by Dr. Shreyasi Jana - Narajole Raj College

(1)

by

Dr. Shreyasi Jana

Asst. Prof., Department of Mathematics Narajole Raj College

WB, India

(2)

Outline

1

Contour Integral

2

Cauchy’s Integral Theorem

3

Cauchy’s Integral Formula

4

Derivatives of analytic functions

5

Some Useful Theorems

6

Problems

7

Exercise

Unit IV(C13T, SEM-VI), Cauchy’s theorem, Cauchy integral formula, derivatives of analytic functions by Dr. Shreyasi Jana

(3)

Contour

Definition

Asimple closed pathis a closed path that does not intersect or touch itself. For example, a circle is simple, but a curve is not simple.

Contour

A simple closed path is sometimes called acontourand an integral over such a path a contour integral.

Simply connected domain

A simply connected domain D in the complex plane is a domain such that every simple closed path in D encloses only points of D. Examples: The interior of a circle. A domain that is not simply connected is called multiply connected.

(4)

Simply connected and Multiply connected domain

Simply Connected

Multiply Connected

(5)

Cauchy’s Integral Theorem

If is analytic in a simply connected domain D, then for every simple closed path C in D I

C

f(z)dz= 0.

Note:

This fundamental theorem, often called Cauchys integral theorem or simply Cauchys theorem, is valid for both simply- and multiply-connected regions.

(6)

Cauchy’s Integral Theorem

(7)

Examples

Example 1. H

Ce^zdz= 0.

2. H

Ccoszdz= 0..

3. H

Csinzdz= 0.

4. H

Czⁿdz= 0,for n= 0, 1, 2, ....

for any closed path C, since these functions are entire.

(8)

Some Consequences of Cauchy’s Theorem

Let f(z) be analytic in a simply-connected region R. Then the following theorems hold Theorem

Suppose a and z are any two points in R. Then

Z z a

f(z)dz is independent of the path in R joining a and z.

Theorem

Suppose a and z are any two points in R and

G(z) = Zz

a

f(z)dz.

Then G(z) is analytic in R and G⁰(z) =f(z).

(9)

Some Consequences of Cauchy’s Theorem

Theorem

Suppose a and z are any two points in R and F⁰(z) =f(z).Then Z b

a

f(z)dz=F(b)−F(a).

is independent of the path in R joining a and z.

Theorem

Let f(z) be analytic in a region bounded by two simple closed curves C and C1[where C1lies inside C as in Fig (a) in the next page and on these curves. Then

I

C

f(z)dz= I

C₁

f(z)dz,

where C and C1are both traversed in the positive sense relative to their interiors (counterclockwise).

(10)

Some Consequences of Cauchy’s Theorem

Y

C

(a)

X C1

(11)

Some Consequences of Cauchy’s Theorem

Theorem

Let f(z) be analytic in a region bounded by two simple closed curves C, C1,C1,C2,..., Cn,[where C1,C2,..., Cn,lies inside C as in Fig (b) in the next page and on these curves. Then

I

C

f(z)dz= I

C1

f(z) + I

C2

f(z)dz+....+ I

Cn

f(z).

This is a generalization of previous theorem.

(12)

Some Consequences of Cauchy’s Theorem

Y

C

(b)

X C2

C1 Cn

(13)

Cauchy’s Integral Formula

Cauchy’s integral theorem leads to Cauchy’s integral formula. This formula is useful for evaluating integrals.

Cauchy’s Integral Formula

Let f(z) be analytic in a simply connected domain D. Then for any pointz0in D and any simple closed path C in D that enclosesz0

I

C

f(z) z−z0

dz= 2πif(z0),

the integration being taken counterclockwise. Alternatively

f(z0) = 1 2πi

I

C

f(z) z−z0

dz.

(14)

Cauchy’s Integral Formula

zo

C D

Cauchy's Integral Formula

(15)

Example

Example 1. EvaluateH

C z³−6

2z−idzwhere C is the unit circle.

Solution: Here

I

C

z³−6 2z−idz=

I

C

z³/2−3 z−i/2 dz.

Sof(z) =z³/2−3 is analytic in C.

Alsoz0=i/2 lies inside C.

Therefore by Cauchy’s integral formula I

C

z³−6

2z−idz= 2πi[z³/2−3]z=i/2=π/8−6πi.

(16)

Derivatives of analytic functions

Complex analytic functions have derivatives of all orders. This differs completely from real calculus. Even if a real function is once differentiable we cannot conclude that it is twice differentiable nor that any of its higher derivatives exist.

Derivatives of analytic functions

If f(z) is analytic in a domain D, then it has derivatives of all orders in D, which are then also analytic functions in D. The values of these derivatives at a pointz0in D are given by the formulas

f⁰(z0) = 1 2πi

I

C

f(z)

(z−z0)²dz (1)

f⁰⁰(z0) = 2!

2πi I

C

f(z)

(z−z0)³dz (2)

and in general

f⁽ⁿ⁾(z0) = n!

2πi I

C

f(z)

(z−z0)ⁿ⁺¹dz n= 1,2,3, ... (3) here C is any simple closed path in D that enclosesz0and whose full interior belongs to D; and we integrate counterclockwise around C.

(17)

Liouville’s Theorem

If an entire function is bounded in absolute value in the whole complex plane, then this function must be a constant.

Note:

As a consequence it is easy to see that sinz,coszare not bounded as being entire functions if they are bounded then they must be constant. But for real variables we know that sinx,cosxboth are bounded.

(18)

Morera’s Theorem

Let f(z) be continuous in a simply-connected region R and suppose that I

C

f(z)dz= 0.

around every simple closed curve C in R. Then f(z) is analytic in R.

(19)

Maximum and Minimum modulus theorem

Maximum modulus theorem

Suppose f(z) is analytic inside and on a simple closed curve C and is not identically equal to a constant. Then the maximum value of|f(z)|occurs on C.

Minimum modulus theorem

Suppose f(z) is analytic inside and on a simple closed curve C andf(z)6= 0 inside C.

Then|f(z)|assumes its minimum value on C.

(20)

Rouche’s theorem

Suppose f(z) and g(z) are analytic inside and on a simple closed curve C and suppose

|g(z)|<|f(z)|on C. Thenf(z) +g(z) and f(z) have the same number of zeros inside C.

The fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

(21)

Example

Example Evaluate 1. H

C

sinπz²+cosπz² (z−1)(z−2) dz 2. H

C e^2z (z+1)⁴dz

where C is the circle|z|= 3.

Solution: 1. Since _{(z−1)(z−2)}¹ =_(z−2)¹ −_(z−1)¹ ,we have I

C

sinπz²+ cosπz² (z−1)(z−2) dz=

I

C

sinπz²+ cosπz² (z−2) dz−

I

C

sinπz²+ cosπz² (z−1) dz.

Since sinπz²+ cosπz²is analytic in C. Alsoz0= 2,1 lies inside C.

(22)

Example continued

Example

Therefore by Cauchy’s integral formula H

C

sinπz²+cosπz²

(z−2) dz= 2πi[sinπ2²+ cosπ2²] = 2πi.

andH

C

sinπz²+cosπz²

(z−1) dz= 2πi[sinπ1²+ cosπ1²] =−2πi.

Therefore the required integral has the value 2πi−(−2πi) = 4πi. Solution: 2. Heref(z) =e^z is analytic in C.

Alsoz0=−1 lies inside C.

Therefore putting n =3 in Cauchy’s integral formula we get I

C

e^2z

(z+ 1)⁴dz=2πi

3! [8e^2z]z=−1= 8πie⁻²/3.

(23)

Exercise

Evaluate 1. H

C sin 3z

(z+^π₂dz,where C is the circle|z|= 5.

2. H

C e^iz

z³dz,where C is the circle|z|= 2.

3. H

C z²

z²+4dz,where C is the square with vertices at±2,±2 + 4i. 4. H

C sin⁶z (z−^π

6)dz,where C is the circle|z|= 1.

5. H

C e^z

(z²+π²)²dz,where C is the circle|z|= 4.

(24)

by Dr. Shreyasi Jana - Narajole Raj College

by

Dr. Shreyasi Jana

Asst. Prof., Department of Mathematics Narajole Raj College

WB, India

Outline

Contour Integral

Cauchy’s Integral Theorem

Cauchy’s Integral Formula

Derivatives of analytic functions

Some Useful Theorems

Problems

Exercise

Contour

Simply connected and Multiply connected domain

Cauchy’s Integral Theorem

Cauchy’s Integral Theorem

Examples

Some Consequences of Cauchy’s Theorem

Some Consequences of Cauchy’s Theorem

Some Consequences of Cauchy’s Theorem

Some Consequences of Cauchy’s Theorem

Some Consequences of Cauchy’s Theorem

Cauchy’s Integral Formula

Cauchy’s Integral Formula

Example

Derivatives of analytic functions

Liouville’s Theorem

Morera’s Theorem

Maximum and Minimum modulus theorem

Rouche’s theorem

Example

Example continued

Exercise

Thank You