Liang_Ling的博客

ArgumentArga=arga+2πZArg \,a=arg\, a +2\pi \mathbb{Z}Stereographic Projectionx1=z+z¯1+|z|2x2=z−z¯1+|z|2x3=|z|2−1|z|2+1x_1 = \frac{z+\bar{z}}{1+|z|^2}\\x_2 = \frac{z-\bar{z}}{1+|z|^2}\\x_3 = \frac{|

Theorem 1(Cauchy-Green formula, Pompeiu formula)Suppose that U⊂CU\subset C is a bounded domain, having C1C^1 boundary. f(z)=u(x,y)+iv(x,y)∈C1(U¯)f(z) = u(x,y)+iv(x,y)\in C^1(\bar{U}), then f(z)=12πi∫

Taylor series and Liouville theoremMany interesting results can be obtained by the Cauchy Integral Theory. Theorem 1Suppose that f(z)f(z) is analytic on U⊂CU\subset C, and continuous on U¯\bar{U}. Then

The zeros of analytic functionsUse Liouville theorem, we can prove the fundamental theorem of algebra.Theorem 1If p(z)p(z) is a polynomial, then there is at least one z0z_0 such that p(z0)=0p(z_0)=0. L

Maximum modulus principle and Schwarz lemmaAverage Vaule Propertiesf(z0)=12πi∫∂D(z0,r)f(ξ)ξ−z0dξ=12πi∫2π0f(z0+reit)ireitreitdt=12π∫2π0f(z0+reit)dt.\begin{align*}f(z_0) &= \frac{1}{2\pi i}\int_{\parti

Laurent SeriesThe special properties of a complex function is much more determined by its singularity, to study the singularity of a function, we first give a useful theorem that does not hold for real