\\ + -i & \text{falls } k \equiv 3 \bmod 4. + \end{cases} + \] + Deshalb ist + \begin{align*} + \exp(i α) &= \begin{pmatrix} \frac{1}{0!} \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ \frac{α}{1!} \end{pmatrix} + \begin{pmatrix} \frac{-α²}{2!} \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ \frac{-α³}{3!} \end{pmatrix} + \begin{pmatrix} \frac{α⁴}{4!} \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ \frac{α⁵}{5!} \end{pmatrix} + …\\ + &= \begin{pmatrix} \cos α \\ \sin α \end{pmatrix} = (\cos α) + i(\sin α) + \end{align*} + ist. Die Aussage für $\exp(-i·α)$ beweist man analog. +\end{proof} + +\begin{kons}[Betrag und Argument von $\exp (i·α)$] + Für jede reelle Zahl $α$ gilt + \[ + |\exp (i·α)| = 1 + \quad\text{und}\quad + \arg \exp (i·α) = α. + \] +\end{kons} + +\begin{kons}[Sinus und Kosinus in Termen der komplexen Exponentialfunktion] + Für jede reelle Zahl $α$ gilt + \begin{align} + \label{eq:1-2-7-1} \cos α &= \frac{1}{2} \left( \exp(iα) + \exp(-iα) \right) \\ + \label{eq:1-2-7-2} \sin α &= \frac{1}{2i} \left( \exp(iα) - \exp(-iα) \right) + \end{align} +\end{kons} + +Die rechten Seiten von \eqref{eq:1-2-7-1} und \eqref{eq:1-2-7-2} ergeben nicht +nur für reelle Zahlen $α$ Sinn, sondern sind für beliebige $α ∈ ℂ$ definiert. + +\begin{definition}[Komplexe Sinus und Cosinus] + Definiere die komplexe Sinus\index{Sinus} und Cosinus\index{Cosinus} + Funktionen wie folgt, + \begin{align*} + \cos: ℂ → ℂ, &\quad z ↦ \frac{1}{2} \left( \exp(i·z) + \exp(-i·z) \right)\\ + \sin: ℂ → ℂ, &\quad z ↦ \frac{1}{2i} \left( \exp(i·z) - \exp(-i·z) \right). + \end{align*} +\end{definition} + +Wie bei der komplexen Exponentialfunktion beachte man, dass der komplexe Sinus +und Cosinus für reelle Argumente mit den bekannten Funktionen übereinstimmen. + + +\subsection{Die Funktionalgleichung der Exponentialfunktion} + +Genau wie in der Vorlesung Analysis rechnet man die folgenden beiden +Eigenschaften der komplexen Exponentialfunktion nach. + +\begin{fakt}[Exponentialfunktion als Grenzwert]\label{fakt:1-2-10} + Für jede komplexe Zahl $z$ gilt + \[ + \exp z = \lim_{n → ∞} \left( 1 + \frac{z}{n} \right)^n. + \] +\end{fakt} + +\begin{fakt}[Die Funktionalgleichung der Exponentialfunktion]\label{fakt:1-2-11} + Für je zwei komplexe Zahlen $z_1$, $z_2$ gilt + \[ + \exp (z_1 + z_2) = (\exp z_1) + (\exp z_2). + \] +\end{fakt} + + +\subsection{Die geometrische Bedeutung der Exponentialfunktion} + +Gegeben irgendeine komplexe Zahl $z = x + i·y$, dan ist +\[ + \exp(z) = \exp(x + i·y) = \exp(x) · \exp(iy) = \exp(x) · (\cos y + i \sin y) +\] +Damit ist die geometrische Bedeutung der Exponentialfunktion klar: $\exp(x + +iy)$ ist die eindeutig bestimmte Zahl mit Betrag $|\exp z| = \exp x > 0$ und +Argument $\arg (\exp z) = y$. + +\begin{kons} + Die Exponentialabbildung nimmt nur Werte in $ℂ^*$ an, kann also als Abbildung + $\exp : ℂ → ℂ^*$ aufgefasst werden. \qed +\end{kons} + +\begin{kons}[Surjektivität der Exponentialabbildung]\label{kons:1-2-13}% + Die Exponentialabbildung $\exp : ℂ → ℂ^*$ ist surjektiv. +\end{kons} +\begin{proof} + Sei $z ∈ ℂ^*$ gegeben. Dann ist $\exp((\log |z|) + i·\arg z)$ eine komplexe + Zahl, deren Betrag und Argument mit $z$ übereinstimmt. Also sind die beiden + Zahlen gleich. +\end{proof} + +\begin{kons}\label{kons:1-2-14}% + Die Exponentialabbildung $\exp : ℂ → ℂ^*$ ist nicht injektiv. Genauer: + Gegeben $z_1, z_2 ∈ ℂ$, dann sind folgende Aussagen äquivalent. + \begin{enumerate} + \item Es ist $\exp(z_1) = \exp(z_2)$. + + \item Es existiert eine ganze Zahl $n ∈ ℤ$, sodass $z_1 - z_2 = (2π i) · n$ + ist. + \end{enumerate} +\end{kons} +\begin{proof} + Schreibe $z_1 = x_1 + i·y_1$ und $z_2 = x_2 + i·y_2$. Dann ist + \begin{align*} + \exp(z_1) = \exp(z_2) & ⇔ \exp(x_1) · (\cos y_1 + i \sin y_1) = \exp(x_2) · (\cos y_2 + i \sin y_2) \\ + & ⇔ x_1 = x_2 \text{ und } (\cos y_1 + i \sin y_1) = (\cos y_2 + i \sin y_2) \\ + & ⇔ x_1 = x_2 \text{ und } \cos y_1 = \cos y_2 \text{ und } \sin y_1 = \sin y_2 \\ + & ⇔ ∃ n ∈ ℤ: z_1 - z_2 = (2π i) · n. \qedhere + \end{align*} +\end{proof} + + +\subsection{Komplexe Logarithmen} + +Konsequenez~\ref{kons:1-2-13} impliziert, dass jede komplexe Zahl $z ∈ ℂ^*$ +einen Logarithmus hat, also eine Zahl $w$ mit $\exp(w) = z$. Aber: Genau wie +bei der Quadratwurzel gibt es keine stetige Logarithmusfunktion + +\begin{lem}[Nichtexistenz einer stetigen Logarithmusfunktion] + Es sei $\log : ℂ^* → ℂ$ eine Funktion, sodass für alle $z ∈ ℂ^*$ gilt: $\exp + \log z = z$. Dann ist die Abbildung $\log$ nicht stetig. +\end{lem} +\begin{proof} + Hausaufgabe! Haben Sie den Beweis von Konsequenz~\ref{kons:1-2-13} wirklich + verstanden? +\end{proof} + +\begin{notation}[Hauptzweig des Logarithmus] + Die unstetige Funktion + \[ + \log: ℂ^* → ℂ, \quad z ↦ \begin{pmatrix} \ln |z| \\ \arg z \end{pmatrix} + \] + wird in der Literatur also ``Hauptzweig des Logarithmus''\index{Hauptzweig des + Logarithmus} bezeichnet. +\end{notation} + + +\subsection{Die Exponentialfunktion als Gruppenmorphismus} + +Wenn ich beachte, dass $\exp(0) = 1$ ist, dann kann ich die +Fakten~\ref{fakt:1-2-10} und \ref{fakt:1-2-10} auch anders ausdrücken: Die +Exponentialabbildung $\exp: ℂ → ℂ^*$ ist ein Gruppenmorphismus zwischen der +additiven Gruppe $(ℂ, +)$ und der multiplikativen Gruppe $(ℂ^*, ·)$. + +Konsequenz~\ref{kons:1-2-13} sagt, dass dieser Gruppenmorphismus surjektiv ist. +Also ist $(ℂ^*, ·)$ isomorph zur Quotientengruppe, +\[ + ℂ^* ≅ \factor{ℂ}{\ker \exp}. +\] +Konsequenz~\ref{kons:1-2-14} sagt, was der Kern des Gruppenmorphismus ist, +\[ + \ker \exp = (2π i) · ℤ ⊂ ℂ. +\] + + % !TEX root = LineareAlgebra2 diff --git a/02-diffbarkeit.tex b/02-diffbarkeit.tex new file mode 100644 index 0000000..b2fe0b1 --- /dev/null +++ b/02-diffbarkeit.tex @@ -0,0 +1,167 @@ +% spell checker language +\selectlanguage{german} + +\chapter{Differenzierbarkeit} + +\section{Holomorphe Funktionen} + +Wir definieren in wenigen Zeilen, komplexe Differenzierbarkeit von Funktionen +$ℂ → ℂ$. Vorher erinnern wir kurz an die relevanten Begriffe aus den +Analysis-Vorlesungen. + +\begin{definition}[Reelle Differenzierbarkeit]\label{def:2-1-1} + Es sei $U ⊂ ℝ$ offen und $p ∈ U$. Eine Funktion $f: U → ℝ$ ist + bei $p$ differenzierbar mit Ableitung $δ ∈ ℝ$, wenn + \[ + \lim_{h → 0} \frac{f(p+h) - f(p)}{h} = δ + \] + ist. +\end{definition} + +\begin{bemerkung} + Reell differenzierbare Funktionen sind stetig. Es gelten die Summenregel, + Produktregel, Quotientenregel, Kettenregel, … +\end{bemerkung} + +\begin{definition}[Komplexe Differenzierbarkeit]\label{def:2-1-3} + Es sei $U ⊂ ℂ$ offen und $p ∈ U$. Eine Funktion $f: U → ℂ$ ist + bei $p$ komplex differenzierbar\index{komplex differenzierbar} mit Ableitung + $δ ∈ ℂ$, wenn + \[ + \lim_{h → 0} \frac{f(p+h) - f(p)}{h} = δ + \] + ist. +\end{definition} + +\begin{bemerkung} + Genau wie in der Vorlesung „Analysis“ beweist man: komplex differenzierbare + Funktionen sind stetig. Es gelten die Summenregel, Produktregel, + Quotientenregel, Kettenregel, … +\end{bemerkung} + +\begin{definition}[Holomorphie an einem Punkt] + Es sei $U ⊂ ℂ$ offen und $p ∈ U$. Eine Funktion $f: U → ℂ$ heißt + „holomorph\index{holomorph} bei $p$“, wenn es eine offene Umgebung $p ∈ V ⊂ U$ + gibt, sodass $f$ bei jedem Punkt aus $V$ komplex differenzierbar ist. Die + Menge der Funktionen, die bei $p$ holomorph sind, wird mit $𝒪_p$ bezeichnet. +\end{definition} + +\begin{definition}[Holomorphie auf einer offenen Menge] + Es sei $U ⊂ ℂ$ offen und $p ∈ U$. Eine Funktion $f: U → ℂ$ heißt „holomorph + auf $U$“, wenn $f$ bei jedem Punkt aus $U$ komplex differenzierbar ist. In + diesem Fall wird die Ableitungsfunktion mit $f' : U → ℂ$ bezeichnet. Die + Menge der Funktionen, die auf $U$ holomorph sind, wird mit $𝒪(U)$ + bezeichnet. +\end{definition} + + +\section{Komplex Differenzierbarkeit und Differenzierbarkeit} + +Definitionen~\ref{def:2-1-1} und \ref{def:2-1-3} sehen völlig gleich aus. Das +wirft die Frage auf: Ist komplexe Differenzierbarkeit wirklich ein neuer +Begriff? Gibt es einen Unterschied zum Begriff der Differenzierbarkeit von +Abbildungen $ℝ² → ℝ²$, den wir schon lange aus der Vorlesung „Analysis“ kennen? + +\begin{erinnerung}[Differenzierbarkeit von Abbildungen $ℝ² → ℝ²$] + Es sei $U ⊂ ℝ²$ offen und $p ∈ U$. Eine Abbildung $f: U → ℝ²$ heißt bei $p$ + differenzierbar mit Ableitungsmatrix $A ∈ \text{Mat}(2 ⨯ 2, ℝ)$, wenn + \[ + \lim_{h → 0} \frac{|f(p+h) - f(p) - A · h|}{|h|} = 0. + \] + Dabei bedeutet der Limes: für alle Nullfolgen $(h_n)$ aus $ℝ²$ ist + \[ + \lim_{n → ∞} \frac{|f(p+h_n) - f(p) - A · h_n|}{|h_n|} = 0. + \] + Falls $f$ bei $p$ differenzierbar ist, dann wissen wir auch genau, wie die + Ableitungsmatrix $A$ aussieht. Dazu schreiben wir die Funktion $f$ zuerst in + Komponenten, + \[ + f(x,y) = \begin{pmatrix} f_1(x,y) \\ f_2(x,y) \end{pmatrix}. + \] + Dann ist $A$ die Matrix der partiellen Ableitungen, + \[ + A = \begin{pmatrix} \frac{∂}{∂ x}f_1(p) & \frac{∂}{∂ y} f_1(p) \\ \frac{∂}{∂ x} f_2(p) & \frac{∂}{∂ y} f_2(p) \end{pmatrix}. + \] +\end{erinnerung} + + +\subsubsection{Proberechnungen} + +Um den Unterschied von reeller und komplexer Differenzierbarkeit zu verstehen, +machen wir in diesem Abschnitt eine große Proberechnung. Es sei $U ⊂ ℂ$ offen +und $p ∈ U$. Weiter sei $f: U → ℂ$ bei $p$ komplex differenzierbar mit +Ableitung $δ = d_1 + i · d_2$. Es gilt also +\[ + \lim_{h → 0} \frac{f(p+h) - f(p)}{h} = δ = d_1 + i · d_2. +\] +Dabei bedeutet der Limes: für alle Nullfolgen $(h_n)$ aus $ℂ$ ist +\begin{equation}\label{eq:2-2-1-1} + \lim_{n → ∞} \frac{f(p+h_n) - f(p)}{h_n} = δ = d_1 + i · d_2. +\end{equation} +Um das jetzt besser zu verstehen, schreiben wir $p$ und $f$ zuerst einmal in +Komponenten, +\[ + p = p_1 + i·p_2 + \quad\text{und}\quad + f(x + iy) = f_1(x,y) + i · f_2(x,y). +\] + +\paragraph{Spezialfall: $h_n$ ist eine rein reelle Folge} + +Die Gleichung~\eqref{eq:2-2-1-1} gilt für alle Nullfolgen. Betrachten wir also +zuerst den Fall, wo $h_n$ eine Folge von reellen Zahlen ist. Dann ist +\begin{align*} +d_1 + i·d_2 &= \frac{f(p+h_n) - f(p)}{h_n}\\ +& = \lim_{n → ∞} \frac{f_1(p_1 + h_n, p_2) + i·f_2(p_1 + h_n, p_2) - f_1(p_1, p_2) - i·f_2(p_1, p_2)}{h_n} \\ +& = \frac{∂}{∂ x}f_1(p_1, p_2) + i \frac{∂}{∂ x} f_2(p_1, p_2). +\end{align*} +Wir sehen: +\begin{equation}\label{eq:2-2-1-2} + \frac{∂}{∂ x} f_1(p) = d_1 + \quad\text{und}\quad + \frac{∂}{∂ x} f_2(p) = d_2. +\end{equation} + +\paragraph{Spezialfall: $h_n$ ist eine rein imaginäre Folge} + +Die Gleichung~\eqref{eq:2-2-1-1} gilt für alle Nullfolgen. Betrachten wir also +als nächstes Folgen der Form $i·h_n$, wo $h_n$ eine Folge von reellen Zahlen +ist. Dann ist +\begin{align*} +d_1 + i·d_2 &= \frac{f(p+h_n) - f(p)}{i·h_n}\\ +& = \lim_{n → ∞} \frac{f_1(p_1, p_2 + h_n) + i·f_2(p_1, p_2 + h_n) - f_1(p_1, p_2) - i·f_2(p_1, p_2)}{i·h_n} \\ +& = -i·\lim_{n → ∞} \frac{f_1(p_1, p_2 + h_n) + i·f_2(p_1, p_2 + h_n) - f_1(p_1, p_2) - i·f_2(p_1, p_2)}{h_n} \\ +& = -i·\left(\frac{∂}{∂ y}f_1(p_1, p_2) + i \frac{∂}{∂ y} f_2(p_1, p_2)\right) \\ +& = \frac{∂}{∂ y} f_2(p_1, p_2)-i·\frac{∂}{∂ y}f_1(p_1, p_2). +\end{align*} +Wir sehen: +\begin{equation}\label{eq:2-2-1-3} + \frac{∂}{∂ y} f_2(p) = d_1 + \quad\text{und}\quad + \frac{∂}{∂ y} f_1(p) = -d_2. +\end{equation} +Wir beenden die Proberechnungen hier. Als Konsequenz halten wir Folgendes fest. + +\begin{prop}[Cauchy-Riemann Gleichungen] + Es sei $U ⊂ ℂ$ offen und $p ∈ U$. Weiter sei $f: U → ℂ$ bei $p$ komplex + differenzierbar. Schreibe $f$ in Komponenten, $f$ in Komponenten, $f = f_1 + + i · f_2$, mit $f_1, f_2 : U → ℝ$. Dann ist + \begin{equation}\label{eq:2-2-2-1} + \frac{∂}{∂ x} f_1(p) = \frac{∂}{∂ y} f_2(p) + \quad\text{und}\quad + \frac{∂}{∂ y} f_1(p) = -\frac{∂}{∂ x} f_2(p). + \end{equation} +\end{prop} +\begin{proof} + Vergleiche~\eqref{eq:2-2-1-2} und \eqref{eq:2-2-1-3}. +\end{proof} + +\begin{notation}[Cauchy-Riemann Gleichungen] + Man nennt das Gleichungssystem~\eqref{eq:2-2-2-1} die „Cauchy-Riemann + partiellen Differenzialgleichungen“\index{Cauchy-Riemann Gleichungen}. +\end{notation} + + + +% !TEX root = LineareAlgebra2 + diff --git a/Funktionentheorie.tex b/Funktionentheorie.tex index ba043e4..5ef844b 100644 --- a/Funktionentheorie.tex +++ b/Funktionentheorie.tex @@ -132,7 +132,8 @@ Link in den Text ein. % \part{Platzhalter} -\input{01-Wiederholung} +\input{01-komplexeZahlen} +\input{02-diffbarkeit} \addchap{Lizenz} diff --git a/Notizen/220429-Vorlesung.pdf b/Notizen/220429-Vorlesung.pdf deleted file mode 100644 index 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