MAT 3730
Complex Variables
Section 4.1
Contours
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Preview
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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
A point set   C given by
z  z  t   x  t   iy  t  , a  t  b
is a sm ooth arc (cur ve) if
(i) z  t  has a continuous derivative on
 ii 
z   t   0 on  a , b 
 iii  z  t 
is 1-1 on  a , b 
 a, b 
Smooth Arcs
(i) z  t  has a continuous derivative on
a, b 
Smooth Arcs
 ii 
z t   0 on
a, b 
Smooth Arcs
 ii 
z t   0 on
a, b 
Smooth Arcs
 iii  z  t 
is 1-1 on
a, b 
Smooth Closed Curves
A point set   C given by
z  z  t   x  t   iy  t  , a  t  b
is a sm ooth closed curve if
(i) z  t  has a continuous derivative on
 ii 
a, b 
z   t   0 on  a , b 
 z ( a )  z (b )
 iii   z ( t ) is 1-1 on [ a , b ) and 
 z ( a )  z ( b )
Admissible Parametrizations
z  z t 
is an adm issible param et rization of   C
if it is a sm ooth arc/close d curve.
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 y  x for
3
0  x 1
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
A contour  is either a single point z 0 or a finite sequence
of directed sm ooth curves
 1 ,  2 ,  ,  n 
such that the
term inal points of  k coincides w ith the initial point of  k  1 ,
for k  1, 2, ...n  1.
N otation :    1   2     n
Opposite Contour
If the directions of all  k are reversed, the resulting
contour is called the O pposite C ontour of 
N otation :  
Definitions
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Closed contour
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Simple closed contour
• The initial and terminal points coincide.
• A closed contour with no multiple points other
than its initial-terminal point.
Example
Orientations
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A simple closed contour separates the
plane into 2 domains: one bounded, and
one unbounded.
Positively oriented
Negatively oriented
Length of a Smooth Curve
  C given by
z  z  t   x  t   iy  t  , a  t  b
b
length of  :
   
a
dz
dt
dt
Next Class
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Read Section 4.2