derivable and holomorphic

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复习提纲

1. stereographic projection (definition and the way to find a projecting point)
2. calculate square root for a given complex number
3. triangle inequality
4. differentiation of a holomorphic function. (definition, Cauchy-Riemann equation, method to calculate a derivative for a given function , find a harmonic conjugate for a given real part)
5. Maximal muduli theorem (proof is not required, just need know how to use it )
6. write a rational function into a sum of partial fractions
7. linear transformation (cross ratio, the way to find a center of a circle decided by three points, how to decide if four points are on a same circle, symmetric points, reflection with respect to a circle, determine a linear transformation which can realize some transformations between circles)

• 疑问：哪些函数连续，连续引申出来的性质有哪些？
• 疑问：Prove fundamental theorem of algbra using maximum mudulii theorem
• 疑问：extended complex plane: C ∪ {∞} |z|>1/ε is the neiborhood of ∞
• Riemann sphere and stereographic projection.

• A function f is continous at piont Zo if all three of the following conditions satisfied: (虽然只是一个点，但是实际上却包含了邻域|Z-Zo|<ζ)< /p>

  1. lim f(Z) exists
  2. f(Zo) exists
  3. lim f(Z) = f(Zo)




• A function of complex variable is said to be continous in the region R if it is continous at each point in R.

  1. two functions are continous at a point,

     1. their sum and product are also continous at that point
     2. their quotient is continous at any such point if the denomonator is not zero there.
     3. a polynomial is continous in the entire plane.



• theorem1：两个连续函数的复合仍然是连续的。


• theorem2; 如果f(Z) 在点Zo连续并且不为0，那么在该店的某个邻域有f(Z)≠0。


• theorem3: 如果f(Z)=u(x,y) + iv(x,y)中的分量函数u和v在点Zo=(xo,yo)连续，那么f也在该点连续。反之，也成立。


• theorem4：设R是一个有界闭区间，如果函数f在整个R上连续，那么存在一个非负实数M，使得

    |f(Z)|≤M 对R中所有的点Z成立，其中等号至少对一个点Z成立
  
  

• Derivatives：let f be a function whose domain of definition contains a neighborhood |Z-Zo|<ε of a point Zo. The derivative of f at Zo is the limit

    f'(Zo)=lim (f(Z)-f(Zo))/(Z-Zo)
  

  or defined as

    f'(Zo)=lim (f(Zo+△Z)-f(Zo))/△Z
  

  and function f is said to be defferentiable at Zo when f'(Zo) exist.
• 复变函数有与实函数相似的求导性质。

    especially, F(Z)=g(f(Z)) has a derivative at Zo, and F'(Zo)=g'(f(Zo))f'(Zo)
  

• Cauchy-riemann equation； if function f(Z)=u(x,y)+v(x,y) is derivable at Zo, then

    f'(Zo)=Ux(xo.yo)+iVx(xo,yo)
    f'(Zo)=-iUy(xo,yo)+Vy(xo,yo) 
  

  by the uniqueness of limits, both the equation provide neccessary conditions for existance of f'(Zo)

    Ux(xo,yo)=Vy(xo,yo) and Uy(xo,yo)=-Vx(xo,yo)
  

• theorem: 设 f(Z)=u(x,y) + iv(x,y) 并且f'(Z) 在点Zo=xo+iyo存在，那么分量u和v的一届偏导数在点(xo,yo)存在，且在该店满足柯西-黎曼方程：

    Ux=Vy, Uy=-Vx
    并且 f'(Zo)=Ux+ iVx
  

• Sufficient consitions for Differentiability； 设函数 f(Z)=u(x,y)+iv(x,y)在点Zo=xo+iyo 的某个ε邻域中处处有定义，并且

  1. 分量u和v关于x和y的一阶偏导数在点Zo=xo+iyo的ε邻域内处处存在。
  2. 这些一阶偏导数在点(xo,yo)连续且在点(xo,yo)满足柯西黎曼方程, 那么f'(Zo)存在。

• 总结：

1. 如果要在某一点dirivable抑或是differentiable，那么需要函数在这点周围有定义，并且分量关于x,y的一阶偏导也要存在并且在周围连续，这样还不能保证derivable，必须还要让这一点满足柯西黎曼方程。
2. 还有第二种方法判断，便是直接用derivative的定义去求函数在那一点的导数，但这种方法基本不可行，因为我们无法去验证它在每一个方向上的极限都相等。倒是这种方法适合证明导数的不存在。
3. 特殊函数，比如ponynomials可以直接用它本身的性质去求。
4. 至于holomorphic function的定义，如果说 "holomorphic at a point Zo", means not just differentiable at Zo, but differentiable everywhere within some neighborhood of Zo in the complex plane.