Theorems and Propositions in Complex Analysis

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Notes of Anton Deitmar's Complex Analysis
$\S\;1.\quad\mathbb{C}$
Th.1.6 If$\,S\subset\mathbb{C}\,$is compact and $f\in\mathscr{C}(S,\mathbb{C}),\;$then
$(a)\quad f(S)\,$is compact, and
$(b)\quad \exists z_1,z_2\in S\,\forall z\in S:\,(|f(z_1)|\le|f(z)|\le|f(z_1)|).$

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Theorem 2.2 (Cauchy-Riemann Equations)
$\quad$Let$\,f=u+iv\,$be complex differentiable at$\;z=x+iy,\;$
$\quad$Then the partial derivatives $u_x,u_y,v_x,v_y\,$all exist and
$\quad$satisfy$\,u_x=v_y,\,u_y=-v_x.$

Proposition 2.3 Suppose$\,f\,$is holomprphic and a disk$\,D.$
$(a)\quad f'(D)=\{0\}\implies f\,$is constant.
$(b)\quad (|f|\,$is constant$)\implies(f\,$is constant$).$

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Lemma 3.0 Suppose $\lim\,\sup\sqrt[n]{|a_n|}=R^{-1}\in(0,\infty),$ Then
$(1)\quad{\small\displaystyle\sum_{n=0}^{\infty}}|a_n||z|^n <\infty\quad(|z|< R);$
$(2)\quad{\small\displaystyle\sum_{n=0}^{\infty}}|a_n||z|^n=\infty\quad(|z|> R).$
Proof. (1) Take$\;\lambda\in(|z|,R),\;\exists M\,\forall n>M:\,(\sqrt[n]{|a_n|}< \lambda^{-1})\small\underset{\,}{\,}$
$\;\;\qquad\qquad\therefore\;\,|a_n z^n|=|({\small\sqrt[n]{|a_n|}\lambda}){\large\frac{z}{\lambda}}|^n <|\large\frac{z}{\lambda}|^n\;(\small n>M)$
$\qquad\quad$(2) Take$\;\eta\in(R,|z|)\,$and then$\,\exists\small M\,$such that$\,\small\sqrt[n]{|a_n|}>\eta^{-1}$
$\qquad\qquad\;\;$and so$\,|a_nz^n|=|(\sqrt[n]{|a_n|}\eta){\large\frac{z}{\eta}}|^n >|\frac{z}{\eta}|^n > 1\;\;\small(n>M)$