diff --git a/refs.bib b/refs.bib index f3e2558..7f7e639 100644 --- a/refs.bib +++ b/refs.bib @@ -111,4 +111,14 @@ number={10}, pages={3211--3212}, year={1996} -} \ No newline at end of file +} +@book{ConwayComplex, + title={Functions of One Complex Variable I}, + author={Conway, J.B.}, + isbn={9780387903286}, + lccn={lc78018836}, + series={Functions of one complex variable / John B. Conway}, + url={https://books.google.ca/books?id=9LtfZr1snG0C}, + year={1978}, + publisher={Springer} +} diff --git a/src/dg/complex/derivative.tex b/src/dg/complex/derivative.tex new file mode 100644 index 0000000..43f1cb6 --- /dev/null +++ b/src/dg/complex/derivative.tex @@ -0,0 +1,34 @@ +\section{Complex Differentiability} +\label{section:complex-derivative} + +\begin{definition}[Complex Analytic] +\label{definition:complex-analytic} + Let $E$ be a separated locally convex space over $\complex$, $U \subset \complex$, and $f: U \to E$, then the following are equivalent: + \begin{enumerate} + \item $f \in C^1(U; E)$. + \item Under the identification of $C = \real^2$, $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \in C(U; E)$ and + \[ + \frac{\partial f}{\partial x} = i\frac{\partial f}{\partial y} + \] + \end{enumerate} +\end{definition} +\begin{proof} + (1) $\Rightarrow$ (2): Let $x_0 \in U$, then + \[ + \frac{\partial f}{\partial x} = \lim_{\substack{h \to 0 \\ h \in \real}}\frac{f(x_0 + h) - f(x_0)}{h} + = \lim_{h \to 0}\lim_{\substack{h \to 0 \\ h \in \real}}\frac{f(x_0 + ih) - f(x_0)}{ih} = \frac{1}{i} \frac{\partial f}{\partial y} + \] + + (2) $\Rightarrow$ (1): Let $x_0 \in U$ and + \[ + L: \complex \to E \quad a + bi \mapsto a \frac{\partial f}{\partial x}(x_0) + b \frac{\partial f}{\partial y}(x_0) + \] + + by assumption and \autoref{proposition:polarisation-linear}, $L \in L(\complex; E)$. By \autoref{proposition:partial-total-derivative}, $f \in C^1(U \subset \real^2; E)$, where for any $(a, b) \in \real^2$, + \[ + Df(x_0)(a, b) = a \frac{\partial f}{\partial x}(x_0) + b \frac{\partial f}{\partial y}(x_0) + \] + + so by definition of differentiability, $f$ is complex-differentiable at $x_0$ with derivative $L$. +\end{proof} + diff --git a/src/dg/complex/index.tex b/src/dg/complex/index.tex new file mode 100644 index 0000000..9a6467a --- /dev/null +++ b/src/dg/complex/index.tex @@ -0,0 +1,8 @@ +\chapter{Complex Analysis} +\label{chap:complex-analysis} + + +\input{./derivative.tex} +\input{./log.tex} + + diff --git a/src/dg/complex/log.tex b/src/dg/complex/log.tex new file mode 100644 index 0000000..c353574 --- /dev/null +++ b/src/dg/complex/log.tex @@ -0,0 +1,35 @@ +\section{The Complex Logarithm} +\label{section:complex-log} + +\begin{definition}[Branch of Logarithm] +\label{definition:branch-of-log} + Let $U \subset \complex$ be a connected open set with $0 \not\in U$ and $f \in C(U; \complex)$, then $f$ is a \textbf{branch of the logarithm} if for every $z \in U$, $z = \exp(f(z))$. +\end{definition} + +\begin{lemma} +\label{lemma:branch-of-log-shift} + Let $U \subset \complex$ be a connected open set with $0 \not\in U$, and $f, g \in C(U; \complex)$ be two branches of the logarithm, then there exists $k \in \integer$ such that $f - g = 2\pi k i$. +\end{lemma} +\begin{proof}[Proof, {{\cite[Proposition 2.19]{ConwayComplex}}}. ] + For each $x \in U$, there exists $k \in \integer$ such that $f(x) - g(x) = 2\pi k i$. Thus $f - g \in C(U; 2\pi i\integer)$. Since $U$ is connected, $(f - g)(U)$ must be a singleton. Therefore there exists $k \in \integer$ such that $f - g = 2\pi k i$. +\end{proof} + +\begin{proposition} +\label{proposition:branch-of-log-analytic} + Let $U \subset \complex$ be a connected open set with $0 \not\in U$, and $f \in C(U; \complex)$ be a branch of the logartihm, then $f$ is analytic. +\end{proposition} +\begin{proof} + By the \autoref{theorem:inverse-function-theorem}. +\end{proof} + + +\begin{definition}[Principal Logarithm] +\label{definition:principal-logarithm} + Let $U = \complex \setminus \bracs{z \in \real|z \le 0}$, then there exists a unique mapping $\ell: U \to \complex$ such that: + \begin{enumerate} + \item $\ell$ is a branch of the complex logarithm. + \item For each $re^{i\theta} \in U$, $\ell(r^{i\theta}) = \ln r + i\theta$. + \end{enumerate} + + The function $\ell$ is the \textbf{principal logarithm} on $U$. +\end{definition} diff --git a/src/dg/derivative/higher.tex b/src/dg/derivative/higher.tex index 9fdd0e8..ec22b11 100644 --- a/src/dg/derivative/higher.tex +++ b/src/dg/derivative/higher.tex @@ -50,6 +50,12 @@ Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, $U \subset E$ be open, and $n \in \natp$, then $D_\sigma^k(U; F)$/$\tilde D_\sigma^k(U; F)$ is the \textbf{space of $n$-fold $\sigma$/$\tilde \sigma$-differentiable functions} from $U$ to $F$. \end{definition} +\begin{definition}[Space of Continuously Differentiable Functions] +\label{definition:continuously-differentiable-space} + Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, $U \subset E$ be open, and $n \in \natp$, then $C_\sigma^k(U; F)$/$\tilde C_\sigma^k(U; F)$ is the \textbf{space of $n$-fold continuously $\sigma$/$\tilde \sigma$-differentiable functions} from $U$ to $F$. +\end{definition} + + \begin{theorem}[Symmetry of Higher Derivatives] \label{theorem:derivative-symmetric-frechet} Let $E, F$ be Banach spaces, $U \subset E$ be open, $n \in \natp$, and $f: U \to F$ be a function $n$-times Fréchet-differentiable at $x \in U$, then $D^nf(x) \in L^n(E; F)$ is symmetric. diff --git a/src/dg/derivative/index.tex b/src/dg/derivative/index.tex index d4ebb80..d57c3ca 100644 --- a/src/dg/derivative/index.tex +++ b/src/dg/derivative/index.tex @@ -6,5 +6,6 @@ \input{./mvt.tex} \input{./higher.tex} \input{./taylor.tex} +\input{./partial.tex} \input{./power.tex} \input{./inverse.tex} diff --git a/src/dg/derivative/partial.tex b/src/dg/derivative/partial.tex new file mode 100644 index 0000000..f2b4256 --- /dev/null +++ b/src/dg/derivative/partial.tex @@ -0,0 +1,61 @@ +\section{Partial Derivatives} +\label{section:partial-derivatives} + +\begin{definition}[Partial Derivative] +\label{definition:partial-derivative} + Let $E_1, E_2$ be TVSs over $K \in \RC$, $\sigma_1 \subset \mathfrak{B}(E_1)$ and $\sigma_2 \subset \mathfrak{B}(E_2)$ be covering ideals, $F$ be a separated TVS over $K$, $U \subset E_1 \times E_2$ be open, and $f: U \to F$. For each $(x_0, y_0) \in E$, let $f_{x_0}(y) = f(x_0, y)$ and $f_{y_0}(x) = f(x, y_0)$ be the partial maps of $f$. If $f_{x_0}$ is $\tilde \sigma_1$-differentiable for each $x_0$, and $f_{y_0}$ is $\tilde \sigma_2$-differentiable for each $y_0$, then + \[ + D_1f: U \to B_{\sigma_1}(E_1; F) \quad (x, y) \mapsto D_{\sigma_1}f_{x}(y) + \] + + and + \[ + D_2f: U \to B_{\sigma_2}(E_2; F) \quad (x, y) \mapsto D_{\sigma_2}f_{y}(x) + \] + + are the \textbf{partial derivatives} of $f$. +\end{definition} + +\begin{proposition} +\label{proposition:partial-total-derivative} + Let $E_1, E_2$ be TVSs over $K \in \RC$, $\sigma_1 \subset \mathfrak{B}(E_1)$ and $\sigma_2 \subset \mathfrak{B}(E_2)$ be covering ideals, $F$ be a separated locally convex space over $K$, $U \subset E_1 \times E_2$ be open, $f: U \to F$, and $p \ge 1$, then the following are equivalent: + \begin{enumerate} + \item $f \in \tilde C_{\sigma_1 \otimes \sigma_2}^p(U; F)$. + \item $D_1 f \in \tilde C_{\sigma_1 \otimes \sigma_2}^{p-1}(U; B_{\sigma_1}(E; F))$ and $D_2 f \in \tilde C_{\sigma_1 \otimes \sigma_2}^{p-1}(U; B_{\sigma_2}(E; F))$ + \end{enumerate} + + If the above holds, then for any $x \in U$ and $(h_1, h_2) \in E_1 \times E_2$, + \[ + D_{\sigma_1 \otimes \sigma_2}f(x)(h_1, h_2) = D_1f(x)(h_1) + D_2f(x)(h_2) + \] + +\end{proposition} +\begin{proof} + (2) $\Rightarrow$ (1): For each $(x, y) \in U$ and $(h_1, h_2) \in E_1 \times E_2$, + \begin{align*} + f(x + h_1, y + h_2) - f(x, y) &= f(x + h_1, y + h_2) - f(x + h_1, y) \\ + &+ f(x + h_1, y) - f(x, y) \\ + &= f(x + h_1, y + h_2) - f(x + h_1, y) \\ + &+ D_1f(x, y)(h_1) + r_1(h_1) + \end{align*} + + where $r_1 \in \mathcal{R}_{\sigma_1}(E_1; F)$. On the other hand, by the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem}, + \begin{align*} + &f(x + h_1, y + h_2) - f(x + h_1, y) - Df_2(x, y)(h_2) \\ + &\in h_2\ol{\text{Conv}}\bracs{D_2f(x + h_1, y + th_2) - Df_2(x, y)|t \in [0, 1]} + \end{align*} + + Since $D_2f$ is continuous and $F$ is locally convex, + \[ + f(x + h_1, y + h_2) - f(x + h_1, y) - Df_2(x, y)(h_2) = r_2(h_1, h_2) + \] + + where $r_2 \in \mathcal{R}_{\sigma_1 \otimes \sigma_2}(E_1 \times E_2; F)$. Therefore + \begin{align*} + f(x + h_1, y + h_2) - f(x, y) &= D_1f(x, y)(h_1) + D_2f(x, y)(h_2) \\ + &+ r_1(h_1) + r_2(h_1, h_2) + \end{align*} + + +\end{proof} + diff --git a/src/dg/index.tex b/src/dg/index.tex index 5f53eb1..ef51ae3 100644 --- a/src/dg/index.tex +++ b/src/dg/index.tex @@ -2,4 +2,5 @@ \label{part:diffgeo} \input{./derivative/index.tex} +\input{./complex/index.tex} \input{./notation.tex} diff --git a/src/dg/notation.tex b/src/dg/notation.tex index 86a1af6..48129cc 100644 --- a/src/dg/notation.tex +++ b/src/dg/notation.tex @@ -12,6 +12,9 @@ Differential geometry is the study of things invariant under change of notation. $D_\sigma f(x_0)$ & $\sigma$-derivative of $f$ at $x_0$. & \autoref{definition:derivative-sets} \\ $D_\sigma^n f$ & $n$-fold $\sigma$-derivative. & \autoref{definition:n-differentiable-sets} \\ $D_\sigma^n(U; F)$ & $n$-fold $\sigma$-differentiable functions. & \autoref{definition:differentiable-space} \\ + $\tilde D_\sigma^n(U; F)$ & $n$-fold $\tilde \sigma$-differentiable functions. & \autoref{definition:differentiable-space} \\ + $C_\sigma^n(U; F)$ & $n$-fold continuously $\sigma$-differentiable functions. & \autoref{definition:continuously-differentiable-space} \\ + $\tilde C_\sigma^n(U; F)$ & $n$-fold continuously $\tilde \sigma$-differentiable functions. & \autoref{definition:continuously-differentiable-space} \\ $L^{(n)}_\sigma(E; F)$ & Codomain of derivatives. $L^{(0)}_\sigma(E; F) = F$, $L^{(n)}_\sigma(E; F) = L(E; L_\sigma^{(n-1)}(E; F))$, equipped with the $\sigma$-uniform topology. & \autoref{definition:higher-derivatives-codomain} \\ $x^{(k)}$ & Tuple of $x$ repeated $k$ times. & \autoref{theorem:taylor-peano} \\ $D^+f(x)$ & Right derivative of $f$ at $x$. & \autoref{definition:right-differentiable-mvt} diff --git a/src/measure/notation.tex b/src/measure/notation.tex index 72856de..36badc1 100644 --- a/src/measure/notation.tex +++ b/src/measure/notation.tex @@ -7,6 +7,7 @@ $\sigma(\mathcal{E})$ & $\sigma$-algebra generated by $\mathcal{E}$. & \autoref{definition:generated-sigma-algebra} \\ $\lambda(\mathcal{E})$ & $\lambda$-system generated by $\mathcal{E}$. & \autoref{definition:generated-lambda-system} \\ + $\sigma \otimes \tau$ & Product of ideals. & \autoref{definition:product-ideal} \\ % ---- Measure Theory ---- $\mathcal{B}_X$ & Borel $\sigma$-algebra on $X$. & \autoref{definition:borel-sigma-algebra} \\ $\sigma(\{f_i \mid i \in I\})$ & $\sigma$-algebra generated by the maps $\{f_i\}$. & \autoref{definition:generated-sigma-algebra-function} \\ diff --git a/src/topology/functions/ideal.tex b/src/topology/functions/ideal.tex index 4c3747f..0bf6dd3 100644 --- a/src/topology/functions/ideal.tex +++ b/src/topology/functions/ideal.tex @@ -47,6 +47,25 @@ (2) $\Rightarrow$ (1): Let $E, F \in \tau$, then $E \cup F \in \sigma$. Since $\tau$ is fundamental, there exists $G \in \tau$ such that $E \cup F \subset G$. \end{proof} +\begin{definition}[Product Ideal] +\label{definition:product-ideal} + Let $X, Y$ be sets, $\sigma \subset 2^X$ and $\tau \subset 2^Y$ be ideals, and + \[ + \beta = \bracs{A \times B|A \in \sigma, B \in \tau} + \] + + then there exists a unique ideal $\sigma \times \tau$ such that $\beta$ is fundamental with respect to $\sigma$. The ideal $\sigma \otimes \tau$ is the \textbf{product} of $\sigma$ and $\tau$. +\end{definition} +\begin{proof} + For each $A_1, A_2 \in \sigma$ and $B_1, B_2 \in \tau$, + \[ + (A_1 \times B_1) \cup (A_2 \times B_2) \subset (A_1 \cup A_2) \times (B_1 \cup B_2) + \] + + By \autoref{proposition:set-ideal-fundamental-criterion}, there exists an ideal $\sigma \otimes \tau$ such that $\beta$ is fundamental with respect to $\sigma \otimes \tau$. +\end{proof} + +