K.L. Vasundhara II Year I Sem M III

7/29/2019 K.L. Vasundhara II Year I Sem M III

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Prepared by K.L.Vasundhara

COMPLEX ANALYSIS

z=x+iy is a complex variable, where x and y are real variables.

Functions of a complex variable:

w=u(x, y) + iv (x,y) is a function of the complex variable z=x+iy.

i.e., w = f(z) = u(x,y) + iv (x,y)

Where u(x,y) is the real part and v(x,y) is the imaginary part of the complex

function f(z). In general we can write f (z) = u+iv.

Example: x2-y

2+ 2ixy = u(x,y) + iv (x,y) is a function of a complex variable z.

For, f(z) = x2-y

2+ 2ixy

z2

= (x+iy) (x+iy)

= x2

-y2

+ 2ixy

Therefore f(z) = z2


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Limit of a Function:

Let f(z) be a function defined in a set S and let z0 be a limit point of S. Then A

is said to be the limit of f(z) at z0 if for any > 0 there exists > 0 such thatf(z) - A < for all z in S other than z0 with 0z - z < .

It is denoted by 0Lt f(z) = A. z- z < z z

0

Continuity of a Function:

Let f(z) be a function defined in S and let z0 be a limit point. If the limit

of f(z) at z0 exists and if it is finite and is equal to f(z0).

0

z z0

Lim f(z) = ( )f z

Then f(z) is said to be continuous at z0.

Derivative of a complex function:

A function f(z) is said to be differentiable at a point z=z0 if the

0 0

( ) ( )

0

f z z f zLim

z

z

exists.

It is denoted by1

0( )f z


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i.e.,

1 0 00

( ) ( )( )

0

f z z f zf z Lim

z

z

We can say a function f(z) is said to be differentiable at a fixed point z if the

limit

0 0( ) ( )

0

f z z f zLim

z

z

exists.

Analytic Function:

A function defined at a point z0 is said to be analytic at z0, if it has a derivative

at z0 and at every point in some neighborhood of z0.

It is said to be analytic in a region R, if it is analytic at every point of R.

The necessary condition for f(z) to be analytic:

CauchyRiemann Equations

The necessary conditions for a complex function f(z) = u(x,y) + iv(x,y) to be

analytic are i.e.,

x y x y

u v v uandx y x y

u v and v u


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Sufficient Condition for f(z) to be analytic

The function f(z) = u(x,y) + iv(x,y) to be analytic in a domain D if

(i) u(x,y) and v(x,y) are differentiable ux,uy, vx and vy are allcontinuous in D.

(ii) The partial derivatives ux, uy, vx and vy are all continuous in D.

Polar Form of CauchyRiemann Equations:

1

1

u v

r r

v u

r r

Example 1. Show that the function f(z) = |z|2

is differentiable only at origin.

Sol. Given f(z) = |z|2 (1)

Since z=x+iy , we have


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|z|2

= x2

+ y2 ..(2)

Substituting (2) in (1), we get

f(z) = x2

+ y2

u+iv = x2

+ y2

( Since f(z) = u+iv )

i.e., u = x2

+ y2

v = 0

2 0

2 0

u vx

x x

u vy

y y

If f(z) is differentiable, then

u v

x y

2x = 0 It implies x = 0

u v

y x


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2y = 0 It implies y = 0

Therefore, CR equations are satisfied only when x = 0, y = 0.

Hence the given function f(z) is differentiable only at the origin (0,0)

Example 2. Test whether the function is Analytical or not.

f(z) = ex

(cos y + i sin y).

Sol: Given f(z) = f(z) = ex

(cos y + i sin y)

= ex

cos y + i ex

sin y

Where u = ex

cos y v = ex

sin y

x xe cos y e sin yu v

andx x

x xe sin y e cos yu v

andy y

Here x y x yu v and v u

i.e., f(z) satisfies CR equations.

Hence the given function is analytic.


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Properties of Analytic Functions:

Property 1: Both the real part and imaginary parts of any analytic function

satisfies Laplaces equation.

Harmonic function: Any function which has continuous second order

partial derivatives and which satisfies Laplaces equation is called Harmonic

function.

Property 2: If w = u + iv is an analytic function, then the curves of the family

u(x,y) = c1 cut orthogonally the curves of the family v(x,y) = c2

Where c1 , c2 are constants.

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K.L. Vasundhara II Year I Sem M III