Complex Analysis pt. I
This is the first in a series of entries on complex analysis. I’m posting as I read the book, Applied Complex Variables by John W. Dettman. So far, I’ve only read up to page 22, but it seems to be a good book; I like the style: definitions and statements followed by proofs. I also like the cheap printing… seriously; it gives it a certain ambience.
Curves and Regions
- simple Jordan arc
- A simple Jordan arc is a set of points defined by a parametric equation
, 
, where
and 
are continuous real-valued functions such that 
implies that 
- simple smooth arc
- A simple smooth arc is a Jordan arc defined by a parametric equation
, , where 
and 
are continuous, and 
.
- simple closed Jordan curve
- A simple closed Jordan curve is a set of points defined by a parametric equation ,
, where 
and are continuous real-valued functions such that 
if and only if 
, or 
.
- Jordan Curve Theorem
- Every simple closed Jordan Curve in the complex plane divides the plane into two disjoint open sets. The curve is the boundary of each of these sets. One set ( the interior of the curve) is bounded and the other (the exterior of the curve) is unbounded.
- simple piecewise smooth curve
- A simple piecewise smooth curve is a simple Jordan arc whose parametric equation
, , has piecewise continuous derivatives and , were 
.
- simple closed piecewise smooth curve
- A simple closed piecewise smooth curve is a simple closed Jordan curve whose parametric equation , , has piecewise continuous derivatives and , where .
- connectedness
- A set
in the complex plane is connected if every pair of points in can be joined by a simple Jordan arc lying entirely in . - domain
- A nonempty open connected set of points is a domain.
- region
- A region is a domain together with all, some, or none of its boundary points
- simply connected domain
- A domain
is simply connected if every simple closed Jordan curve lying in has its interior lying in .
Possibly relevant posts:
- Q: convexity of level sets (1/29/2008)
- Cauchy’s theorem via Homotopy (8/8/2007)
- Exactness of differential forms (7/28/2007)