mat 3730 complex variables section 4.1 contours
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Chapter 4: Complex Integration Very similar to line integrals in
Multivariable Calculus 4.1: Set up the notations:
• Parametrizations
• Contours
Smooth Arcs
smooth arc (cur
A point set given by
,
is a if
(i) has a continuous derivative on ,
ii 0 on ,
iii is 1-1 on
ve)
,
C
z z t x t iy t a t b
z t a b
z t a b
z t a b
Smooth Closed Curves
A point set given by
,
is a if
(i) has a continuous derivative on ,
ii 0 on
smooth closed curve
( ) ( )iii ( ) is 1-1 on [ , ) and
( )
,
( )
C
z z t x t iy t a t
z a z bz t
b
z t a b
z t
z b
b
a ba z
a
Admissible Parametrizations
admissible paramet
is an of
if it is
rization
smooth arc/closea d curve.
z z t
C
Example 1 (a)
Find an admissible parametrization for the following smooth curve
The straight-line segment from
z1=-2-3i to z2=5+6i
Example 1 (b)
Find an admissible parametrization for the following smooth curve
The circle with radius 2 centered at 1-i
Example 1 (c)
Find an admissible parametrization for the following smooth curve
The graph of the function for3y x 0 1x
Directed Smooth Curves
A smooth arc/closed curve is directed if its points have a specific ordering.
(All curves in example 1 are directed with the order induced by the parametrization)
Contours
0
1 2
1
A is either a single point or a finite sequence
of directed smooth curves , , , such that the
terminal points of coincides with the initial point of ,
for 1,2,... 1.
contour
n
k k
z
k n
Notation
1 2: n
Opposite Contour
If the directions of all are reversed, the resulting
Oppositecontour is called the of Contour
:
k
Notation
Definitions
Closed contour• The initial and terminal points coincide.
Simple closed contour• A closed contour with no multiple points other
than its initial-terminal point.
Example
Orientations
A simple closed contour separates the plane into 2 domains: one bounded, and one unbounded.
Positively oriented Negatively oriented