Complex Analysis(1)

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最新推荐文章于 2025-05-08 14:47:49 发布

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分类专栏：数学笔记文章标签： Complex Analysis Stein 复分析

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In this series, I will summarize some meaningful conclusions as well as the poofs in the book Complex Analysis by Elias M. Stein and Rami Shakarchi.

I start from the first chapter, preliminaries to complex analysis.

1. Some basic properties of complex number

$\bullet$ Two different ways to represent complex number:

By definition of complex number: $z=x+iy$ where $x,y\in\mathbb{R}$ , $i^2=-1$ . Comparing it with the plane coordinate we can use real and imaginary axes to represent all complex number.

Analogue to the polar coordinate we get the other form of complex number, since every vector in the plane determined by both length and direction. For the first form, we have $z=x+iy=\sqrt{x^2+y^2}(\frac{x}{\sqrt{x^2+y^2}}+i\frac{y}{\sqrt{x^2+y^2}})$ , mark $r=\sqrt{x^2+y^2}$ is the module of , by the definition of cosine and sine, we derive $z=re^{i\theta}$ , where $r,\theta\in\mathbb{R}, r\geq0,e^{i\theta}=cos\theta+isin\theta$ .