diff --git a/README.md b/README.md index e7dec1f..66ce9f1 100644 --- a/README.md +++ b/README.md @@ -28,9 +28,8 @@ The topics covered in this script will be: 5.1 Metric and Normed spaces 5.2 Sequences, Series and Limits 5.3 Open and Closed Sets - 5.4 ???? - 5.5 Continuiuty - 5.6 Convergence of Function Sequences + 5.4 Continuiuty + 5.5 Convergence of Function Sequences 6. Differential Calculus for Functions with multiple Variables 6.1 Partial and Total Differentiability @@ -42,9 +41,8 @@ The topics covered in this script will be: 7.1 Contents and Measures 7.2 Integrals 7.3 Integrals over the real numbers - 7.4 ???? - 7.5 Product Measures and the Fubini Theorem - 7.6 The Transformation Theorem + 7.4 Product Measures and the Fubini Theorem + 7.5 The Transformation Theorem 8. Ordinary Differential Equations 8.1 Solution Methods diff --git a/chapters/complex_analysis.tex b/chapters/complex_analysis.tex index 3224d2c..1a3f46b 100644 --- a/chapters/complex_analysis.tex +++ b/chapters/complex_analysis.tex @@ -7,5 +7,5 @@ \pagebreak \subfile{sections/complex_diff.tex} - \subfile{sections/complex_line_int.tex} + \subfile{sections/contour_integrals.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/complex_line_int.tex b/chapters/sections/complex_line_int.tex deleted file mode 100644 index 65219be..0000000 --- a/chapters/sections/complex_line_int.tex +++ /dev/null @@ -1,6 +0,0 @@ -% !TeX root = ../../script.tex -\documentclass[../../script.tex]{subfiles} - -\begin{document} -\section{Complex Line Integrals} -\end{document} \ No newline at end of file diff --git a/chapters/sections/contour_integrals.tex b/chapters/sections/contour_integrals.tex new file mode 100644 index 0000000..231855a --- /dev/null +++ b/chapters/sections/contour_integrals.tex @@ -0,0 +1,510 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section{Contour Integrals} + +\begin{defi}[Contour integrals] + Let $U \subset \cmpln$ be open, $\gamma = C([a, b], U)$ a curve in $U$ and $f: U \rightarrow \cmpln$ continuous. Then + \[ + \int_{\gamma} f(z) \dd{z} := \int_a^b f(\gamma(t)) \gamma'(t) \dd{t} + \] +\end{defi} + +\begin{eg} + Consider the path + \[ + \gamma(t) = re^{it}, ~\quad t \in [0, 2\pi], r > 0 + \] + we want to take the contout integral along the path $\gamma$ of the function $z^n$ + \begin{align*} + \int_{\gamma} z^n \dd{z} &= \int_0^{2\pi} (r e^{it})^n ire^{it} \dd{t}\\ + &= ir^{n+1} \int_0^{2\pi} e^{it(n+1)} \dd{t} = ir^{n+1} \begin{cases} + 2\pi, & n = -1 \\ + 0, & n \ne -1 + \end{cases} + \end{align*} +\end{eg} + +\begin{lem}[Estimation Lemma] + For every curve $\gamma \in C([0, 1], U)$ and every continuous function $f: U \rightarrow \cmpln$ we have + \[ + \abs{\int_{\gamma} f(z) \dd{z}} \le \sup_{z \in \gamma} \abs{f(z)} \int_0^1 \abs{\gamma'(t)} \dd{t} + \] +\end{lem} +\begin{proof} + \begin{equation} + \begin{split} + \abs{\int_{\gamma} f(z) \dd{z}} = \abs{\int_0^1 f(\gamma(t)) \gamma'(t) \dd{t}} &\le \int_0^1 \abs{f(\gamma(t))} \abs{\gamma'(t)} \dd{t} \\ + &\le \sup_{t \in [0, 1]} \abs{f(\gamma(t))} \int_0^1 \abs{\gamma'(t)} \dd{t} + \end{split} + \end{equation} +\end{proof} + +\begin{cor} + Let $\gamma \in C([0, 1], U)$ be a simple closed curve, $U \subset \cmpln$, and let $f: U \rightarrow \cmpln$ a holomorphic function with + \begin{align*} + u = \Re f && v = \Im f + \end{align*} + Then + \[ + \oint_{\gamma} f(z) \dd{z} = 0 + \] +\end{cor} +\begin{proof} + Let $A \subset U$ be the surface bounded by $\gamma$. Then + \begin{equation} + \oint_{\gamma} f(z) \dd{z} = \int_0^1 f(\gamma(t)) \gamma'(t) \dd{t} + \end{equation} + We can split $\gamma$ into a real and an imaginary part, like this + \begin{equation} + \gamma(t) = \gamma_1(t) + i\gamma_2(t), \quad \gamma_1, \gamma_2: [0, 1] \rightarrow \realn + \end{equation} + Then we can calculate + \begin{equation} + \begin{split} + \oint_{\gamma} f(z) \dd{z} &= \int_0^1 \left(u(\gamma_1(t), \gamma_2(t)) + iv(\gamma_1(t), \gamma_2(t)) (\gamma_1'(t) + i\gamma_2'(t))\right) \dd{t} \\ + &= \int_0^1 u(\gamma_1(t), \gamma_2(t)) \gamma_1'(t) - v(\gamma_1(t), \gamma_2(t)) \gamma_2'(t) \dd{t} \\ + &\quad + i\int_0^1 u(\gamma_1(t), \gamma_2(t))\gamma_2'(t) + v(\gamma_1(t), \gamma_2(t))\gamma_1'(t) \dd{t} \\ + &= \int_0^1 \begin{pmatrix} + u(\gamma(t)) \\ -v(\gamma(t)) + \end{pmatrix} + \begin{pmatrix} + \gamma_1'(t) \\ \gamma_2'(t) + \end{pmatrix} + \dd{t} + i \int_0^1 \begin{pmatrix} + v(\gamma(t)) \\ u(\gamma(t)) + \end{pmatrix} + \begin{pmatrix} + \gamma_1'(t) \\ \gamma_2'(t) + \end{pmatrix} + \dd{t} \\ + &= \oint_{\boundary{A}} \begin{pmatrix} + u \\ -v + \end{pmatrix} \dd{s} + i \oint_{\boundary{A}} \begin{pmatrix} + v \\ u + \end{pmatrix} \\ + &= \int_A (-\partial_x v - \partial_y u) \dd{\lambda^2} + i \int_A (\partial_x u - \partial_y v) \dd{\lambda^2} \\ + \end{split} + \end{equation} + Because $f$ is holomorphic we can apply the Cauchy-Riemann equation + \begin{equation} + \oint_{\gamma} f(z) \dd{z} = 0 + \end{equation} +\end{proof} + +\begin{defi} + \begin{enumerate}[(i)] + \item A closed curve $\gamma: [a, b] \rightarrow U$ with $U \subset \cmpln$ is said to be null-homotopic, + if it can be continuously deformed into a point within the set $U$. + + \item Two curves $\gamma_1, \gamma_2: [0, 1] \rightarrow U$ with identical boundary points + \begin{align*} + \gamma_1(0) = \gamma_2(0) ~\wedge~ \gamma_1(1) = \gamma_2(1) + \end{align*} + is said to be homotopic in $U$ if the concatenation + \begin{align*} + \gamma: [0, 2] &\longrightarrow U \\ + \gamma(t) &= \begin{cases} + \gamma_1(t), & t \in [0, 1] \\ + \gamma_2(2 - t) & t \in [1, 2] + \end{cases} + \end{align*} + is null-homotopic. + + \item Two closed surves $\gamma_0, \gamma_1$ are said to be free-homotopic in $U$ if they can be continuously transformed into each other. + \end{enumerate} +\end{defi} + +\begin{defi} + A non-empty set $U \subset \cmpln$ is said to be + \begin{enumerate}[(i)] + \item \textit{connected} if any two points in $U$ can be connected by a curve in $U$. + \item \textit{simply connected} if $U$ is connected and every closed surve in $U$ is null-homotopic. + \item a \textit{domain} if it is open and connected. + \end{enumerate} +\end{defi} + +\begin{thm}[Cauchy's Integral Theorem] + Let $f: U \rightarrow \cmpln$ be holomorphic and $\gamma$ a closed, null-homotopic curve in $U \subset \cmpln$ open. Then + \[ + \oint_{\gamma} f(z) \dd{z} = 0 + \] +\end{thm} +\begin{proof} + \noproof +\end{proof} + +\begin{cor} + \begin{enumerate}[(i)] + \item Let $\gamma_1, \gamma_2$ be holomorphic curves with the same endpoints on the open set $U \subset \cmpln$. Then + \[ + \int_{\gamma_1} f(z) \dd{z} = \int_{\gamma_2} f(z) \dd{z} + \] + for all holomorphic $f: U \rightarrow \cmpln$. + + \item For $f: U \rightarrow \cmpln$ holomorphic, with $U \subset \cmpln$ open and simple connected. Then $\forall z_0 \in U$ + \[ + F(z) := \int_{z_0}^{z} f(\zeta) \dd{\zeta} = \int_{\gamma = \gamma_0} f(\zeta) \dd{\zeta} + \] + is a holomorphic anti-derivative of $f$, i.e. + \[ + F'(z) = f(z) \quad \forall z \in U + \] + \end{enumerate} +\end{cor} + +\begin{proof} + First we prove (i). The concatenation $\gamma := \gamma_1 \gamma_2$ is a null-homotopic curve, so together with the holomorphy of $f$ we can apply the Cauchy integral theorem + \begin{equation} + \begin{split} + 0 = \oint_{\gamma} f(z) \dd{z} &= \int_0^2 f(\gamma(t)) \dot{\gamma}(t) \dd{t} \\ + &= \int_0^1 f(\gamma_1(t))\dot{\gamma_1}(t) \dd{t} - \int_1^2 f(\gamma_2(2 - t)) \dot{\gamma_2}(2 - t) \dd{t} + \end{split} + \end{equation} + Substitute $s = 2 - t$ with $\dd{s} = -\dd{t}$: + \begin{equation} + \begin{split} + &= \int_0^1 f(\gamma_1(t)) \dot{\gamma}(t) \dd{t} - \gamma_0^1 f(\gamma_2(s)) \dot{\gamma_2}(s) \dd{s} \\ + &= \int_{\gamma_1} f(z) \dd{z} - \int_{\gamma_2} f(z) \dd{z} + \end{split} + \end{equation} + + Now we prove (ii). According to (i), we have + \begin{equation} + F(z + h) = F(z) + \int_{\gamma_{z+h, z}} f(z) \dd{z} + \end{equation} + We choose $\gamma_{z+h, z}$ to be a straight line, i.e. + \begin{equation} + \gamma_{z+h, z}(t) = t(z + h) + (1 - t)z, \quad t \in [0, 1] + \end{equation} + Then + \begin{equation} + \int_{\gamma_{z+h, z}} 1 \dd{\zeta} = \int_0^1 \dot{\gamma}(t) \dd{t} = h + \end{equation} + Thus follows + \begin{equation} + F(z + h) - F(z) = \int_{\gamma_{z + h, z}} f(\zeta) \dd{\zeta} \iff \frac{F(z+h) - F(z)}{h} = \frac{1}{h} \in f(\zeta) \dd{\zeta} + \end{equation} + and therefore + \begin{equation} + \begin{split} + \left| \frac{F(z+h) - F(z)}{h} - f(z) \right| = &\left| \frac{1}{h} \int_{\gamma_{z+h, z}} f(\zeta) \dd{\zeta} - f(z) \right| \\ + = &\left| \frac{1}{h} \int_{\gamma_{z+h, z}} f(\zeta) - f(z) \dd{\zeta} \right| \\ + = &\frac{1}{\abs{h}} \int_0^1 \left| f(\gamma_{z+h, z}(t)) - f(z)\right| \left| \dot{\gamma_{z+h, z}}(t) \right| \dd{t} \\ + \le &\frac{1}{h} \sup_{t \in [0, 1]} \left| f(\gamma_{z+h,z}(t)) - f(z) \right| \cdot \underbrace{\int \left| \dot{\gamma_{z+h, z}}(t) \right| \dd{t}}_{\abs{h}} \\ + = &\sup_{t \in [0, 1]} \left| f(\gamma_{z+h, z}(t)) - f(z) \right| \\ + \conv{k \rightarrow 0} &0 + \end{split} + \end{equation} +\end{proof} + +\begin{eg}[The complex logarithm] + Consider $t \mapsto e^{it}, ~t \in \realn$. This is a $2\pi$-periodic function, that means + \[ + e^{it} = e^{i(t + 2\pi n)}, \quad n \in \intn + \] + The function + \begin{align*} + f: \cmpln \setminus \set{0} &\longrightarrow \cmpln \\ + z &\longmapsto \frac{1}{z} + \end{align*} + is holomorphic, and does not have an anti-derivative on $\cmpln \setminus \set{0}$. If it did, then + \[ + \int_{\gamma} f(z) \dd{z} = F(\gamma(2\pi)) - F(\gamma(0)) = 0 + \] + would have to hold, but we know that + \[ + \int_{\gamma} \frac{\dd{z}}{z} = 2\pi i + \] + This is a contradiction. However $f$ does have an anti-derivative on $\cmpln_{-}$ (the complex numbers without the negative real axis) , since $\cmpln_{-}$ is simple connected and $f$ is holomorphic. + Thus we can define + \begin{align*} + \Log: \cmpln_{-} &\longrightarrow \cmpln \\ + z &\longmapsto \int_{\gamma: [0, 1] \rightarrow z} \frac{\dd{\zeta}}{\zeta} + \end{align*} + It can also be defined as + \begin{align*} + \Log z = \begin{cases} + 0, & 1 \\ + \log\abs{z} + i \arg(z), & \text{else} + \end{cases} + \end{align*} + The function $\arg$ is defined as + \begin{align*} + \arg: \cmpln_{-} &\longrightarrow (-\pi, \pi) \\ + z &\longmapsto \phi \text{ for } z = \abs{z} e^{i\phi} + \end{align*} + $\Log$ is said to be the main branch of the complex logarithm, and + \[ + \Log z = \log\abs{z} + i (\arg(z) + 2\pi n), \quad n \in \intn + \] + the secondary branches. +\end{eg} + +\begin{eg}[Fresnel Integrals] + Consider the integrals + \begin{align*} + \int_0^{\infty} \cos(t^2) \dd{t} && \int_0^{\infty} \sin(t^2) \dd{t} + \end{align*} + The way these integrals are supposed to be interpreted is as + \[ + \int_0^{\infty} f(t) \dd{t} = \lim_{N \rightarrow \infty} \int_0^N f(t) \dd{t} + \] + We realize that + \begin{align*} + \cos(t^2) = \Re e^{-it^2} && \sin(t^2) = -\Im e^{-it^2} + \end{align*} + Now, consider these paths + \begin{center} + \begin{tikzpicture} + \draw[->, >=stealth] (0, -1) -- (0, 5) node[above] {$\Im$}; + \draw[->, >=stealth] (-1, 0) -- (5, 0) node[right] {$\Re$}; + + \draw[->, >=stealth, very thick] (0, 0) -- node[below] {$\gamma_1$} (4, 0) node[below] {$R$}; + \draw[->, >=stealth, very thick] (4, 0) -- node[right] {$\gamma_2$} (4, 4); + + \draw[->, >=stealth, very thick] (0, 0) -- node[above left] {$\gamma$} (4, 4); + + \draw[dashed] (0, 4) node[left] {$R$} -- (4, 4); + \end{tikzpicture} + \end{center} + So it becomes apparent that + \[ + \int_0^R \cos(t^2) \dd{t} = \Re \int_0^R e^{-it^2} \dd{t} = \Re \int_{\gamma} e^{-z^2} \dd{z} + \] + We can define a new (closed) path + \[ + \Gamma = \gamma_1 \gamma_2 (-\gamma) + \] + and with Cauchy's theorem we can realize that + \begin{align*} + 0 = \oint_{\Gamma} e^{-z^2} \dd{z} = \int_{\gamma_1} e^{-z^2} \dd{z} + \int_{\gamma_2} e^{-z^2} \dd{z} - \int_{\gamma} e^{-z^2} \dd{z} + \end{align*} + The next step is to evaluate each of the integrals in the last term, starting with the integral over $\gamma$. + \begin{align*} + \int_{\gamma} e^{-z^2} \dd{z} = &\int_0^R e^{-((1+i)t)^2} (1+i) \dd{t} \\ + = &(1 + i) \int_0^R e^{-2it^2} \dd{t} \\ + = &\frac{1+i}{\sqrt{2}} \int_0^{\sqrt{2} R} e^{-is^2} \dd{s} + \end{align*} + The integrall over $\gamma_1$ evaluates to + \begin{align*} + \int_{\gamma_1} e^{-z^2} \dd{z} = \int_0^R e^{-t^2} \dd{t} \conv{R \rightarrow \infty} \int_0^{\infty} e^{-t^2} \dd{t} = \frac{1}{2} \int_{-\infty}^{\infty} e^{-t^2} \dd{t} = \frac{\sqrt{\pi}}{2} + \end{align*} + And the one over $\gamma_2$ to + \begin{align*} + \int_{\gamma_2} e^{-z^2} \dd{z} = \int_0^R e^{-(r +it)^2} i \dd{t} = i \int_0^R e^{-R^2 + t^2} e^{-2irt} \dd{t} + \end{align*} + To evaluate this we need to consider the absolute value of this integral + \begin{align*} + \abs{\int_{\gamma_2} e^{-z^2} \dd{z}} \le &e^{-R^2} \int_0^R e^{t^2} \underbrace{\abs{e^{-2iRt}}}_{=1} \dd{t} \\ + = &e^{-R^2} \int_0^R e^{t^2} \dd{t} \le e^{-R^2} \int_0^R e^{tR} \dd{t} \\ + = &e^{-R^2} \left[\frac{1}{R} e^{tR}\right]_0^R = \frac{e^{-R^2}}{R} \left(e^{R^2} - 1\right) + \end{align*} + so + \[ + \abs{\int_{\gamma_2} e^{-z^2} \dd{z}} \le \frac{1}{R} \left(1 - e^{-R^2}\right) \conv{R \rightarrow \infty} 0 + \] + Thus we can calculate + \[ + \int_{\gamma} e^{-z^2} \dd{z} = \frac{1+i}{\sqrt{2}} \int_0^{\sqrt{2} R} e^{-it^2} \dd{t} = \int_{\gamma_1} e^{-z^2} \dd{z} + \int_{\gamma_2} e^{-z^2} \dd{z} + \] + And finally + \begin{align*} + \lim_{R \rightarrow \infty} \int_0^{\infty} e^{-it^2} \dd{t} = &\lim_{R \rightarrow \infty} \int_0^{\sqrt{2} R} e^{-t^2} \dd{t} \\ + = &\frac{\sqrt{2}}{1+i} \left(\lim_{R \rightarrow \infty} \left( \int_{\gamma_1} e^{-z^2} \dd{z} + \int_{\gamma_2} e^{-z^2} \dd{z} \right) \right) \\ + = &\frac{\sqrt{2}}{1+i} \left(\frac{\pi}{2} + 0\right) \\ + = &\sqrt{\frac{\pi}{2}} \frac{1-i}{2} = \sqrt{\frac{\pi}{8}} (1 - i) + \end{align*} + So we can calculate the Fresnel integrals + \begin{align*} + \int_0^{\infty} \cos(t^2) \dd{t} &= \sqrt{\frac{\pi}{8}} \\ + \int_0^{\infty} \sin(t^2) \dd{t} &= \sqrt{\frac{\pi}{8}} + \end{align*} +\end{eg} + +\begin{thm}[Cauchy's Theorem for circular disks] + Let $f: U \rightarrow \cmpln$ be holomorphic, $U \subset \cmpln$ open and $\cball[r](a) \subset U$. Then + \[ + f(a) = \frac{1}{2\pi i} \int_{\abs{z - a} = r} \frac{f(z)}{z-a} \dd{z} + \] +\end{thm} +\begin{proof} + Consider the following path + \begin{center} + \begin{tikzpicture}[scale=7.5] + \draw[fill] (5, 0.7) node[below] {$a$} circle [radius=0.004]; + + \begin{scope}[decoration={ + markings, + mark=at position 0.5 with {\arrow{latex}}} + ] + \draw[thick, postaction={decorate}, domain=435:105] plot ({0.1*cos(\x) + 5}, {0.1*sin(\x) + 0.7}); + \draw[postaction={decorate}, domain=435:105] plot ({0.3*cos(\x) + 5}, {0.3*sin(\x) + 0.7}); + \draw[thick, postaction={decorate}] ({0.1*cos(105) + 5}, {0.1*sin(105) + 0.7}) -- node[left] {$\alpha_{\delta}^1$} ({0.3*cos(105) + 5}, {0.3*sin(105) + 0.7}); + \draw[thick, postaction={decorate}] ({0.3*cos(435) + 5}, {0.3*sin(435) + 0.7}) -- node[right] {$\alpha_{\delta}^2$} ({0.1*cos(435) + 5}, {0.1*sin(435) + 0.7}); + + \draw[thick, postaction={decorate}, domain=105:75] plot ({0.3*cos(\x) + 5}, {0.3*sin(\x) + 0.7}); + \end{scope} + + \node[below] at ({0.1*cos(270) + 5}, {0.1*sin(270) + 0.7}) {$\gamma_{\epsilon}$}; + \node[below] at ({0.3*cos(270) + 5}, {0.3*sin(270) + 0.7}) {$|z - a| = r$}; + + \draw[dashed] (5, 0.7) -- ({0.1*cos(150) + 5}, {0.1*sin(150) + 0.7}) node[above left=-0.1cm] {$\epsilon$} ; + \draw[dashed] (5, 0.7) -- ({0.3*cos(5) + 5}, {0.3*sin(5) + 0.7}) node[right] {$r$}; + + \node[above] at ({0.3*cos(90) + 5}, {0.3*sin(90) + 0.7}) {$\gamma_{\delta}$}; + \end{tikzpicture} + \end{center} + According to the first corollary of Cauchy's theorem we have + \begin{equation} + \begin{split} + \int_{\abs{z-a}=r} \frac{f(z)}{z-a} \dd{z} = &\lim_{\delta \rightarrow 0} \int_{\gamma_{\epsilon, \delta}} \frac{f(z)}{z-a} \dd{z} + \underbrace{\int_{\alpha_{\delta}^1} \frac{f(z)}{z-a} \dd{z} + \int_{\alpha_{\delta}^2} \frac{f(z)}{z-a} \dd{z}}_{\conv{\delta \rightarrow 0} 0} \\ + = &\int_{\gamma_{\epsilon}} \frac{f(z)}{z-a} \dd{z} + \end{split} + \end{equation} + Thus we conclude + \begin{equation} + \begin{split} + \int_{\abs{z-a}=r} \frac{f(z)}{z-a} \dd{z} = \int_{\gamma_{\epsilon}} \frac{f(z)}{z-a} \dd{z} = &\int_{\gamma_{\epsilon}} \frac{f(z) - f(a)}{z-a} \dd{z} + \int_{\gamma_{\epsilon}} \frac{f(a)}{z-a} \dd{z} \\ + = &\int_{\gamma_{\epsilon}} \frac{f(z) - f(a)}{z-a} \dd{z} + f(a) \int_{\gamma_{\epsilon}} \frac{\dd{z}}{z-a} + \end{split} + \end{equation} + We also know that + \begin{equation} + \int_{\gamma_{\epsilon}} \frac{\dd{z}}{z-a} = 2\pi i + \end{equation} + Since $f$ is holomorphic we can realize + \begin{equation} + \sup_{\cball[r](a)} \left| \frac{f(z) - f(a)}{z - a} \right| = M_r < \infty + \end{equation} + Which results in + \begin{equation} + \abs{\int_{\gamma_{\epsilon}} \frac{f(z) - f(a)}{z - a} \dd{z}} \le M_r \underbrace{\int_0^{2\pi} \abs{\dot{\gamma_{\epsilon}}(t)} \dd{t}}_{2\pi\epsilon} \conv{\epsilon \rightarrow 0} 0 + \end{equation} + Thus follows + \begin{equation} + \int_{|z - a| = r} \frac{f(z)}{z - a} \dd{z} = \int_{\gamma_{\epsilon}} \frac{f(z) - f(a)}{z - a} \dd{z} + 2\pi i f(a) \conv{\epsilon \rightarrow 0} 2\pi i f(a) + \end{equation} + Or short + \begin{equation} + \int_{\abs{z - a} = r} \frac{f(z)}{z - a} \dd{z} = 2\pi i f(a) + \end{equation} +\end{proof} + +\begin{cor} + Let $f: U \rightarrow \cmpln$ be a holomorphic function and $U \subset \cmpln$ an open set such that $\cball[r](a) \subset U$. Then + \[ + f(a) = \frac{1}{2\pi} \int_0^{2\pi} f(a + re^{it}) \dd{t} + \] +\end{cor} +\begin{proof} + \reader +\end{proof} + +\begin{defi}[Analytic functions] + Let $f: U \rightarrow \cmpln$ be a function and $U \subset \cmpln$ a domain. $f$ is said to be analytic in $z_0 \in U$ if and only if there exists a power series + \[ + \sum_{n=0}^{\infty} a_n \zeta^n + \] + with convergence radius + \[ + \rho = \left(\limsup |a_n|^{\frac{1}{n}}\right)^{-1} > 0 + \] + and $\delta \in (0, \rho)$ such that $\oball[\delta](z_0) \subset U$ and + \[ + f(z) = \int_{k=0}^{\infty} a_n (z - z_0)^n, \quad \forall z \in \oball[\delta](z_0) + \] + $f$ is said to be analytic on $U$ if $f$ is analytic $\forall z_0 \in U$. +\end{defi} + +\begin{thm}[Power series expansion] + If $f$ is holomorphic on a circular disk $\oball[r](z_0)$ for some $r > 0$, then $f$ is analytic in $z_0$. + $f$ can be represented with the on $\oball[\rho](z_0)$ convergent power series + \[ + f(z) = \sum_{n=0}^{\infty} c_n (z - z_0)^n, \quad z \in \oball[\rho](z_0) + \] + with + \[ + c_n = \frac{1}{2\pi i} \int_{|z - z_0| = r} \frac{f(z)}{(z - z_0)^{n+1}} \dd{z}, \quad \forall \rho \in (0, r) + \] +\end{thm} +\begin{proof} + \noproof +\end{proof} + +\begin{rem} + If $f$ is holomorphic then $f$ can be infinitely often differentiated on $\cmpln$ with + \[ + f^{(n)}(z) = n! c_n = \frac{n!}{2\pi i} \int_{\abs{z - z_0} = \rho} \frac{f(z)}{(z- z_0)^{n+1}} \dd{z} + \] + By employing the estimation lemma we can then find that + \begin{align*} + \abs{c_n} \le \frac{1}{2\pi} \abs{\int_{\abs{z - z_0} = \rho} \frac{f(z)}{(z - z_0)^{n+1}} \dd{z}} \le &\frac{1}{2\pi} \sup_{\abs{z - z_0} = \rho} \abs{\frac{f(z)}{\abs{z - z_0}^{n+1}}} \cdot 2\pi \rho \\ + = &\frac{1}{\rho^n} \sup_{z \in \oball[r](z_0)} \abs{f(z)} \\ + = &\frac{M_r}{\rho^n}, \quad M_r < \infty + \end{align*} + This is Cauchy's estimate. +\end{rem} + +\begin{thm}[Liouville's Theorem] + Every bounded entire function is constant. +\end{thm} +\begin{proof} + According to the power series expansion theorem, $f$ can be represented by a power series on all of $\cmpln$: + \begin{equation} + f(z) = \sum_{n=0}^{\infty} c_n z^n + \end{equation} + and the coefficients satisfy the Cauchy estimate + \begin{equation} + \abs{c_n} \le \frac{1}{\rho^n} \sup_{\abs{z} = \rho} \abs{f(z)} \le \frac{1}{\rho^n} \underbrace{\sup_{z \in \cmpln} \abs{f(z)}}_{< \infty} + \end{equation} + This inequality tends to $0$ if $\rho$ tends to $\infty$ for all $n \ge 1$, thus we can find + \begin{equation} + c_n = 0, \quad \forall n \ge 1 + \end{equation} + Thus + \begin{equation} + f(z) = c_0 = \const. + \end{equation} +\end{proof} + +\begin{thm}[Fundamental Theorem of Algebra] + Every polynomial of degree $n \ge 1$ + \[ + f(z) = \sum_{k=0}^n c_k z^k, \quad c_n \ne 0 + \] + has a root, i.e. + \[ + \exists z_0 \in \cmpln: \quad f(z_0) = 0 + \] +\end{thm} +\begin{proof} + Assume there exists no root. Then the function + \begin{equation} + z \longmapsto \frac{1}{f(z)} + \end{equation} + would be holomorphic on all of $\cmpln$, since $z \mapsto \rec{z}$ is holomorphic on $\cmpln \setminus \set{0}$. + Furthermore we find that + \begin{equation} + \exists R \ge 0: ~~\abs{z} \ge R \implies \abs{f(z)} \ge \abs{f(0)} > 0 + \end{equation} + which implies + \begin{equation} + \sup_{z \in \cmpln} \frac{1}{\abs{f(z)}} = \sup_{\abs{z} < R} \frac{1}{\abs{f(z)}} = \max_{\abs{z} \le R} \frac{1}{\abs{f(z)}} < \infty + \end{equation} + since $f$ doesn't have a root. According to Liouville's theorem $\rec{f}$ has to be constant, and thus $f$ must be constant. + This implies that $c_n = 0$, which contradicts the assumption. So $f$ has to have a root. +\end{proof} + +\begin{cor}[Polynomial Decomposition] + Let + \[ + f(z) = \sum_{k=0}^n c_k z^k, \quad n \in \natn, c_k \in \cmpln, c_n = 1 + \] + Then $\exists z_j \in \cmpln, ~~j = 1, \cdots, n$ such that + \[ + f(z) = \prod_{j=1}^n (z - z_j) + \] +\end{cor} +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 0309c85..500ad2c 100644 --- a/script.pdf +++ b/script.pdf @@ -274,7 +274,7 @@ endobj << /S /GoTo /D (section.10.2) >> endobj 187 0 obj -(\376\377\000C\000o\000m\000p\000l\000e\000x\000\040\000L\000i\000n\000e\000\040\000I\000n\000t\000e\000g\000r\000a\000l\000s) +(\376\377\000C\000o\000n\000t\000o\000u\000r\000\040\000I\000n\000t\000e\000g\000r\000a\000l\000s) endobj 188 0 obj << /S /GoTo /D [189 0 R /Fit] >> @@ -285,10 +285,10 @@ endobj /Filter /FlateDecode >> stream -xڅRMo0WXqx:էDOnLqH䶝 -yb* -˵1jh'bꀩW+ -ŝO_j%.(`Bтm.Y'?:V< xJd7=wcο7A +xڅMo0'؉nKHEV8YdY3Z2<++WIz%C#maU ,H93%J Ȫ_u}fߏS n yy|Z10RY&hHv˗i~A>Rfz߄wMӓ | +KYd :v}MRI$upin + +J\-VjEɁڵކ@sTgNRd-b&#ji@SCcv+b|Ȟu9efmaLnqzr?W^{ UQ endstream endobj 189 0 obj @@ -651,12 +651,15 @@ endobj endobj 259 0 obj << -/Length 959 +/Length 953 /Filter /FlateDecode >> stream -xXMo8WR6iYhm1Ң9_/)Rvl4Nrhy3cE0x~zT@29OQaMW]p{fф0b"g8MQi*NRh6{ "x|Y$QHʨR'b nc̭Z#rp 1'{eTNtk|/4Ngy8U9 tA8 J vДZWi7[Zگ!p1lП, e)+ZP͕J{XV/RՋ=m3:N:Y4{ CJnw~2tSZY(w'QGkQ|+EgM"[= } Q1o%&1H7a~PRF)k3ro|^YeW~mTWړE,N2}34 F2 ڭ,*.Ev=޶OquwvFnUCrpͷEnR.?zgp.Q˷%o +ܹH&uPşwcq/#iʘٻvu7vաtuM!g\§퍦3`tҘ [B[̤eUO$D)xpڊm:VXd86'Yʹ~3u"a?r(IP{'aK"'OdZZ7A{V}j,eqdo{> >> endobj @@ -7294,21 +7297,27 @@ endobj endobj 1267 0 obj << -/Length 2054 +/Length 2064 /Filter /FlateDecode >> stream -xZIw6Wү&}I4qRIS#ӱRrοP(/I/X24 0|3N>&8ѱ` `CyB@Dje,yrDaN_v2y{1ҟO729GT -WHq U4B' qItֈgL -EaFr^Jyynz(W1pQCPtv/fV"ZKѵ](܇׳ܭJ^.3x=LFQ-Z.ֹz7aLaڔg$OC+AIFdI-(qMLDFSSQp_/FRN/qo;)J=a?1EqX܃&3J%ʄ|^v^*g3,p[ӭqu)籫%IQ'E E&ԛL bA'n F˝(U,6*N|^{PbS/n">]PWڭ-Σ.aC{> --|f nj$HuwXa -՟7!7jw 0Sd|0}Fo"T`Ydԟ>uWYY٨euf\aU]uU ySO+TWU$;9(DlF-}jMu%VzJZQ/Uq}$P@C*c&f,K+5b 4z7ռfy҂BsCybK2Rx!}.6r(Cj-ϯmN-4k\NrVG~17" -嚄!ok6ojO|$'h/ζ -%Ix(_i(3^,er1c -zpz׆,1.Z/34Bhkݓ8Լ-^4Q(as_qyJ<^/ /J#{5*/q889_&q@u%zȄug|K?6  Sڇi|OzԎѣx5i;]-6 5jSH(sZEN@3U\aQy{-,n .0yg aaWf;0oF xl`n˫|qBM H|_ړ #Dyǻݝn7lI6 E*qn N]ߌ S0a -՟d;|?\_Gjq=jHMcDndd e|@޷@}ҷH3xm +\<4md/y"CQ -msM1zѥvj]bYG2b<wg!#gQ q9Kk,}ڣ[=ڲ뛙0{ -,VzV~ttU|eE ߯H"c!F꩝-9B* ;q@lhd5l^< +xZIw6Wү}I4qRIScӱRrοP )NzŒ!h0曅'<<:,!lH2HHH,'R:(W'LNcϧݛ?O~c}+JQM' qII}R4N]+8V4+ݛCMq-ULL(!wR8ח3+2^Kэ]݇7ܭ +^.'z9kAbάN ֱBqP!C+UU;y*\nԈ*LL]aA`>9.2Q1`p +~j_T\ +eߡbvQORU%֎ ~M`LG;wIJ%ʄ|t^JO8f.XBAᱚ!^ RT՛LK(һ$n.4ivdlX\M3|V2I,QHFimbF2Nϲ}LYj +بe3L/.;ʅ']ҵ[2ͪb BRAGEHCHoXm.LW|CJ˫ŬL/Hi +p)" Jxrvu=N× F |Sn`kW{tXbEeE$|VP +H/ny +« m +Bf>Ģd,s.Fm,p.ù[IzdˁEz#~▋Amj;Hϲ*ɜ.$LiMrW*_d&'`ygۀMM +)0X<_߯R%6ZUȈr1c +zpzK  t&@'*h$^jޤ^bla/m,*W8`mˎ96iDt'g2$\ílH2IUYlL0`;!- +B/{)ϡ)wz9G/$]kՠ"ҩ)pUx -z>U/kI |aG )z K@*e}ZuLF`t`/nX'uD/h+YՄ#YFf|`)$=Ǹ'TƐQp`X`^툎~QIwMwaT!pp6fEUgkG<[UoE5Zc/o=K ;mg LUC% 퇸kd)] 5b۲v^!u-U4ʖG`dZ06^HSNIwGL;mJ-OJ=@;qOQGF,P;"4F?UDSڅil2Gj0 +56 JSH(}00㇊&QL +@x 3d!*0uoKݩn-? 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[(equation.3.1.1) 949 0 R (equation.3.1.10) 962 0 R (equation.3.1.11) 963 0 R (equation.3.1.12) 972 0 R (equation.3.1.13) 974 0 R (equation.3.1.14) 975 0 R] +/Limits [(equation.3.1.1) (equation.3.1.14)] >> endobj 3397 0 obj << -/Names [(page.53) 944 0 R (page.54) 957 0 R (page.55) 971 0 R (page.56) 984 0 R (page.57) 996 0 R (page.58) 1007 0 R] -/Limits [(page.53) (page.58)] +/Names [(equation.3.1.15) 976 0 R (equation.3.1.16) 977 0 R (equation.3.1.17) 988 0 R (equation.3.1.18) 989 0 R (equation.3.1.2) 950 0 R (equation.3.1.3) 951 0 R] +/Limits [(equation.3.1.15) (equation.3.1.3)] >> endobj 3398 0 obj << -/Names [(page.59) 1021 0 R (page.6) 302 0 R (page.60) 1030 0 R (page.61) 1043 0 R (page.62) 1050 0 R (page.63) 1066 0 R] -/Limits [(page.59) (page.63)] +/Names [(equation.3.1.4) 952 0 R (equation.3.1.5) 953 0 R (equation.3.1.6) 958 0 R (equation.3.1.7) 959 0 R (equation.3.1.8) 960 0 R (equation.3.1.9) 961 0 R] +/Limits [(equation.3.1.4) (equation.3.1.9)] >> endobj 3399 0 obj << -/Names [(page.64) 1078 0 R (page.65) 1088 0 R (page.66) 1098 0 R (page.67) 1111 0 R (page.68) 1121 0 R (page.69) 1135 0 R] -/Limits [(page.64) (page.69)] +/Names [(equation.3.2.19) 1022 0 R (equation.3.2.20) 1023 0 R (equation.3.2.21) 1034 0 R (equation.3.2.22) 1035 0 R (equation.3.2.23) 1036 0 R (equation.3.2.24) 1037 0 R] +/Limits [(equation.3.2.19) (equation.3.2.24)] >> endobj 3400 0 obj << -/Names [(page.7) 321 0 R (page.70) 1149 0 R (page.71) 1157 0 R (page.72) 1164 0 R (page.73) 1176 0 R (page.74) 1187 0 R] -/Limits [(page.7) (page.74)] +/Names [(equation.3.2.25) 1070 0 R (equation.3.2.26) 1072 0 R (equation.3.2.27) 1073 0 R (equation.3.2.28) 1079 0 R (equation.3.2.29) 1080 0 R (equation.3.2.30) 1082 0 R] +/Limits [(equation.3.2.25) (equation.3.2.30)] >> endobj 3401 0 obj << -/Names [(page.75) 1199 0 R (page.76) 1209 0 R (page.77) 1218 0 R (page.78) 1225 0 R (page.79) 1233 0 R (page.8) 335 0 R] -/Limits [(page.75) (page.8)] +/Names [(equation.3.2.31) 1092 0 R (equation.3.2.32) 1093 0 R (equation.3.3.33) 1122 0 R (equation.3.3.34) 1123 0 R (equation.3.3.35) 1126 0 R (equation.3.3.35a) 1127 0 R] +/Limits [(equation.3.2.31) (equation.3.3.35a)] >> endobj 3402 0 obj << -/Names [(page.80) 1248 0 R (page.81) 1255 0 R (page.82) 1264 0 R (page.83) 1268 0 R (page.84) 1281 0 R (page.85) 1293 0 R] -/Limits [(page.80) (page.85)] +/Names [(equation.3.3.35b) 1128 0 R (equation.3.3.36) 1129 0 R (equation.3.3.37) 1130 0 R (equation.3.3.38) 1136 0 R (equation.3.3.39) 1138 0 R (equation.3.3.39a) 1139 0 R] +/Limits [(equation.3.3.35b) (equation.3.3.39a)] >> endobj 3403 0 obj << -/Names [(page.86) 1305 0 R (page.87) 1312 0 R (page.88) 1323 0 R (page.89) 1331 0 R (page.9) 340 0 R (page.90) 1343 0 R] -/Limits [(page.86) (page.90)] +/Names [(equation.3.3.39b) 1140 0 R (equation.3.3.40) 1141 0 R (equation.3.3.40a) 1142 0 R (equation.3.3.40b) 1143 0 R (equation.3.3.41) 1144 0 R (equation.3.3.42) 1145 0 R] +/Limits [(equation.3.3.39b) (equation.3.3.42)] >> endobj 3404 0 obj << -/Names [(page.91) 1357 0 R (page.92) 1363 0 R (page.93) 1373 0 R (page.94) 1382 0 R (page.95) 1395 0 R (page.96) 1407 0 R] -/Limits [(page.91) (page.96)] +/Names [(equation.3.3.43) 1150 0 R (equation.3.3.44) 1151 0 R (equation.3.4.45) 1166 0 R (equation.3.4.46) 1167 0 R (equation.3.4.47) 1168 0 R (equation.3.4.48) 1169 0 R] +/Limits [(equation.3.3.43) (equation.3.4.48)] >> endobj 3405 0 obj << -/Names [(page.97) 1420 0 R (page.98) 1431 0 R (page.99) 1438 0 R (section.1.1) 9 0 R (section.1.2) 13 0 R (section.1.3) 17 0 R] -/Limits [(page.97) (section.1.3)] +/Names [(equation.3.4.49) 1170 0 R (equation.3.4.50) 1178 0 R (equation.3.4.51) 1179 0 R (equation.3.4.52) 1188 0 R (equation.3.4.53) 1189 0 R (equation.3.4.54) 1191 0 R] +/Limits [(equation.3.4.49) (equation.3.4.54)] >> endobj 3406 0 obj << -/Names [(section.10.1) 181 0 R (section.10.2) 185 0 R (section.2.1) 25 0 R (section.2.2) 29 0 R (section.2.3) 33 0 R (section.3.1) 41 0 R] -/Limits [(section.10.1) (section.3.1)] +/Names [(equation.3.4.54a) 1192 0 R (equation.3.4.54b) 1193 0 R (equation.3.4.55) 1194 0 R (equation.3.5.56) 1234 0 R (equation.3.5.57) 1235 0 R (equation.3.5.58) 1236 0 R] +/Limits [(equation.3.4.54a) (equation.3.5.58)] >> endobj 3407 0 obj << -/Names [(section.3.2) 45 0 R (section.3.3) 49 0 R (section.3.4) 53 0 R (section.3.5) 57 0 R (section.4.1) 65 0 R (section.4.2) 69 0 R] -/Limits [(section.3.2) (section.4.2)] +/Names [(equation.3.5.59) 1237 0 R (equation.3.5.59a) 1238 0 R (equation.3.5.59b) 1239 0 R (equation.3.5.60) 1240 0 R (equation.3.5.61) 1241 0 R (equation.3.5.62) 1242 0 R] +/Limits [(equation.3.5.59) (equation.3.5.62)] >> endobj 3408 0 obj << -/Names [(section.5.1) 77 0 R (section.5.2) 81 0 R (section.5.3) 85 0 R (section.5.4) 89 0 R (section.5.5) 93 0 R (section.6.1) 101 0 R] -/Limits [(section.5.1) (section.6.1)] +/Names [(equation.3.5.63) 1256 0 R (equation.3.5.64) 1257 0 R (equation.3.5.65) 1258 0 R (equation.3.5.66) 1259 0 R (equation.3.5.67) 1260 0 R (equation.4.1.1) 1276 0 R] +/Limits [(equation.3.5.63) (equation.4.1.1)] >> endobj 3409 0 obj << -/Names [(section.6.2) 105 0 R (section.6.3) 109 0 R (section.6.4) 113 0 R (section.7.1) 121 0 R (section.7.2) 125 0 R (section.7.3) 129 0 R] -/Limits [(section.6.2) (section.7.3)] +/Names [(equation.4.1.10) 1300 0 R (equation.4.1.11) 1307 0 R (equation.4.1.12) 1308 0 R (equation.4.1.12a) 1313 0 R (equation.4.1.12b) 1314 0 R (equation.4.1.13) 1315 0 R] +/Limits [(equation.4.1.10) (equation.4.1.13)] >> endobj 3410 0 obj << -/Names [(section.7.4) 133 0 R (section.7.5) 137 0 R (section.7.6) 141 0 R (section.8.1) 149 0 R (section.8.2) 153 0 R (section.8.3) 157 0 R] -/Limits [(section.7.4) (section.8.3)] +/Names [(equation.4.1.14) 1333 0 R (equation.4.1.15) 1334 0 R (equation.4.1.16) 1346 0 R (equation.4.1.16a) 1347 0 R (equation.4.1.16b) 1348 0 R (equation.4.1.17) 1349 0 R] +/Limits [(equation.4.1.14) (equation.4.1.17)] >> endobj 3411 0 obj << -/Names [(section.9.1) 165 0 R (section.9.2) 169 0 R (section.9.3) 173 0 R] -/Limits [(section.9.1) (section.9.3)] +/Names [(equation.4.1.18) 1350 0 R (equation.4.1.19) 1351 0 R (equation.4.1.2) 1277 0 R (equation.4.1.20) 1364 0 R (equation.4.1.21) 1366 0 R (equation.4.1.22) 1367 0 R] +/Limits [(equation.4.1.18) (equation.4.1.22)] >> endobj 3412 0 obj << -/Kids [3094 0 R 3095 0 R 3096 0 R 3097 0 R 3098 0 R 3099 0 R] -/Limits [(Doc-Start) (Item.13)] +/Names [(equation.4.1.23) 1368 0 R (equation.4.1.24) 1369 0 R (equation.4.1.25) 1374 0 R (equation.4.1.26) 1375 0 R (equation.4.1.27) 1384 0 R (equation.4.1.28) 1385 0 R] +/Limits [(equation.4.1.23) (equation.4.1.28)] >> endobj 3413 0 obj << -/Kids [3100 0 R 3101 0 R 3102 0 R 3103 0 R 3104 0 R 3105 0 R] -/Limits [(Item.130) (Item.162)] +/Names [(equation.4.1.29) 1386 0 R (equation.4.1.3) 1282 0 R (equation.4.1.30) 1387 0 R (equation.4.1.31) 1388 0 R (equation.4.1.32) 1389 0 R (equation.4.1.4) 1283 0 R] +/Limits [(equation.4.1.29) (equation.4.1.4)] >> endobj 3414 0 obj << -/Kids [3106 0 R 3107 0 R 3108 0 R 3109 0 R 3110 0 R 3111 0 R] -/Limits [(Item.163) (Item.195)] +/Names [(equation.4.1.5) 1295 0 R (equation.4.1.6) 1296 0 R (equation.4.1.7) 1297 0 R (equation.4.1.8) 1298 0 R (equation.4.1.9) 1299 0 R (equation.4.2.33) 1408 0 R] +/Limits [(equation.4.1.5) (equation.4.2.33)] >> endobj 3415 0 obj << -/Kids [3112 0 R 3113 0 R 3114 0 R 3115 0 R 3116 0 R 3117 0 R] -/Limits [(Item.196) (Item.227)] +/Names [(equation.4.2.34) 1409 0 R (equation.4.2.35) 1410 0 R (equation.4.2.36) 1411 0 R (equation.4.2.37) 1412 0 R (equation.4.2.38) 1421 0 R (equation.4.2.39) 1422 0 R] +/Limits [(equation.4.2.34) (equation.4.2.39)] >> endobj 3416 0 obj << -/Kids [3118 0 R 3119 0 R 3120 0 R 3121 0 R 3122 0 R 3123 0 R] -/Limits [(Item.228) (Item.26)] +/Names [(equation.4.2.40) 1423 0 R (equation.4.2.41) 1424 0 R (equation.4.2.42) 1439 0 R (equation.4.2.43) 1440 0 R (equation.4.2.44) 1443 0 R (equation.4.2.45) 1444 0 R] +/Limits [(equation.4.2.40) (equation.4.2.45)] >> endobj 3417 0 obj << -/Kids [3124 0 R 3125 0 R 3126 0 R 3127 0 R 3128 0 R 3129 0 R] -/Limits [(Item.260) (Item.292)] +/Names [(equation.4.2.46) 1455 0 R (equation.4.2.47) 1462 0 R (equation.4.2.48) 1463 0 R (equation.4.2.49) 1464 0 R (equation.4.2.50) 1478 0 R (equation.4.2.51) 1479 0 R] +/Limits [(equation.4.2.46) (equation.4.2.51)] >> endobj 3418 0 obj << -/Kids [3130 0 R 3131 0 R 3132 0 R 3133 0 R 3134 0 R 3135 0 R] -/Limits [(Item.293) (Item.324)] +/Names [(equation.4.2.52) 1480 0 R (equation.4.2.53) 1481 0 R (equation.4.2.54) 1498 0 R (equation.4.2.55) 1499 0 R (equation.4.2.56) 1500 0 R (equation.4.2.57) 1511 0 R] +/Limits [(equation.4.2.52) (equation.4.2.57)] >> endobj 3419 0 obj << -/Kids [3136 0 R 3137 0 R 3138 0 R 3139 0 R 3140 0 R 3141 0 R] -/Limits [(Item.325) (Item.357)] +/Names [(equation.5.2.1) 1554 0 R (equation.5.2.10) 1579 0 R (equation.5.2.11) 1580 0 R (equation.5.2.12) 1582 0 R (equation.5.2.13) 1583 0 R (equation.5.2.14) 1591 0 R] +/Limits [(equation.5.2.1) (equation.5.2.14)] >> endobj 3420 0 obj << -/Kids [3142 0 R 3143 0 R 3144 0 R 3145 0 R 3146 0 R 3147 0 R] -/Limits [(Item.358) (Item.39)] +/Names [(equation.5.2.15) 1592 0 R (equation.5.2.2) 1556 0 R (equation.5.2.3) 1557 0 R (equation.5.2.4) 1558 0 R (equation.5.2.5) 1559 0 R (equation.5.2.6) 1565 0 R] +/Limits [(equation.5.2.15) (equation.5.2.6)] >> endobj 3421 0 obj << -/Kids [3148 0 R 3149 0 R 3150 0 R 3151 0 R 3152 0 R 3153 0 R] -/Limits [(Item.390) (Item.421)] +/Names [(equation.5.2.6a) 1566 0 R (equation.5.2.6b) 1567 0 R (equation.5.2.7) 1568 0 R (equation.5.2.8) 1569 0 R (equation.5.2.9) 1570 0 R (equation.5.3.16) 1616 0 R] +/Limits [(equation.5.2.6a) (equation.5.3.16)] >> endobj 3422 0 obj << -/Kids [3154 0 R 3155 0 R 3156 0 R 3157 0 R 3158 0 R 3159 0 R] -/Limits [(Item.422) (Item.65)] +/Names [(equation.5.3.17) 1617 0 R (equation.5.3.18) 1618 0 R (equation.5.3.19) 1619 0 R (equation.5.3.20) 1625 0 R (equation.5.3.20a) 1626 0 R (equation.5.3.20b) 1627 0 R] +/Limits [(equation.5.3.17) (equation.5.3.20b)] >> endobj 3423 0 obj << -/Kids [3160 0 R 3161 0 R 3162 0 R 3163 0 R 3164 0 R 3165 0 R] -/Limits [(Item.66) (Item.98)] +/Names [(equation.5.3.20c) 1628 0 R (equation.5.3.20d) 1629 0 R (equation.5.3.20e) 1630 0 R (equation.5.3.21) 1639 0 R (equation.5.3.22) 1640 0 R (equation.5.3.23) 1654 0 R] +/Limits [(equation.5.3.20c) (equation.5.3.23)] >> endobj 3424 0 obj << -/Kids [3166 0 R 3167 0 R 3168 0 R 3169 0 R 3170 0 R 3171 0 R] -/Limits [(Item.99) (defi.1.30)] +/Names [(equation.5.3.24) 1655 0 R (equation.5.3.25) 1656 0 R (equation.5.3.26) 1657 0 R (equation.5.3.26a) 1658 0 R (equation.5.3.26b) 1659 0 R (equation.5.3.27) 1660 0 R] +/Limits [(equation.5.3.24) (equation.5.3.27)] >> endobj 3425 0 obj << -/Kids [3172 0 R 3173 0 R 3174 0 R 3175 0 R 3176 0 R 3177 0 R] -/Limits [(defi.1.31) (defi.2.13)] +/Names [(equation.5.3.28) 1661 0 R (equation.5.3.29) 1668 0 R (equation.5.3.30) 1669 0 R (equation.5.3.31) 1683 0 R (equation.5.3.32) 1684 0 R (equation.5.3.33) 1685 0 R] +/Limits [(equation.5.3.28) (equation.5.3.33)] >> endobj 3426 0 obj << -/Kids [3178 0 R 3179 0 R 3180 0 R 3181 0 R 3182 0 R 3183 0 R] -/Limits [(defi.2.14) (defi.2.46)] +/Names [(equation.5.3.34) 1686 0 R (equation.5.3.35) 1687 0 R (equation.5.4.36) 1707 0 R (equation.5.4.36a) 1708 0 R (equation.5.4.36b) 1709 0 R (equation.5.4.37) 1710 0 R] +/Limits [(equation.5.3.34) (equation.5.4.37)] >> endobj 3427 0 obj << -/Kids [3184 0 R 3185 0 R 3186 0 R 3187 0 R 3188 0 R 3189 0 R] -/Limits [(defi.2.47) (defi.3.17)] +/Names [(equation.5.4.38) 1718 0 R (equation.5.4.39) 1728 0 R (equation.5.4.40) 1729 0 R (equation.5.4.41) 1730 0 R (equation.5.4.42) 1731 0 R (equation.5.4.43) 1745 0 R] +/Limits [(equation.5.4.38) (equation.5.4.43)] >> endobj 3428 0 obj << -/Kids [3190 0 R 3191 0 R 3192 0 R 3193 0 R 3194 0 R 3195 0 R] -/Limits [(defi.3.18) (defi.3.5)] +/Names [(equation.5.4.44) 1746 0 R (equation.5.4.45) 1751 0 R (equation.5.4.46) 1752 0 R (equation.5.4.47) 1757 0 R (equation.5.4.48) 1762 0 R (equation.5.4.49) 1763 0 R] +/Limits [(equation.5.4.44) (equation.5.4.49)] >> endobj 3429 0 obj << -/Kids [3196 0 R 3197 0 R 3198 0 R 3199 0 R 3200 0 R 3201 0 R] -/Limits [(defi.3.50) (defi.4.14)] +/Names [(equation.5.4.49a) 1764 0 R (equation.5.4.50) 1765 0 R (equation.5.4.51) 1766 0 R (equation.5.4.52) 1768 0 R (equation.5.4.53) 1769 0 R (equation.5.4.54) 1778 0 R] +/Limits [(equation.5.4.49a) (equation.5.4.54)] >> endobj 3430 0 obj << -/Kids [3202 0 R 3203 0 R 3204 0 R 3205 0 R 3206 0 R 3207 0 R] -/Limits [(defi.4.15) (defi.4.47)] +/Names [(equation.5.5.55) 1786 0 R (equation.5.5.56) 1787 0 R (equation.5.5.57) 1788 0 R (equation.5.5.58) 1793 0 R (equation.5.5.59) 1794 0 R (equation.5.5.60) 1798 0 R] +/Limits [(equation.5.5.55) (equation.5.5.60)] >> endobj 3431 0 obj << -/Kids [3208 0 R 3209 0 R 3210 0 R 3211 0 R 3212 0 R 3213 0 R] -/Limits [(defi.4.48) (defi.5.3)] +/Names [(equation.5.5.61) 1799 0 R (equation.5.5.62) 1800 0 R (equation.5.5.63) 1801 0 R (equation.5.5.64) 1806 0 R (equation.5.5.65) 1807 0 R (equation.5.5.66) 1808 0 R] +/Limits [(equation.5.5.61) (equation.5.5.66)] >> endobj 3432 0 obj << -/Kids [3214 0 R 3215 0 R 3216 0 R 3217 0 R 3218 0 R 3219 0 R] -/Limits [(defi.5.30) (defi.5.62)] +/Names [(equation.5.5.67) 1809 0 R (equation.5.5.68) 1813 0 R (equation.5.5.69) 1814 0 R (equation.5.5.70) 1815 0 R (equation.5.5.71) 1823 0 R (equation.5.5.72) 1838 0 R] +/Limits [(equation.5.5.67) (equation.5.5.72)] >> endobj 3433 0 obj << -/Kids [3220 0 R 3221 0 R 3222 0 R 3223 0 R 3224 0 R 3225 0 R] -/Limits [(defi.5.63) (defi.6.20)] +/Names [(equation.5.5.73) 1839 0 R (equation.5.5.74) 1840 0 R (equation.6.1.1) 1858 0 R (equation.6.1.10) 1899 0 R (equation.6.1.11) 1900 0 R (equation.6.1.12) 1901 0 R] +/Limits [(equation.5.5.73) (equation.6.1.12)] >> endobj 3434 0 obj << -/Kids [3226 0 R 3227 0 R 3228 0 R 3229 0 R 3230 0 R 3231 0 R] -/Limits [(defi.6.21) (defi.6.9)] +/Names [(equation.6.1.13) 1902 0 R (equation.6.1.14) 1903 0 R (equation.6.1.15) 1904 0 R (equation.6.1.16) 1905 0 R (equation.6.1.17) 1920 0 R (equation.6.1.18) 1921 0 R] +/Limits [(equation.6.1.13) (equation.6.1.18)] >> endobj 3435 0 obj << -/Kids [3232 0 R 3233 0 R 3234 0 R 3235 0 R 3236 0 R 3237 0 R] -/Limits [(defi.7.1) (defi.7.41)] +/Names [(equation.6.1.19) 1923 0 R (equation.6.1.2) 1863 0 R (equation.6.1.20) 1924 0 R (equation.6.1.21) 1925 0 R (equation.6.1.22) 1931 0 R (equation.6.1.23) 1932 0 R] +/Limits [(equation.6.1.19) (equation.6.1.23)] >> endobj 3436 0 obj << -/Kids [3238 0 R 3239 0 R 3240 0 R 3241 0 R 3242 0 R 3243 0 R] -/Limits [(defi.7.42) (defi.7.74)] +/Names [(equation.6.1.3) 1864 0 R (equation.6.1.4) 1867 0 R (equation.6.1.5) 1868 0 R (equation.6.1.6) 1891 0 R (equation.6.1.7) 1892 0 R (equation.6.1.8) 1893 0 R] +/Limits [(equation.6.1.3) (equation.6.1.8)] >> endobj 3437 0 obj << -/Kids [3244 0 R 3245 0 R 3246 0 R 3247 0 R 3248 0 R 3249 0 R] -/Limits [(defi.7.75) (defi.8.35)] +/Names [(equation.6.1.9) 1898 0 R (equation.6.2.24) 1956 0 R (equation.6.2.25) 1964 0 R (equation.6.2.26) 1965 0 R (equation.6.2.27) 1970 0 R (equation.6.2.28) 1971 0 R] +/Limits [(equation.6.1.9) (equation.6.2.28)] >> endobj 3438 0 obj << -/Kids [3250 0 R 3251 0 R 3252 0 R 3253 0 R 3254 0 R 3255 0 R] -/Limits [(defi.8.36) (defi.9.17)] +/Names [(equation.6.2.29) 1972 0 R (equation.6.2.30) 1973 0 R (equation.6.2.31) 1974 0 R (equation.6.2.32) 1983 0 R (equation.6.3.33) 1994 0 R (equation.6.3.34) 1995 0 R] +/Limits [(equation.6.2.29) (equation.6.3.34)] >> endobj 3439 0 obj << -/Kids [3256 0 R 3257 0 R 3258 0 R 3259 0 R 3260 0 R 3261 0 R] -/Limits [(defi.9.18) (equation.1.2.3)] +/Names [(equation.6.3.35) 1996 0 R (equation.6.3.36) 1997 0 R (equation.6.3.37) 2002 0 R (equation.6.3.38) 2003 0 R (equation.6.3.39) 2004 0 R (equation.6.3.40) 2007 0 R] +/Limits [(equation.6.3.35) (equation.6.3.40)] >> endobj 3440 0 obj << -/Kids [3262 0 R 3263 0 R 3264 0 R 3265 0 R 3266 0 R 3267 0 R] -/Limits [(equation.1.3.10) (equation.2.1.6)] +/Names [(equation.6.3.41) 2013 0 R (equation.6.3.42) 2014 0 R (equation.6.3.43) 2015 0 R (equation.6.3.44) 2016 0 R (equation.6.3.45) 2018 0 R (equation.6.3.46) 2019 0 R] +/Limits [(equation.6.3.41) (equation.6.3.46)] >> endobj 3441 0 obj << -/Kids [3268 0 R 3269 0 R 3270 0 R 3271 0 R 3272 0 R 3273 0 R] -/Limits [(equation.2.1.7) (equation.2.2.44)] +/Names [(equation.6.4.47) 2057 0 R (equation.6.4.48) 2058 0 R (equation.6.4.49) 2059 0 R (equation.6.4.50) 2060 0 R (equation.6.4.51) 2061 0 R (equation.6.4.52) 2062 0 R] +/Limits [(equation.6.4.47) (equation.6.4.52)] >> endobj 3442 0 obj << -/Kids [3274 0 R 3275 0 R 3276 0 R 3277 0 R 3278 0 R 3279 0 R] -/Limits [(equation.2.2.45) (equation.2.3.101)] +/Names [(equation.6.4.53) 2063 0 R (equation.6.4.54) 2064 0 R (equation.6.4.55) 2073 0 R (equation.6.4.56) 2083 0 R (equation.6.4.57) 2084 0 R (equation.6.4.58) 2085 0 R] +/Limits [(equation.6.4.53) (equation.6.4.58)] >> endobj 3443 0 obj << -/Kids [3280 0 R 3281 0 R 3282 0 R 3283 0 R 3284 0 R 3285 0 R] -/Limits [(equation.2.3.102) (equation.3.1.12)] +/Names [(equation.6.4.59) 2086 0 R (equation.6.4.60) 2087 0 R (equation.6.4.61) 2088 0 R (equation.6.4.61a) 2089 0 R (equation.6.4.61b) 2090 0 R (equation.6.4.62) 2091 0 R] +/Limits [(equation.6.4.59) (equation.6.4.62)] >> endobj 3444 0 obj << -/Kids [3286 0 R 3287 0 R 3288 0 R 3289 0 R 3290 0 R 3291 0 R] -/Limits [(equation.3.1.13) (equation.3.3.38)] +/Names [(equation.6.4.63) 2092 0 R (equation.6.4.64) 2093 0 R (equation.6.4.65) 2098 0 R (equation.6.4.66) 2099 0 R (equation.6.4.67) 2100 0 R (equation.6.4.68) 2101 0 R] +/Limits [(equation.6.4.63) (equation.6.4.68)] >> endobj 3445 0 obj << -/Kids [3292 0 R 3293 0 R 3294 0 R 3295 0 R 3296 0 R 3297 0 R] -/Limits [(equation.3.3.39) (equation.3.5.66)] +/Names [(equation.6.4.69) 2102 0 R (equation.6.4.70) 2110 0 R (equation.7.1.1) 2162 0 R (equation.7.1.10) 2177 0 R (equation.7.1.11) 2178 0 R (equation.7.1.12) 2179 0 R] +/Limits [(equation.6.4.69) (equation.7.1.12)] >> endobj 3446 0 obj << -/Kids [3298 0 R 3299 0 R 3300 0 R 3301 0 R 3302 0 R 3303 0 R] -/Limits [(equation.3.5.67) (equation.4.1.8)] +/Names [(equation.7.1.12a) 2180 0 R (equation.7.1.12b) 2181 0 R (equation.7.1.12c) 2182 0 R (equation.7.1.13) 2183 0 R (equation.7.1.14) 2189 0 R (equation.7.1.15) 2190 0 R] +/Limits [(equation.7.1.12a) (equation.7.1.15)] >> endobj 3447 0 obj << -/Kids [3304 0 R 3305 0 R 3306 0 R 3307 0 R 3308 0 R 3309 0 R] -/Limits [(equation.4.1.9) (equation.5.2.4)] +/Names [(equation.7.1.16) 2191 0 R (equation.7.1.17) 2192 0 R (equation.7.1.18) 2194 0 R (equation.7.1.19) 2195 0 R (equation.7.1.2) 2163 0 R (equation.7.1.20) 2196 0 R] +/Limits [(equation.7.1.16) (equation.7.1.20)] >> endobj 3448 0 obj << -/Kids [3310 0 R 3311 0 R 3312 0 R 3313 0 R 3314 0 R 3315 0 R] -/Limits [(equation.5.2.5) (equation.5.4.36a)] +/Names [(equation.7.1.21) 2204 0 R (equation.7.1.22) 2205 0 R (equation.7.1.23) 2206 0 R (equation.7.1.24) 2207 0 R (equation.7.1.25) 2208 0 R (equation.7.1.26) 2214 0 R] +/Limits [(equation.7.1.21) (equation.7.1.26)] >> endobj 3449 0 obj << -/Kids [3316 0 R 3317 0 R 3318 0 R 3319 0 R 3320 0 R 3321 0 R] -/Limits [(equation.5.4.36b) (equation.5.5.70)] +/Names [(equation.7.1.27) 2215 0 R (equation.7.1.28) 2216 0 R (equation.7.1.29) 2217 0 R (equation.7.1.3) 2164 0 R (equation.7.1.30) 2218 0 R (equation.7.1.31) 2219 0 R] +/Limits [(equation.7.1.27) (equation.7.1.31)] >> endobj 3450 0 obj << -/Kids [3322 0 R 3323 0 R 3324 0 R 3325 0 R 3326 0 R 3327 0 R] -/Limits [(equation.5.5.71) (equation.6.2.32)] +/Names [(equation.7.1.32) 2220 0 R (equation.7.1.33) 2222 0 R (equation.7.1.34) 2227 0 R (equation.7.1.35) 2228 0 R (equation.7.1.36) 2229 0 R (equation.7.1.37) 2237 0 R] +/Limits [(equation.7.1.32) (equation.7.1.37)] >> endobj 3451 0 obj << -/Kids [3328 0 R 3329 0 R 3330 0 R 3331 0 R 3332 0 R 3333 0 R] -/Limits [(equation.6.3.33) (equation.6.4.66)] +/Names [(equation.7.1.38) 2238 0 R (equation.7.1.39) 2239 0 R (equation.7.1.4) 2165 0 R (equation.7.1.40) 2240 0 R (equation.7.1.41) 2241 0 R (equation.7.1.42) 2242 0 R] +/Limits [(equation.7.1.38) (equation.7.1.42)] >> endobj 3452 0 obj << -/Kids [3334 0 R 3335 0 R 3336 0 R 3337 0 R 3338 0 R 3339 0 R] -/Limits [(equation.6.4.67) (equation.7.1.35)] +/Names [(equation.7.1.5) 2166 0 R (equation.7.1.6) 2167 0 R (equation.7.1.7) 2174 0 R (equation.7.1.8) 2175 0 R (equation.7.1.9) 2176 0 R (equation.7.2.43) 2269 0 R] +/Limits [(equation.7.1.5) (equation.7.2.43)] >> endobj 3453 0 obj << -/Kids [3340 0 R 3341 0 R 3342 0 R 3343 0 R 3344 0 R 3345 0 R] -/Limits [(equation.7.1.36) (equation.7.2.63)] +/Names [(equation.7.2.43a) 2270 0 R (equation.7.2.43b) 2271 0 R (equation.7.2.44) 2277 0 R (equation.7.2.45) 2279 0 R (equation.7.2.46) 2280 0 R (equation.7.2.47) 2281 0 R] +/Limits [(equation.7.2.43a) (equation.7.2.47)] >> endobj 3454 0 obj << -/Kids [3346 0 R 3347 0 R 3348 0 R 3349 0 R 3350 0 R 3351 0 R] -/Limits [(equation.7.2.64) (equation.8.1.9)] +/Names [(equation.7.2.48) 2317 0 R (equation.7.2.49) 2318 0 R (equation.7.2.50) 2319 0 R (equation.7.2.51) 2320 0 R (equation.7.2.52) 2321 0 R (equation.7.2.53) 2322 0 R] +/Limits [(equation.7.2.48) (equation.7.2.53)] >> endobj 3455 0 obj << -/Kids [3352 0 R 3353 0 R 3354 0 R 3355 0 R 3356 0 R 3357 0 R] -/Limits [(equation.8.2.10) (equation.8.3.45)] +/Names [(equation.7.2.54) 2328 0 R (equation.7.2.55) 2329 0 R (equation.7.2.56) 2330 0 R (equation.7.2.57) 2331 0 R (equation.7.2.58) 2332 0 R (equation.7.2.59) 2333 0 R] +/Limits [(equation.7.2.54) (equation.7.2.59)] >> endobj 3456 0 obj << -/Kids [3358 0 R 3359 0 R 3360 0 R 3361 0 R 3362 0 R 3363 0 R] -/Limits [(equation.8.3.46) (page.103)] +/Names [(equation.7.2.60) 2334 0 R (equation.7.2.61) 2335 0 R (equation.7.2.62) 2336 0 R (equation.7.2.63) 2351 0 R (equation.7.2.64) 2352 0 R (equation.7.2.65) 2380 0 R] +/Limits [(equation.7.2.60) (equation.7.2.65)] >> endobj 3457 0 obj << -/Kids [3364 0 R 3365 0 R 3366 0 R 3367 0 R 3368 0 R 3369 0 R] -/Limits [(page.104) (page.136)] +/Names [(equation.7.2.66) 2382 0 R (equation.7.2.67) 2383 0 R (equation.7.2.68) 2384 0 R (equation.7.2.69) 2385 0 R (equation.7.3.70) 2400 0 R (equation.7.3.71) 2401 0 R] +/Limits [(equation.7.2.66) (equation.7.3.71)] >> endobj 3458 0 obj << -/Kids [3370 0 R 3371 0 R 3372 0 R 3373 0 R 3374 0 R 3375 0 R] -/Limits [(page.137) (page.169)] +/Names [(equation.7.3.72) 2402 0 R (equation.7.3.73) 2403 0 R (equation.7.3.74) 2404 0 R (equation.7.3.75) 2412 0 R (equation.7.3.76) 2413 0 R (equation.7.3.77) 2414 0 R] +/Limits [(equation.7.3.72) (equation.7.3.77)] >> endobj 3459 0 obj << -/Kids [3376 0 R 3377 0 R 3378 0 R 3379 0 R 3380 0 R 3381 0 R] -/Limits [(page.17) (page.200)] +/Names [(equation.7.3.78) 2415 0 R (equation.7.3.79) 2420 0 R (equation.7.3.80) 2423 0 R (equation.7.3.81) 2424 0 R (equation.7.3.82) 2430 0 R (equation.7.3.83) 2431 0 R] +/Limits [(equation.7.3.78) (equation.7.3.83)] >> endobj 3460 0 obj << -/Kids [3382 0 R 3383 0 R 3384 0 R 3385 0 R 3386 0 R 3387 0 R] -/Limits [(page.201) (page.233)] +/Names [(equation.7.3.84) 2439 0 R (equation.7.3.85) 2440 0 R (equation.7.3.86) 2441 0 R (equation.7.4.87) 2473 0 R (equation.7.4.88) 2474 0 R (equation.7.4.89) 2475 0 R] +/Limits [(equation.7.3.84) (equation.7.4.89)] >> endobj 3461 0 obj << -/Kids [3388 0 R 3389 0 R 3390 0 R 3391 0 R 3392 0 R 3393 0 R] -/Limits [(page.234) (page.36)] +/Names [(equation.7.4.90) 2482 0 R (equation.8.1.1) 2557 0 R (equation.8.1.2) 2558 0 R (equation.8.1.3) 2559 0 R (equation.8.1.4) 2560 0 R (equation.8.1.5) 2565 0 R] +/Limits [(equation.7.4.90) (equation.8.1.5)] >> endobj 3462 0 obj << -/Kids [3394 0 R 3395 0 R 3396 0 R 3397 0 R 3398 0 R 3399 0 R] -/Limits [(page.37) (page.69)] +/Names [(equation.8.1.6) 2566 0 R (equation.8.1.7) 2575 0 R (equation.8.1.8) 2582 0 R (equation.8.1.9) 2584 0 R (equation.8.2.10) 2621 0 R (equation.8.2.11) 2622 0 R] +/Limits [(equation.8.1.6) (equation.8.2.11)] >> endobj 3463 0 obj << -/Kids [3400 0 R 3401 0 R 3402 0 R 3403 0 R 3404 0 R 3405 0 R] -/Limits [(page.7) (section.1.3)] +/Names [(equation.8.2.12) 2623 0 R (equation.8.2.13) 2624 0 R (equation.8.2.14) 2625 0 R (equation.8.2.15) 2626 0 R (equation.8.2.16) 2632 0 R (equation.8.2.17) 2633 0 R] +/Limits [(equation.8.2.12) (equation.8.2.17)] >> endobj 3464 0 obj << -/Kids [3406 0 R 3407 0 R 3408 0 R 3409 0 R 3410 0 R 3411 0 R] -/Limits [(section.10.1) (section.9.3)] +/Names [(equation.8.2.18) 2634 0 R (equation.8.2.19) 2635 0 R (equation.8.2.20) 2643 0 R (equation.8.2.21) 2644 0 R (equation.8.2.22) 2645 0 R (equation.8.2.23) 2654 0 R] +/Limits [(equation.8.2.18) (equation.8.2.23)] >> endobj 3465 0 obj << -/Kids [3412 0 R 3413 0 R 3414 0 R 3415 0 R 3416 0 R 3417 0 R] -/Limits [(Doc-Start) (Item.292)] +/Names [(equation.8.2.24) 2655 0 R (equation.8.2.25) 2656 0 R (equation.8.2.26) 2657 0 R (equation.8.2.27) 2658 0 R (equation.8.2.28) 2659 0 R (equation.8.3.29) 2681 0 R] +/Limits [(equation.8.2.24) (equation.8.3.29)] >> endobj 3466 0 obj << -/Kids [3418 0 R 3419 0 R 3420 0 R 3421 0 R 3422 0 R 3423 0 R] -/Limits [(Item.293) (Item.98)] +/Names [(equation.8.3.30) 2682 0 R (equation.8.3.31) 2683 0 R (equation.8.3.32) 2684 0 R (equation.8.3.33) 2685 0 R (equation.8.3.34) 2686 0 R (equation.8.3.35) 2687 0 R] +/Limits [(equation.8.3.30) (equation.8.3.35)] >> endobj 3467 0 obj << -/Kids [3424 0 R 3425 0 R 3426 0 R 3427 0 R 3428 0 R 3429 0 R] -/Limits [(Item.99) (defi.4.14)] +/Names [(equation.8.3.36) 2694 0 R (equation.8.3.37) 2695 0 R (equation.8.3.38) 2696 0 R (equation.8.3.39) 2697 0 R (equation.8.3.40) 2698 0 R (equation.8.3.41) 2708 0 R] +/Limits [(equation.8.3.36) (equation.8.3.41)] >> endobj 3468 0 obj << -/Kids [3430 0 R 3431 0 R 3432 0 R 3433 0 R 3434 0 R 3435 0 R] -/Limits [(defi.4.15) (defi.7.41)] +/Names [(equation.8.3.42) 2709 0 R (equation.8.3.43) 2711 0 R (equation.8.3.44) 2712 0 R (equation.8.3.45) 2713 0 R (equation.8.3.46) 2714 0 R (equation.8.3.47) 2720 0 R] +/Limits [(equation.8.3.42) (equation.8.3.47)] >> endobj 3469 0 obj << -/Kids [3436 0 R 3437 0 R 3438 0 R 3439 0 R 3440 0 R 3441 0 R] -/Limits [(defi.7.42) (equation.2.2.44)] +/Names [(equation.8.3.48) 2728 0 R (equation.8.3.49) 2729 0 R (equation.8.3.50) 2730 0 R (equation.8.3.51) 2757 0 R (equation.8.3.52) 2758 0 R (equation.9.1.1) 2803 0 R] +/Limits [(equation.8.3.48) (equation.9.1.1)] >> endobj 3470 0 obj << -/Kids [3442 0 R 3443 0 R 3444 0 R 3445 0 R 3446 0 R 3447 0 R] -/Limits [(equation.2.2.45) (equation.5.2.4)] +/Names [(equation.9.1.2) 2847 0 R (equation.9.1.3) 2848 0 R (equation.9.1.4) 2849 0 R (equation.9.1.5) 2850 0 R (equation.9.1.6) 2851 0 R (equation.9.1.7) 2852 0 R] +/Limits [(equation.9.1.2) (equation.9.1.7)] >> endobj 3471 0 obj << -/Kids [3448 0 R 3449 0 R 3450 0 R 3451 0 R 3452 0 R 3453 0 R] -/Limits [(equation.5.2.5) (equation.7.2.63)] +/Names [(equation.9.1.8) 2853 0 R (equation.9.1.9) 2854 0 R (equation.9.2.10) 2893 0 R (equation.9.2.11) 2894 0 R (equation.9.2.11a) 2895 0 R (equation.9.2.11b) 2896 0 R] +/Limits [(equation.9.1.8) (equation.9.2.11b)] >> endobj 3472 0 obj << -/Kids [3454 0 R 3455 0 R 3456 0 R 3457 0 R 3458 0 R 3459 0 R] -/Limits [(equation.7.2.64) (page.200)] +/Names [(equation.9.2.12) 2897 0 R (equation.9.3.13) 2930 0 R (equation.9.3.14) 2931 0 R (equation.9.3.15) 2932 0 R (equation.9.3.16) 2938 0 R (equation.9.3.17) 2940 0 R] +/Limits [(equation.9.2.12) (equation.9.3.17)] >> endobj 3473 0 obj << -/Kids [3460 0 R 3461 0 R 3462 0 R 3463 0 R 3464 0 R] -/Limits [(page.201) (section.9.3)] +/Names [(equation.9.3.18) 2955 0 R (equation.9.3.19) 2956 0 R (figure.caption.2) 336 0 R (figure.caption.3) 356 0 R (page.1) 196 0 R (page.10) 355 0 R] +/Limits [(equation.9.3.18) (page.10)] >> endobj 3474 0 obj << -/Kids [3465 0 R 3466 0 R 3467 0 R 3468 0 R 3469 0 R 3470 0 R] -/Limits [(Doc-Start) (equation.5.2.4)] +/Names [(page.100) 1448 0 R (page.101) 1461 0 R (page.102) 1476 0 R (page.103) 1487 0 R (page.104) 1497 0 R (page.105) 1510 0 R] +/Limits [(page.100) (page.105)] >> endobj 3475 0 obj << -/Kids [3471 0 R 3472 0 R 3473 0 R] -/Limits 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00000 n trailer -<< /Size 3480 -/Root 3478 0 R -/Info 3479 0 R -/ID [ ] >> +<< /Size 3596 +/Root 3594 0 R +/Info 3595 0 R +/ID [<983E776EA8312F1096E209B7D8C98AC6> <983E776EA8312F1096E209B7D8C98AC6>] >> startxref -1245956 +1287669 %%EOF diff --git a/script.tex b/script.tex index b4e101a..5e6faf2 100644 --- a/script.tex +++ b/script.tex @@ -11,7 +11,7 @@ \usepackage{fancyhdr} \usepackage{pdfpages} \usepackage[arrowdel]{physics} -\usetikzlibrary{calc,trees,positioning,arrows,fit,shapes,angles,patterns} +\usetikzlibrary{calc,trees,decorations.markings,positioning,arrows,fit,shapes,angles,patterns} \DeclareDocumentCommand\vnabla{}{\vectorarrow{\nabla}} \graphicspath{assets} @@ -38,6 +38,7 @@ \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\const}{const} +\DeclareMathOperator{\Log}{Log} \newcommand{\natn}{\mathbb{N}} \newcommand{\intn}{\mathbb{Z}} @@ -86,7 +87,7 @@ \newcommand{\interior}[1]{\mathring{#1}} \newcommand{\boundary}[1]{\partial #1} -\newcommand{\closure}[1]{\bar{#1}} +\newcommand{\closure}[1]{\overline{#1}} \newcommand{\conv}[1]{\xrightarrow{\makebox[2em][c]{$\scriptstyle#1$}}} \newcommand{\convinf}{\conv{n \rightarrow \infty}}