Regular Function
Taylor expansion
If
is regular for 
, then
- with
Cauchy's evaluation formula
If a function
is regular for 
, then
Maximum principle
If
is a regular (but not constant) function in a region 
, 
takes the maximum value on the boundary and not in the interior of 
.Liouville's theorem
If
is regular in the complex plane 
and 
(
:constant), then
is constant.Algebra's fundamental theorem
The
-th order, complex-coefficient algebraic equation
has
roots in the complex domain.Analytic prolongation
Consider that
is regular in 
, and 
is regular in 
.If
is true in 
, then
is an analytic prolongation of 
.Gauss' mean value theorem
If
is regular in the interior and along a circle 
centered on 
with radius 
, then
Poisson's integral
If
is regular in the interior and along circle 
described by 
, then
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