some possible implications

Solution

(i) The parametric representation z=e^iεx (0≤∉≤i) and since

( )

^{e d∉=iei∉}

d ^iQ /

i d

e i e dz z

I ^{i i}

c =− =−π

=

∫

^∧

∫

^{−∉ ∉ 0}

0

(ii) I=

∫

zdz=

∫

_∧^2∧e^−1Qie^i∉ dφ =πi

Example: Evaluate

∫

zdz^{from z=0to z =4+2i}along the curve Cgiven by (a) z=t² +it (b) the line from z=0and z=2iand then the line from z=2i to z=4+2i

Solution

(a) The given integral equal,

( )( ) ∫ ∫

∫

_c^{n−iy dn+idy =} _c^ndk+ydy+i _c^ndy−ydn

The parametric equations of Care n=t²,y=tfrom t =0to t=2 Then the line integral equal

( )

^t

^{( ) ( )( )}

^{tdt t dt i} _t

( )

^t

^{( ) ( )(}

^{dt t dt dt}

⁾

t + +

∫

− −

∫

²₌₀ ^{2 2 2}₌₀ ²

=

( ) ( )

3 10 8

2 ²

0 2 2

0

3 i

dt t i dt t

i + +

∫

− = −

∫

(b)

∫

_c

(

^x−iy

)(

^dx+idy

)

⁼

∫

_c^xdx+ydy+i

∫

_c^ndy−ydx

The line from Z =0to Z =2iis the same as

( )

^{0,0 to}

( )

^0,2 for which 0

,

0 =

= dn

x and the line integral equals.

( )( ) ∫ ( ) ( ) ∫

∫

_y⁴₌₀ ^{Q 0 +ydy+i} _y²₌₀ ^{0dy−y0 =} _y²₌₀^ydy=2.

y

-1 1^c

The line from z=2i to z =4+2i is the same as the line from

( )

^0,2

to

( )

^4,2 for which y=2,dy=0 and the line integral equals i i

i xdx xdx dn

n i

xdx x 8 8

8 2 2

0 0

2

4 0

4 0

4 0

4

0 − = −

+

=

−

• +

•

∫

+

∫

₌

∫ ∫

Then the requires value = ²+

(

⁸−⁸i

)

=¹⁰.

3.4 Cauchy-Goursat Theorem

Suppose that two real-valued function ^{P ,}

( )

^{n y and Q ,}

( )

^{n y} together with their partial derivatives of the first order, are continuous throughout a closed region Rconsisting of points interior to and on a simple closed contour C in the nyplane. By Green’s theorem, for line integrals,

(

^P

)

^dndy.

Qdy

cPdn R x y

∫

^{+ =}

∫∫

^{φ −}

Consider a function

( ) ( )

^{z u x y i v}

( )

^{x y}

f = , + ,

Which is analytic throughout such a region Rin the ny, or Z, plane, the line integral of f along Ccan be written

( ) ∫ ∫

∫

_c^{f zdz=} _c^ndn−vdy+i _c^vdx+udy………(1) Since f is its inR, the functions uand v are also its theorem i and if the derivative f¹ of f is its in R, so are the first order partial derivatives of u and v. By Green’s theorem, (1) could be written as

( )

^zdz

(

^{v u}

)

^{dndy i}

(

^{u v}

)

^dndy

f R x y R x y

c

∫∫ ∫∫

∫

^{= − − + −} ………. (2)

But in view of the Cauchy-Goursat equations

x y y

x V U V

U = , =−

The integrals of these two double integral are zero throughoutR. So Theorem: If f is analytic in Rand f¹ is continuous then,

∫

_c^f

( )

^zdz=0.

This is known as Cauchy theorem.

Goursat proved that the condition of continuity of f¹in the above Cauchy theorem can be omitted.

Theorem: (Cauchy-Goursat theorem)

If a function f is analytic at all points interior to and in a simple closed contour C, then

( )

=⁰

∫

_c^{f zdz .}

Cauchy-Goursat theorem can also be modified for the ……Bof a multiply connected domain.

Theorem: Let Cbe a simple closed contour and let Cj (j =1,2,....n)be a finite number of simple closed contours inside Csuch that the regions interior to each Cj have no points in common. Let Rbe the closed region consisting of all points within and on Cexcept for points interior to each Cj. Let Band all the contours oriented boundary of Rconsisting of C and all the contoursCj, described in a direction such that the interior points of

Rlie to the left ofB. Then, if f is analytic throughoutR₁.

( )

=⁰

∫

_B ^{f zdz}

As a consequence of Cauchy’s theorem, we have the following

Theorem: If ^f

( )

^z is analytic in a simply-connected region R,then

∫

_a^{b f}

( )

^zdz

is independent of the path in Rjoining any two points a and b in R.

C2

C1

y C

x

Proof

Consider the figure below

By Cauchy’s theorem

( )

=⁰

∫

^{f zdz}

ADBCA

Or

∫

^f

( )

^zdz+

∫

^f

( )

^zdz = ⁰

BEA ADB

Hence

( )

^{zdz f}

( )

^{zdz f}

( )

^zdz

f

AEB BEA

ADB

∫

∫

∫

^{=− =}

Thus

( ) ∫ ( ) ∫ ( )

∫

_C ^{f zdz=} _C ^{f zdz=} _a^{b f z dz}

2 1

This yields the required result.

Example: If Cis the curve y=x³−3x²+4x−1joining the points (1, 1) and (2, 3), show that

(

^{z iz}

)

^dz

∫

c^{12 2 −4} is independent of the path joining (1, 1) and (2, 3) Solution:

A (1, 1) → B (2, 1) → C (2, 3)

Along A (1, 1) to B (2, 1), y = 1, dy = 0. So that z=x4i anddz=dx. Then

( ) ( )

{

^{x i i x i}

}

^{dx i}

x

30 20 4

4 12

2

1

2 − + = +

∫

−

y A a

C1

D E

x C2

B b

Along B (2, 1) to C (2, 3), x=2, dx=0 so that z=2+iy anddz=idy. Then

( ) ( )

{

^{iy i iy}

}

^{idy i}

y

8 176 2

4 2

12

3

1

2 − + =− +

∫

+

=

=1 y So that

i i

i dz

z

c(12z² −41 ) = 20+30 −176+8 =−156+38

∫

The given integral equals

(

^{z iz}

) (

^{dz z iz}

)

_i ^{i i}

i

i 12 4 4 2 ^{2 3} 156 38

1 3

2 1

2 3

2 − = −

∫

=− +

∫

₊⁺ ₊⁺

Morera’s Theorem

Let ^f

( )

^z be continuous in a simply connected region Rand suppose that

( )

⁰

1 =

∫

_c^{f zdz}

Around every simple closed curve C in R. Then ^f

( )

^z is analytic in R. This theorem is called the converse of Cauchy’s theorem and it can be extended to multiply-connected regions.

Indefinite Integrals (Anti-derivatives)

Let ^f

( )

^z be a function which is continuous throughout a domainD, and suppose that there is an analytic function Fsuch that ^F1

( ) ( )

^{z = f z} at each point inD. The function Fis said to be an anti derivative of f in the domainD.

Cauchy Integral Formula

Theorem: Let f be analytic everywhere within and in a simple closed contour Ctaken in the positive sense. If Z₀ is any point interior to C, then

( )

⁼

∫

_c

( )

₋

z Z

dz z f z i

f

0

0 2

1 π

This formula is called the Cauchy integral formula. It says that that if a function f is to be analytic within and on a simple closed contourC, then the values of f interior to Care completely determined by the values of

f in C.

When the Cauchy integral formula is written as

( ) ( )

0

0

2 i f z z

Z dz a f

c = π

∫

− ………..(4)

It can be used to evaluate certain integrals along.

Simple closed contours

Example: Let C be the positively oriented circle z = since the function

( )

^{z 2 /}

(

^{9 z2}

)

f = − is analytic within and in Cand the point Z0 =−iis interior to C, then by Cauchy Integral formula

( ) ^{( )} ( )

( )

δ

π =π



 



−

− =

−

= − +

−

∫

∫

/9 2 10

9

2 2

c i i

z z z i z z

zdz

c c

Proof

Since f is its at Z₀, there corresponds to any positive number ε , however small, a positive number δ such that

( ) ( )

z − f z0 <ε

f whenever z−z0 =ρ………(1)

Observe that the function ^{( )}₍z z₀₎ z

f − is analytic at all points within and in Cexcept at the pointz₀. Hence, by Cauchy-Goursat theorem for multiply connected domain, it’s integral around the oriented boundary of the region between Cand C₀ has value zero.

( ) ( )

₀

0 0

0

− =

− −

∫

∫

_{c C}

z Z

dz z f z

Z dz z f

That is

( ) ∫ ( )

∫

₋ ⁼ ₋

0 0

0 C

c Z Z

dz z f Z

Z dz z f

y

1 ε

z0

p ε c

This allows us to write

( ) ( ) ( ) ( )

Z dz Z

z f z f Z

Z z dz Z f

Z dz z f

C C

C

∫ ∫

∫

₋ ⁻ ₋ ⁼ ₋⁻

0

0 0

0 0

0 0

………(2)

Z i Z

dz

C 2π

0 0

− =

∫

And so equation (Z) becomes

( ) ( ) ( ) ( )

Z dz Z

z f z z f

f Z i

Z dz z f

C

C

∫

∫

₋ ^{− =} ₋⁻

0 0

0 0

0

2π ……….(3)

By (1) and noting that the length of C₀ is 2∧ρ, by properties of integrals

( ) ( )

_{πρ πε}

ρ

ε 2 2

0 0

0 < =

−

∫

_C ^{f z}_Z−_Z^{f z dz} In view of (3) then

( )

^2π

( )

0 ^2πε 0

<

− −

∫

_C ^{i f z}

Z Z

dz z

f .

Since the left hand side of this inequality is a non negative constant which is less than an arbitrary small positive number, it must be equal to zero.

Hence, equation for it valid and the theorem is proved.

Cauchy’s integral formula can also be extended to a multiply connected region. With the understanding that

( )

z

f v denotes ^f

( )

^z and that 0! = 1, we can use mathematical induction to verify that

( )

( ) ( )

( ) (

^,1,2.

)

2

!

1 0

0 n v

Z Z

dz z f i z n

f C n

n =

= π

∫

− +

When n = 0, this is just the Cauchy integral formula stated earlier.

Example: Find the value of

(

^{ZSin Z}

)

^dz

∫

C _∧

− 6 ³ 6

Where Cis a circle z =1

Solution:

( ) ( )

1 6 6 2 3

6 6

2

2 ^π

π

πi f Sin Z dz

z Sin

C =

∫

−

=

[

^{5 4 6 2 6 6 6}

]

2 2

6 π _{π π π}

Sin Cos

i Sin

x −

= 21πi 16

Other Important Theorems 1. Cauchy’s inequality:

If ^f

( )

^z is analytic inside and on a circle Cof radius r and centre at ,

a

z≠ then ^{( )}

( )

≤ Μ• ^! n=^{0,1, 2,.....}

r a n

f ⁿ _n

Where M is a constant such that ^f

( )

^z <^M on C, i.e. M is an upper bound of ^f

( )

^{z on C.}

2. Lowville’s Theorem:

Suppose that for all Z in the entire complex plane, (i) ^f

( )

^{z is}

analytic and (ii) ^f

( )

^z is bounded, i.e. ^f

( )

^z <^Mfor some constant M, then ^f

( )

^z must be a constant

3. Fundamental Theorem of Algebra:

Every polynomial P

( )

z =a0 +a1z+a2z² + anZⁿ =⁰with degree ,

≥1

π and a_n ≠0 has at least one root.

4. Maximum Modulus Theorem:

If ^f

( )

^z is analytic inside and on a simple closed curve Cand is not identically equal to a constant, then the maximum values of ^f

( )

^z

occurs on C.

SELF ASSESSMENT EXERCISES

1. Evaluate

∫

_{( )}^{( )}₀²_,1^,5

(

^3x+y

)

^dx+

(

^xy−x

)

^{dy along}

(a) the curve y=x² +1

(b) the straight line joining (0, 1) and (2, 5)

(c) the straight line from (0, 1) to (0, 5) and then (0, 5) to (2, 5)

2. Evaluate

∫

_C

(

^{x2 −iy2}

)

^dz

(a) along the parabola……y=4n² from (1,4) to (2, 16)

(b) straight line from (1, 1) to (1, 8) and then from (1, 8) to (2, 8).

3. Evaluate

∫

₋²₂⁻₊ⁱ_i

(

^3xy+iy2

)

^dz

(a) along the curve x=2t−2i−y=1+t−t²

(b) along the straight line joining x=−2+i and z=2−i 4. Evaluate

(a)

∫

^CSin

(

^ZZ₋ ⁺

)(

^ZCos₋

)

^{Z dz}

2 1

2

2 π

π , where Cis the circle Z =3.

(b) ^C

(

^Ze

)

^dz

∫

₊z 4 2

1 where Cis the circle Z =3

5. Evaluate dz

Z z Sin

∫

C ₊ 2

3

π if Cis the circle Z =5

4.0 CONCLUSION

The materials in this unit must be learnt properly because they will keep on re occurring as progress in the study of mathematics at higher level.

5.0 SUMMARY

We recap what we have learnt in this unit as follows:

You learnt about Cauchy-Goursat equations, Moreras Theorem and applied it to indefinite integrals. We also consider Cauchy integral Formula

We considered some solved examples to illustrate the theory we have learnt in this unit. You may which to answer the following tutor-marked assignment

In document Design Discourse: Composing and Revising Programs in Professional and Technical Writing (Page 181-185)