Complex functions C – Integration

7/29/2019 Complex functions C Integration

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1.6 Contour integration

(pp. 61-79; 86-88)

NotationA line in the complex plane can be

smooth

piecewise smooth

closed

* simple closed * not simple

closed

A contour is asimple closed curve with

apositive (counterclockwise) orientation.

Contour deformation: The process used

to remove singularities from inside acontour.

1

A

B

A

B

A

A


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1.6.1 Contour integrals

If Cis a piecewise smooth curve in the

complex plane withz x jy= +

and

( ) ( ; ) ( ; )f z u x y jv x y= + , we use

( ) [ ( , ) ( , )]( C C

f z dz u x y jv x y dx jd= + +

The integral is called a complex lineintegral.

Example 1

Evaluate

2( 3 )C

z z dz+along Cwhere Cis

the curve shown in

the sketch.

1.6.2 Cauchys theoremIff(z) is an analytic function withderivativef'(z) that is continuous at all

2

2

2 + 2j2


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points inside and on a simple closed

curve C, then

( ) 0C

f z dz= No singularities in C!The integral is called a contour integral.

Example 2

Evaluatez

C

e dz

where Cis the curve in the sketch.

Example 3

Evaluate

2 5 6C

dz

z z +

where Cis the unit circle.

Cauchy integral theorem: p. 69USE PARTIAL FRACTIONS!Letf(z) be analytic within and on a

simple closed curve C. Then

3


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0

0

( )2 ( )

C

f zdz j f z

z z

=

0 0( ) ( )2 ( )C

f z dz z z jf z = forz0a simple pole and

( )

( )

01

0

( ) 2( )!

n

n

C

f z jdz f z nz z

+ =.

1 ( )

0 0( ) ( ) 2 ( )n n

Cf z dz z z jf z+ =

forz0a pole of ordern + 1.Example 4

Evaluate

2

z

C

edz

zfor each of the following contours.

a) Cis any contour enclosingz= 2

b) Cis any contour excludingz= 2

4


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Example 5

Evaluate

2(2 1)( 2)C

zdzz z +

for the given contourC.

a) C:1z =

b) C: 3z =Example 6

Evaluate3

3

(2 1)C

z zdz

z

+

+


Example 7

Evaluate

2

4

( 1)( 2)C

z

dzz z + where Cis the curve

3z =

5


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1.6.4 The residue theorem: p. 73DONT USE PARTIAL FRACTIONS!

Iff(z) is an analytic function within andon a simple closed curve C, apart from a

finite number of poles, then

( )C

f z dz= 2j[sum of residues off(z) at the

poles inside C]Example 8

Evaluate

2( 1) ( 3)C

dz

z z where Cis the circle 2z =

Example 9

Evaluate

22 6

4C

z dzz

++

C:2z j =

.

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Example 10

Evaluate

2( 1) ( 3)C

dzz z

for each contourC.

a) C:

1

2z =

b) C:2z =

Example 11

Evaluate

3( 1) ( 1)( 2)C

dz

z z z+ where Cis the rectangle with vertices

jand 3 j .

1.6.5 Evaluation of definite real

integralsType 1: Infinite real

integrals

7

- R - R

x

y

O

C


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-Cconsists of the real axis between R

andR, and the semicircle in the upper

half of the complex plane.

- In calculationsR is chosen to include

all poles in the upper half plane.

Example 12

Evaluate

2 2( 1)( 9

dx

x x

+ + using contour integration.Example 13

Evaluate

4,

1

dxx

x

+

Example 14

Evaluate

2 2 20 ( 1)( 4)

dx

x x

+ +

8


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Type 2: Integrals of trig. functions

Suppose (sin , cos )G G = is a function

containing sin and cos . To evaluate2

0Gd

, let , 0 2 .jz e =

Thus,

j

j

dz dz

dz je d d je jz

= = = .Furthermore, p.30

( ) ( )11 1

cos

2 2

j j

Ze e z = + = +

and

( ) ( )11 1

sin2 2

j j

Ze e z

j j

= =

Substitute these expressions and then2

0( )

C

Gd f z dz

=

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Example 15Using contour integration, evaluate

2 cosC

d

+

Example 16

Show that

22 cosC

d

=

Work through every example from p 61

up to and including p 79. Then do the

exercises on p 71 (#53(b), 54(b), 56, 57,

58, 59), p 78 79 (#60 65) and pp 87 88 (#7, 8, 11 16,19 24).

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Complex functions C – Integration

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