$ E $ is provided a set of complex numbers, each complex if the $ z $ $ E $, are uniquely determined corresponding plurality $ $ W, called a single-valued function is determined on $ E $.
Multi-valued function
Provided $ w = f (z) $ is defined in the single or multi-valued function on a set of points $ E $, and let $ z = x + iy $, $ w = u + iv $, $ u, v $ are with $ x, y $ determined, so $ w = f (z) $ and often written $ w = u (x, y) + iv (x, y) $
As will be shown as $ z $ exponent $ z = re ^ {i \ theta} $, the function $ w = f (z) $ turn table to $ w = P (r, \ theta) + iQ (r, \ theta) $
The transformation : The point set $ E $ on $ z $ plane at any point $ z $, there point set $ F $ on $ W $ plane point $ w $, such that $ w = f (z) $ , called $ w = f (z) $ the $ E $ variant (enantiomers) F. $ $ into, abbreviated as $ f (E) \ subseteq F $
Or said $ w = f (z) $ $ E $ is to transform into $ F $.
Full conversion : If $ f (E) \ subseteq F $, and for any point $ F $ $ w $, $ E $ has a point $ z $, such that $ w = f (z) $ , $ w called = f (z) $ the $ E $ variations (enantiomers) into $ F $, (abbreviated as: $ f (E) = F $), also known as $ w = f (z) $ is $ E $ and $ full conversion of F $.
If $ w = f (z) $ point is set to $ E $ $ F $ full conversion, and to each of $ F $ $ W $ point, there is at least one or two points with $ E $ the correspondence is determined in $ F $ a single or multi-valued function, denoted as $ z = f ^ {- 1} (w) $, it is called a function $ w = f (z) $ trans function, referred to as transform or $ w = f (z) $ inverse transform, if $ z = f ^ {- 1} (w) $ $ F $ is converted to a single value of $ E $, called $ w = f (z) $ $ E $ is $ F $ to both the single-valued conversion, or every other transform.
Limit, continuity:
Provided function $ w = f (z) $ has on the point set $ E $ defined, $ z_ {0} $ to $ E $ accumulation point, such as the presence of a complex $ w_ {0} $, such that for any given $ \ varepsilon> 0 $ with a $ \ delta> 0 $, as long as the $ 0 <| z-z_ {0} | <\ delta, z \ in E $ there
$ | F (z) -f (z_ {0}) | <\ varepsilon $ called function $ f (z) $ $ E $ along at $ z_ {0} $ there is a limit $ w_ {0} $
记为：$\underset{z \rightarrow z_{0},z\in E}{lim}f(z)=w_{0}$
1. If there are limits, will be the only
2. $ f (z), g (z) $ in the point set $ E $ $ z_ {0} $ limit at present, it still limits arithmetic, arithmetic and equal to their limit.
Theorem: Let the function $ f (z) = u (x, y) + iv (x, y) $ is defined on the set of points E, $ z_ {0} = x_ {0} + iy_ {0} $ is E poly point,
$\underset{z \rightarrow z_{0},z\in E}{lim}f(z)=\eta =a+ib$
The necessary and sufficient condition
$\underset{(x,y)\rightarrow (x_{0},y_{0}),(x,y)\in E}{lim}u(x,y)=a$
$\underset{(x,y)\rightarrow (x_{0},y_{0}),(x,y)\in E}{lim}v(x,y)=b$
Continuity: set function $ w = f (z) $ is defined, $ z_ {0} $ $ E $ is poly point set to the point $ E $, and $ z_ {0} \ in E $, if
$\underset{z\rightarrow z_{0},z_{0}\in E}{lim}f(z)=f(z_{0})$
I.e., to any $ \ varepsilon> 0 $, there $ \ delta> 0 $ long $ | z-z_ {0} | <\ delta, z \ in E $, there
$ | F (z) -f (Z_ {0}) | <\ varepsilon $
Called $ f (z) $ $ E $ along at $ z_ {0} $ continuously.
Theorem: Let the function $ f (z) = u (x, y) + iv (x, y) $ is defined on the set of points $ E $, $ z_ {0} \ in E $, then $ f (z) $ $ E $ point along $ z_ {0} = x_ {0} + iy_ {0} $ continuous iff:
Binary function $ u (x, y), v (x, y) $ $ E $ along at point $ (x_ {0}, y_ {0}) $ continuously.
Definitions: The function $ f (z) $ at the point set $ E $ are successive points, called $ f (z) $ $ E $ continuously on.