講義概要

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

取り扱う内容は以下の通り．

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

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

参考書：
吹田・新保 著「理工系の微分積分学」学術図書出版
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，
演習に相当するものは，各自で行うべき段階に皆さんは入っていることを意味しています．
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．