1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an...

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(1) Indefinite Integration

(2) Cauchy’s Integral Formula

(3) Formulas for the derivatives of an analytic function

Section 5

SECTION 5Complex Integration II

value of the integralbetween two pointsdepends on the path

1no real meaning to

Section 5

integrate the function along the path Cjoining 2 to 12j as shown

2)( zzf

Example

1022)( ttjttz

)219(3

)3/8(1)21(

)84()443()21(

)21()22(

dtttjttj

dtjtjt

dztzfdzzf

Section 5

integrate the function along the path CC1 C2 joining 2 to 12j as shown

2)( zzf

Example

202)( tttz

)1()2()(

dttdzzfC

Along C1:

alongreal axis !

102)( ttjttzAlong C2:

dtjtjtdzzfC

11)211(

)21()2()(

Section 5

3/)219(2 jdzzC

value of the integralalong both paths is

the same

coincidence ??

Section 5

Dependence of Path

2C0)()(

dzzfdzzf

Suppose f (z) is analytic ina simply connected domain D

by the Cauchy Integral Theorem

alongalong

dzzfdzzf

note:if they intersect,we just do thisto each “loop”,one at a time

Section 5

Section 5Integration (independence of path)

Consider the integral dzzfz

If f (z) is analytic in a simply connected domain D, and z0

and z1 are in D, then the integral is independent of path in D

)()()( 01

zFzFdzzfz

where )(zfdz

)219(3

dzzzjz

Not only that, but.......

Section 5Examples

097.23

sinh2)sin(2

sincos

the wholecomplex plane

(2) ?1

( f (z) not analytic anywhere - dependent on path )

(3) jz

f (z) analytic in

this domain

(both 1z2 and 1z are not analytic at z0 - the path of integration C must bypass this point)

Section 5

Question:

Can you evaluate the definite integral

Section 5More Integration around Closed Contours ...We can use Cauchy’s Integral Theorem to integrate aroundclosed contours functions which are

(a) analytic, or (b) analytic in certain regions

For example,

f (z) is analytic everywhereexcept at z0

But what if the contour surrounds a singular point ?

zfjdzzz

Section 5Cauchy’s Integral Formula

Let f (z) be analytic in a simply connected domain D. Then forany point z0 in D and any closed contour C in D that encloses z0

Section 5Cauchy’s Integral Formula

zfjdzzz

Let f (z) be analytic in a simply connected domain D. Then forany point z0 in D and any closed contour C in D that encloses z0

Section 5Example

Evaluate the integral where C is

Singular point inside !

becomes

or jdzz

zfjdzzz

The Cauchy Integral formula

Section 5Example

becomes

or jdzz

zfjdzzz

Section 5Example

becomes

or jdzz

zfjdzzz

Section 5Example

becomes

or jdzz

zfjdzzz

Section 5Illustration of Cauchy’s Integral Formula

zfjdzzz

Let us illustrate Cauchy’s Integral formulafor the case of f (z)z and z0 1

C DSo the Cauchy Integral formula

becomes

1)1( f

or jdzz

f (z) is analytic everywhere,so C can be any contour in thecomplex plane surrounding thepoint z1

Section 5Another Example

zfjdzzz

becomes

ejdzjz

jedzjz

eEvaluate where C is any closed contour

surrounding zj

f (z) is analytic everywhere

zfjdzzz

becomes

ejdzjz

jedzjz

eEvaluate where C is any closed contour

surrounding zj

f (z) is analytic everywhere

zfjdzzz

Let us illustrate Cauchy’s Integral formulafor the case of f (z)1 and z0 0

becomes

or jdzzC

f (z) is a constant, and so is entire,so C can be any contour in thecomplex plane surrounding theorigin z0

zfjdzzz

Let us illustrate Cauchy’s Integral formulafor the case of f (z)1 and z0 0

becomes

or jdzzC

f (z) is a constant, and so is entire,so C can be any contour in thecomplex plane surrounding theorigin z0

Section 5

Cut out the point z0 from the simply connected domain by introducinga small circle of radius r - this creates a doubly connected domain inwhich 1z is everywhere analytic.

From the Cauchy Integral Theorem as appliedto Doubly Connected Domains, we have

note: see section 4, slide 6

Let us now prove Cauchy’s Integral formulafor this same case: f (z)1 and z0 0

But the second integral, around C*, is given by

jdtjdtrjeer

dztzfdzzf jtjt

What does the equation mean ?

Section 5Equations involving the modulus

equation of a circle22

0 )()( ryyxx

mathematically:

(these are used so that we can describe paths(circles) of integration more concisely)

Section 5

Example

1)2( 22 yx

equation of a circle22

0 )()( ryyxx

Section 5

centre

Section 5

centre

radius

Section 5

231 jz

Question:

Section 5

zfjdzzz

Examples

Evaluate the following integrals:

dz(1) where C is the circle z 2

jz 0let

1)( zflet

f (z) is analytic in D and C encloses z0

1)( 0 zf

Section 5

zfjdzzz

12(2) where C is the circle zj1

We need a term in the form 1(z z0) so we rewrite the integral as:

First of all, note that 1(z21) hassingular points at zj.

The path C encloses one of these points, zj.We make this our point z0 in the formula

CC jzjz

))((12

Section 5

zfjdzzz

jz 0let

CC jzjz

))((12

Section 5

zfjdzzz

jz 0letjz

CC jzjz

))((12

Section 5

zfjdzzz

jz 0letjz

)(let2/)( 0 jzf

CC jzjz

))((12

Section 5

14(3) where C is the circle zj1C

1 1Here we have

CC jzjzzz

))()(1)(1(14

The path C encloses one of the four singular points, zj.We make this our point z0 in the formula

14 ))(1)(1(

jzzzzf

4)2)(1)(1(

1)()( 0

jjjjfzf

zfjdzzz

Section 5

Question:

Evaluate the integral

1where C is the circle z 2

(i) Where is C ?

(ii) where are the singular point(s) ?

(ii) what’s z0 and what’s f (z) ? Is f (z) analytic on and inside C ?

(iii) Use the Cauchy Integral Formula.........

Section 5

tan2(4) where C is the circle z3/2

tanz is not analytic at /2, 3/2, , but thesepoints all lie outside the contour of integration

The path C encloses two singular points, z1.To be able to use Cauchy’s Integral Formula we mustonly have one singular point z0 inside C.

112/3 2/

Use Partial Fractions:

)1)(1(

)1()1(

2/1,2/11

Section 5

1tan)(

)1tan()(

785.9)1tan()1tan(2

Section 5

For example,

More complicated functions, having powers of zz0, can betreated using the following formula:

Note: when n0 we haveCauchy’s Integral Formula: 0

zfjdzzz

Generalisation of Cauchy’s Integral Formula

C zzC dz

zzdjdz

f analytic on andinside C, z0 inside C

This formula is also called the “formula for the derivatives of an analytic function”

Section 5

zfjdzzz

Example

where C is the circle z 2

00 zlet

zezf )(let

f (z) is analytic in D, and C encloses z0

Section 5

zfjdzzz

Example

00 zlet

zezf )(let

Section 5

zfjdzzz

Example

00 zlet

zezf )(let

Section 5

Another Example

jz 0let

2)( zzf let

Section 5Summary of what we can Integrate

dzzf )( with f (z) analytic inside and on C - equals 0(1)

zf )(with f (z) analytic inside and on C, except at zzo - equals

( Cauchy’s Integral Theorem )

( Cauchy’s Integral Formula )

zf )(with f (z) analytic inside and on C, except at zzo

( The Formula for Derivatives )

)(2 0zfj

Section 5

What can’t we Integrate ?

(singularities at 2 inside C)

zdzze / where C is the unit circle

(singularity at 0 inside C)

Functions we can’t put in the form of our formulas:

zdztan where C is e.g. 2z

Section 5Topics not Covered

(2) Proof of Cauchy’s Integral Formula

(3) Proof that the derivatives of all orders of an analytic function exist - and the derivation of the formulas for these derivatives

zfjdzzz

(use the MLinequality in the proof)

(1) Proof that the antiderivative of an analytic function exists

)()()( 01

zFzFdzzfz

where )(zfdz

(use Cauchy’s Integral Formula and the MLinequality in the proof)

Section 5(4) Morera’s Theorem (a “converse” of Cauchy’s Integral Theorem)

(5) Cauchy’s Inequality

“If f (z) is continuous in a simply connected domain D and if 0)( C

for every closed path in D, then f (z) is analytic in D”

)( 0zr

CMzf on)(

(proved using the formula for the derivatives of an analytic function and the MLinequality)

(6) Liouville’s Theorem“If an entire function f (z) is bounded in absolute value for all z, then f (z) must be a constant” - proved using Cauchy’s inequality

1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an...

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