一〇二上複變函數論期末考-2014.01.07

Final Exam
考試日期： ~ 15:10:00

考試地點：新生教學館 302

Conformal mappings

 * Schwatz Lemma (proof)
 * Automorphisms of $\mathbb{D}, \mathbb{C}, \mathbb{P}^1$
 * Explict isomorphisms between simply connected open subsets of $\mathbb{C}$
 * $\mathbb{H} \cong \mathbb{D}$
 * §7.4 Examples 1~3, 5~9
 * §7.4 Theorem 4.1 (proof)
 * Proof of Riemann uniformization theorem
 * Statement
 * Montel's theorem
 * Diagonal argument

Harmonic functions

 * Definition
 * Examples
 * $u\!: U \to \mathbb{R}$ harmonic
 * $U$: simply connected, $u = \Re{f}$, $f$ holomorphic
 * $u = \log|f(z)|$
 * Mean value theorem / Maximum modules principle
 * $\sup_{z \in U} u(z) < \sup\limits_{z \in \partial{U}} u(z)$, $U$: bounded open subset
 * Dirichlet problem on a disk
 * Poisson ketnel (formula)
 * Harmonic functions on an annulus
 * Proves
 * ($\bigstar$) Some Dirichlet problems on a simply connected domain
 * §7.2 Examples 2, 3, 6

Schwarz reflection

 * Statement and proof (use Morera's theorem)
 * Application to Dirichlet problems
 * §9.2 Theorem 2.5
 * §9.2 Example 1~5
 * ($\bigstar$) $f\!: U \to \mathbb{C}$ holomorphic, if $f$ extends to $\partial{U}$ continuously, then can $f$ extend to $\overline{U}$ holomorphically?

Infinite product

 * Weierstrass product (proof)
 * Mittay-Laffler theorem
 * Hadamard factorization
 * Definition of "order of growth" of entire functions
 * Example: $\sin{\pi z} = \pi z \prod\limits_{n \in \mathbb{Z}, n \neq 0} \left( 1 - \frac{z}{n} e^{z/n} \right)$

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複變函數論期末考解答-2014.01.07