MAT 3730Complex 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 Arcs

(i) has a continuous derivative on ,z t a b

Smooth Arcs

ii 0 on ,z t a b

Smooth Arcs

ii 0 on ,z t a b

Smooth Arcs

iii is 1-1 on ,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

Length of a Smooth Curve

given by

,

length of : b

a

C

z z t x t iy t a t b

dzdt

dt

Next Class

Read Section 4.2