Contour Integrals

Derivatives of functions w(t) (1) Let w( t ) = u ( t ) + i v( t ) be a complex - valued function of a real variable t, where u and v are real - valued functions of t.

dw Then = w′( t ) = u ′( t ) + i v′( t ) dt provided each of the derivatives u ′ & v′ exists at t

( 2) If z 0 is a complex constant, then d dw ( z0 w( t ) ) = z0 . dt dt

( 3)

( )

d z 0t z0 t e = z0 e . dt

4. Mean Value Theorem for derivatives is NOT true. Suppose that (i) w(t) = u(t) + i v(t), a ≤ t ≤ b. be continuous, i.e. u and v are

continuous on [ a, b] (ii ) w ′ (t) exists in a < t < b.

Then there may NOT exist any c in (a, b) such that w(b) - w(a) w ′(c) = b−a

Example : Let w(t) = e , 0 ≤ t ≤ 2π it

⇒ w ′(t) = i e

it

⇒ w ′(t) = 1 for all t ∈ [ 0,2π ] ⇒ w ′(t ) ≠ 0 for all t ∈ [ 0,2π ]

i2π

But w(2π ) - w(0) = e =0

−e

i. 0

⇒ MVT for derivative in NOT true in the complex plane.

Sec 37 : Definite Integrals of w(t) Let w(t) = u(t) + i v(t) be a complex - valued function of a real - variable t. u(t), v(t) : real - valued functions over a ≤ t ≤ b.

Then definite integral of w(t) over an integral a ≤ t ≤ b is defined as

∫

b

a

b

b

a

a

w(t ) dt = ∫ u(t ) dt + i ∫ v(t ) dt ,

where the individual integrals on the right exist.

b

b

a

a

⇒ Re ∫ w(t ) dt = ∫ Re w(t ) dt , b

b

a

a

Im ∫ w (t ) dt = ∫ Im w(t ) dt .

Example :

( ) 1 − i t ∫0 1

1

(

2

1

(

)

0

1

= ∫ 1 − t dt − i ∫ 2 t dt 0

2 = −i 3

2

)

dt = ∫ 1 − t − 2 i t dt 2

0

Property : Let w(t) be a complex - valued function integrable on [ a, b]. Then b

∫a

b

w(t ) dt ≤ ∫ w(t ) dt a

Sec38 : contours Definitions (1) Curve : A set of points z = (x, y) in the complex plane is said to be a curve C if x = x(t), y = y(t) are continuous functions of a real parameter t.

We write C : x = x(t), y = y(t) OR C : z(t) = x(t) + i y(t).

(2) Arc : The portion between any two points of a curve is called an arc of the curve, i.e. C : x(t) + i y(t), a ≤ t ≤ b is an arc.

For simplicity, we shall use the single term " curve" to denote the entire curve as well as an arc of the curve.

(3) Differentiable Curve : The curve C : z(t) = x(t) + i y(t) is said to be differentiable if x ′(t) & y′(t) exist & they are continuous in a ≤ t ≤ b, and we write z′(t) = x′(t) + i y′(t )

If z′(t) ≠ 0, then such a curve (arc) is said to be regular or smooth.

(4) Piecewise differentiable curve/ arc : The curve C is said to be piecewise differentiable if among its various representations, ∃ at least one representation, say C : z(t) = x(t) + i y(t), a ≤ t ≤ b,

such that [ a, b] can be divided into a finite no. of sub intervals [a, a1 ], [a1 , a2 ],...., [an−1 , b] and on each sub - interval z′(t) exists.

If in addition z′(t) ≠ 0 in any of the sub - intervals, then such a curve C is said to be piecewise smooth or piecewise regular.

Jordan arc / curve or simple curve : A curve may have points at which it intersects or touches itself. Such a point is called multiple point of the curve.

A curve having NO MULTIPLE POINTS is called a simple curve, i.e., a curve is said to be simple if it neither touches itself nor crosses itself,

i.e., the curve C : z(t) = x(t) + i y(t) is said to be simple if z(t1 ) ≠ z(t 2 ) whenever t1 ≠ t 2 .

If the curve C : z(t) = x(t) + i y(t), a ≤ t ≤ b is simple except for the fact that z(a) = z(b), then C is said to be a simple closed curve or a Jordan curve.

6. Length of a differentiable curve Let c : (z) = x(t) + i y(t), a ≤ t ≤ b be a differeniable curve (arc). ⇒ z' (t) = x ′(t) + i y′(t) and z′(t) = ( x′(t )) + ( y′( t ) ) 2

2

Then b

L = ∫ z' (t) dt a

is called the length of the curve C.

7. Contour : A Contour is a piecewise smooth arc, i.e. an arc consisting of finite number of smooth arcs joined end to end.

Sec 39 & 40

Contour Integral or line integral : Let z = z(t), a ≤ t ≤ b denotes a contour C extending from a point z1 = z ( a ) to a point z 2 = z ( b ).

Let f(z) be piecewise continuous on C, i.e. f(z(t)) is piecewise continuous on a ≤ t ≤ b.

Then we define the line integral or contour integral of f along C as follows :

∫

C

f ( z ) dz = ∫ f ( z (t ) ) z ′(t ) dt b

a

Properties :

(1) ∫ f ( z )dz ≤ ∫a

b

C

c : z (t ), a ≤ t ≤ b

f ( z (t )) z ′(t ) dt ,

( 2) If z 0 is a constant, then z f ( z ) dz = z f ( z ) dz 0 0 ∫ ∫

C

C

(3) ∫ [ f ( z ) + g ( z )]dz C

= ∫ f ( z ) dz + ∫ g ( z ) dz C

C

(4) If the contour C : z = z(t), a ≤ t ≤ b in extended from z1 to z 2 then − C is extended from z 2 to z1 i.e. − C : Z = Z (−t ),−b ≤ t ≤ −a And

∫ f ( z ) dz = − ∫ f ( z ) dz y

−C

C

C

z2

-C z1 O

x

(5) Let C = C1 U C 2 , where C : z = z(t); a ≤ t ≤ b

Y

c1

C1 : z = z (t ) , a ≤ t ≤ c

z2 c2

& C 2 : z = z (t ), c ≤ t ≤ b Then

∫

C

f ( z ) dz =

O

∫

C1

f ( z ) dz +

∫

C2

f ( z ) dz

X

Ex.1 Let f(z) = Re z, then evaluate

∫

f ( z ) dz , where

C

C : z (t ) = t + it , 0 ≤ t ≤ 1

∫

C

b

f ( z )d z = ∫ f ( z (t )) z (t )dt a

= ∫ Re z ( t ) . z (t ) dt 1

0

= ∫ t (1 + i ) dt

y

(1,1)

1

0

1+ i = 2

c

x

Ex.2 z +2 Let f ( z ) = & z iθ C : z = 2 e , π ≤ θ ≤ 2π . Then evaluate ∫ f ( z ) dz C

Soln : z = 2 e

iθ iθ

⇒ dz = 2 e idθ z = 2 ei θ

θ=π

θ = 2π C

∴I = ∫ f ( z ) dz C

iθ

2e + 2 iθ . 2 e . i d θ iθ 2e

2π

=∫

π

2π

= 2 i∫

π

(e

= 4 + 2πi

iθ

)

+ 1 dθ

Ex.3  1, y < 0 f(z) = 4y, y > 0

Let

& C is the arc from z = −1 −i to z =1 + i along the curve 3

y = x . Then evaluate

∫

C

f ( z ) dz.

B 1+i O

We have

∫

C

-1-i

f ( z ) dz =

∫

f ( z ) dz +

AO

∫

f ( z ) dz

OB

Along AO : 3

A

z = x + iy = x + ix , − 1 ≤ x ≤ 0

0

∴ ∫ f ( z )dz = ∫ 1(1 + i 3 x ) dx A0

2

−1

0 3

 3x  =  x + i  3   −1 = 1+ i

3

Along OB, z = x + i x , 0 ≤ x ≤ 1

∫

f ( z ) dz =

OB

( ∫ 1

0

1

2

)

4 y 1 + 3 i x dx

(

)

= ∫ 4 x 1 + 3ix dx 3

0

2

=1+ 2 i ∴∫ f ( z ) dz = 1 + i +1 + 2 i = 2 + 3 i C