Dr. Nirav Vyas complexnumbers3.pdf

8/20/2019 Dr. Nirav Vyas complexnumbers3.pdf

1/39

Complex Function

N. B. Vyas

Department of Mathematics,Atmiya Institute of Tech. and Science,

Rajkot (Guj.)

N.B.V yas−Department of M athematics, AIT S − Rajkot (2)


2/39

Curves and Regions in Complex Plane

Distance between two complex numbers

The distance between two complex numbers z 1 and z 2 is given by|z 1 − z 2| or |z 2 − z 1|



3/39




Circles

A circle with centre z 0 = (x0, y0)C and radius pR+ isrepresented by |z − z 0| = p



4/39




Circles


Interior and exterior part of the circle |z − z 0| = pThe set {zC, pR+/|z − z 0| < p} indicates the interior part of the circle |z − z 0| = p



5/39




Circles


Interior and exterior part of the circle |z − z 0| = pThe set {zC, pR+/|z − z 0| < p} indicates the interior part of the circle |z − z 0| = p whereas {zC, pR+/|z − z 0| > p} indicatesexterior part of it.



6/39


Circular Disk

The open circular disk with centre z 0 and radius p is given byzC, pR+/|z − z 0| < p.


C R C P


7/39


Circular Disk

The open circular disk with centre z 0 and radius p is given byzC, pR+/|z − z 0| < p. The close circular disk is given by{zC, pR+/|z − z 0| ≤ p}


C R C P


8/39


Circular Disk


NeighbourhoodAn open neighbourhood of a point z 0 is a subset of C containing an open circular disk centered at z 0 .


C R C P


9/39


Circular Disk


NeighbourhoodAn open neighbourhood of a point z 0 is a subset of C containing an open circular disk centered at z 0 . MathematicallyN p(z 0) = {zC, pR+/|z − z 0| < p}


C R C P


10/39


Circular Disk


NeighbourhoodAn open neighbourhood of a point z 0 is a subset of C containing an open circular disk centered at z 0 . MathematicallyN p(z 0) = {zC, pR+/|z − z 0| < p}A punctured or deleted neighbourhood of a point z 0 containall the points of a neighbourhood of z 0, excepted z 0 itself.Mathematically {zC, pR+/0


11/39


Annulus

The region between two concentric circles with centre z 0 of radii

p1 and p2(> p1) can be represented by p1


12/39


Annulus




13/39


Annulus




14/39


Open Set

Let S be a subset of C . It is called an open set if for eachpoints z 0S , there exists an open circular disk centered at z 0which included in S .




15/39


Open Set


Closed SetA set S is called closed if its complement is open.




16/39


Open Set


Closed SetA set S is called closed if its complement is open.

Connected Set

A set A is said to be connected if any two points of A can be joined by finitely many line segments such that each point on theline segment is a point of A




17/39


Domain

An open connected set is called a domain.




18/39


Domain


Region

It is a domain with some of its boundary points.




19/39


Domain


Region


Closed region

It is a region together with the boundary points (all boundarypoints included).




20/39


Domain


Region


Closed region

It is a region together with the boundary points (all boundarypoints included).

Bounded region

A region is said to be bounded if it can be enclosed in a circleof finite radius.


Function of a Complex Variable


21/39


If z = x + iy and w = u + iw are two complex variables andif to each point z of region R there corresponds at least on

point w of a region R we say that w is a function of z andwe write w = f (z )




22/39

u c o o Co V r



If for each value of z in a region R of the z -plane therecorresponds a unique value for w then w is called singlevalued function.




23/39




E.g.: w = z 2 is a single valued function of z .




24/39





If for each value of z if more than one value of w exists thenw is called multi-valued function.




25/39






E.g.: w =√

Z




26/39






E.g.: w =√

Z

w = f (z ) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) areknown as real and imaginary parts of the function w.




27/39






E.g.: w =√

Z


E.g.: f (z ) = z 2 = (x + iy)2 = (x2 − y2) + i(2xy)




28/39






E.g.: w =√

Z


E.g.: f (z ) = z 2 = (x + iy)2 = (x2 − y2) + i(2xy)∴ u(x, y) = x2

−y2 and v(x, y) = 2xy

N.B.V yas−

Department of M athematics, AIT S−

Rajkot (14)

Limit and Continuity of f (z )


29/39

( )

A function w = f (z ) is said to have the limit l as z approaches a point z 0 if for given small positive number ε wecan find positive number δ such that for all z = z 0 in a disk|z − z 0| < δ we have |f (z ) − l| < ε

N.B.V yas−


Rajkot (15)



30/39

A function w = f (z ) is said to have the limit l as z approaches a point z 0 if for given small positive number ε wecan find positive number δ such that for all z = z 0 in a disk|z − z 0| < δ we have |f (z ) − l| < εSymbolically, we write lim

z→z0f (z ) = l

N.B.V yas−


Rajkot (15)



31/39


z→z0f (z ) = l

A function w = f (z ) = u(x, y) + iv(x, y) is said to becontinuous at z = z 0 if f (z 0) is defined andlimz→z0

f (z ) = f (z 0)

N.B.V yas−


Rajkot (15)



32/39


z→z0f (z ) = l


f (z ) = f (z 0)

In other words if w = f (z ) = u(x, y) + iv(x, y) is continuousat z = z 0 then u(x, y) and v(x, y) both are continuous at(x0, y0)

N.B.V yas−


Rajkot (15)



33/39


z→z0f (z ) = l


f (z ) = f (z 0)

In other words if w = f (z ) = u(x, y) + iv(x, y) is continuousat z = z 0 then u(x, y) and v(x, y) both are continuous at(x0, y0)

And conversely if u(x, y) and v(x, y) both are continuous at(x0, y0) then f (z ) is continuous at z = z 0.

N.B.V yas−


Rajkot (16)

Differentiation of f (z )


34/39

The derivative of a complex function w = f (z ) a point z 0 iswritten as f (z 0) and is defined bydw

dz = f (z 0) = lim

δz→0

f (z 0 + δz ) − f (z 0)δz

provided limit exists.

N.B.V yas−


Rajkot (17)

Differentiation of f (z )


35/39

The derivative of a complex function w = f (z ) a point z 0 iswritten as f (z 0) and is defined bydw

dz = f (z 0) = lim

δz→0

f (z 0 + δz ) − f (z 0)δz

provided limit exists.

Then f is said to be differentiable at z 0 if we write thechange δz = z − z 0 since z = z 0 + δz ∴ f (z 0) = lim

z→z0

f (z ) − f (z 0)z

−z 0

N.B.V yas−


Rajkot (18)

Analytic Functions


36/39

A single - valued complex function f (z ) is said to beanalytic at a point z 0 in the domain D of the z −plane, if f (z ) is differentiable at z 0 and at every point in someneighbourhood of z 0.

Point where function is not analytic (i.e. it is not singlevalued or not) are called singular points or singularities.From the definition of analytic function

1 To every point z of R, corresponds a definite value of f (z).2 f (z) is continuous function of z in the region R.

3 At every point of z in R, f (z) has a unique derivative.

N.B.V yas−


Rajkot (19)

Cauchy-Riemann Equation


37/39

f is analytic in domain D if and only if the first partialderivative of u and v satisfy the two equations

∂u∂x = ∂v∂y , ∂u∂y = −∂v∂x − − − − − (1)

The equation (1) are called C-R equations.

N.B.V yas−


Rajkot (20)

Example


38/39

Ex. Find domain of the following functions:

1 1

z 2

+ 1Sol. Here f (z ) =

1

z 2 + 1f (z ) is undefined if z = i and z = −i∴ Domain is a complex plane except z =

±i

2 arg

1z

Sol. Here f (z ) = arg

1

z

1

z = 1

x + iy = 1

x + iy xx

−iy

x − iy = x

−iy

x2 + y2

∴1

z is undefined for z = 0

Domain of arg 1z is a complex plane except z = 0.

N.B.V yas−


Rajkot (21)

Example


39/39

3

z

z + z̄

Sol. Here f (z ) = z

z + z̄ f (z ) is undefined if z + z̄ = 0

i.e. (x + iy) + (x − iy) = 0∴ 2x = 0

∴ x = 0

f (z ) is undefined if x = 0

Domain is complex plane except x = 0

N.B.V yas−


Rajkot (22)

Dr. Nirav Vyas complexnumbers3.pdf

Documents

Transcript of Dr. Nirav Vyas complexnumbers3.pdf