5115
シラバス更新日
2015年10月5日

2015年9月16日

### 講義の目的

「複素解析第一」に引き続き，複素解析学に関するより進んだ内容を講義する．

### 講義計画

１． Sequences and Series of Holomorphic Functions：we will see that under the appropriate notion of convergence of a sequence of holomorphic functions, the limit function inherits sev- eral properties that the approximating functions have, such as being holomorphic, and in the second part, we show that the space of holomorphic functions on a domain can be given the structure of a complete metric space.we show that the space of holomorphic functions on a domain can be given the structure of a complete metric space.
２． Conformal Equivalence: we study conformal maps between domains in the extended complex plane. These maps are one-to-one meromorphic functions. Our goal is characterize all simply connected domains in the complex plane. We also study the action of a quotient of the group of two-by-two nonsingular complex matrices on the extended complex plane, namely the projective special linear group.
３． Harmonic Functions : this topic is devoted to the study of harmonic functions. These functions are closely connected to holomorphic maps since the real and imaginary parts of a holomorphic function are harmonic functions. One of the most important aspects of harmonic functions is that they arise as functions that solve a boundary value problem for holomorphic functions, known as the Dirichlet problem. An example is the problem of finding a function continuous in a closed disk that assumes certain known values on the boundary of the disk and is harmonic in the in- terior of the disk. An important tool in the solution is the Poisson formula.

### 教科書・参考書等

J. P. Gilman, I. Kra, R. E. Rodrigues, Complex Analysis, Springer GTM 245, ISBN 9780387747149

J. Ahlfors 著「複素解析」現代数学者（邦訳）

### 関連科目・履修の条件等

「複素解析第一」を習得していること．

### 成績評価

レポート及び試験により総合的に評価する．

### 担当教員の一言

この講義には対応する演習科目がありません．しかし，それは演習が必要ない，ということではなく，

どのように勉強を進めればよいか，分からないことがあればいつでも質問にくること．