IndexSerial no. Contents

(1) Introduction

(2) Definition

(3) Theorem

(4) Implicit Formula

(5) Laplace Operator in Polar Coordinates

(6) Two-dimensional problems complex Analytical Function

(7) Conclusion

INTRODUCTIONIn mathematics, a conformal map is a function which

preserves cangels. In the most common case the function

is between domains in the complex plane. More formally,

a map

f : U V is called conformal (or angel preserving) at u0 if

it preserves oriented angles between curves through u0

with respect to their orientation. Conformal mappings is

very

important in Complex Analysis, as well as in many areas

of physics and engineering. In this project, I study about

the some definitions and information about conformal

mapping and i am

going to explain with some examples.

DEFINATION -1

The mapping f ( z )=f ( x+ iy)=x , y )+iv ( x , y )is called conformal if

1. It is one to one. (Bijection)

2. It and its inverse continuously differentiable.

3. Image of every smooth curve of G1 is a smooth curve

of G2.

4. Angle between any two curves equals angle between

their images.

DEFINATION -2

A Bilinear Transformation is a mapping of the form

T ( z )=az+bcz+d

where a, b, c, d are fixed complex numbers and ad−bc≠0because otherwise T would be a constant. T is also called

a

mobius transformation, or fractional linear

transformation.

Theorem

Let f : A → B be analytic and let

f '( z ο)≠0 for each z 0 ∈ A .

Then f is conformal.

Implicit Formula

Theorem

There exist a unique bilinear transformation that maps

three

disctinct points, z1, z2 and z3, onto three distinct points,

w1,

w2 and w3, respectively. An implicit formula for the

mapping is

given by Mustafa.

Definition

The function w=f=exp z is a one to one mapping

of the fundamental period strip −π< y≤π in the z plane onto the w plane with the point w=0 deleted.

Because f'( z )≠0 , the mapping w=exp z is a

conformal mapping at each point z in the complex plane.

The family of horizontal lines y=c for −π<c<π and the segments x=a for −π< y<π from an orthogonal grid in the fundamental period strip. Their images under the

mapping w=exp z are the rays p>0 and

φ=c and the circles |w|=ea , respectively. The images from an orthogonal curvilinear grid in the w plane, as shown in the shape.

DefinitionThe trigonometric functions can be expressed with

compositions that involve the exponential function

followed by a bilinear function. We can find images of

certain regions by following the shapes of successive

images in the composite mapping.

w= f=exp z−π< y≤πw=0f '( z )≠0w=exp zy=c−π<c<πx=a−π< y<πw=exp zp>0φ=c|w|=ea

Laplace Operator In Polar Coordinates

Φ (φ)=Cncosnφ+Dn sinnφ n=0,1,2 , .. ..

V=R (r )Φ(φ)

1r

∂∂ r

(r ∂V∂r

)+ 1r2

∂2V∂φ2 =0

For Two-dimensional Problems Complex Analytical Function

Are a Powerful Tool Of Much Elegance.

ConclusionIn this project, we have studied the bilinear

transformations, mappings involving elementary

functions, mapping by trigonometric functions. Before

giving details of bilinear

eiyxz

xiyMaps (x,y) plane onto (u,v) plane.For analytical functions the derivative exists.Examples:

zezzz znn ln,,sin,, /1

transformation, we've shown basic properties and defined

conformal mapping theorem and riemann mapping

theorem. Then, we have given the definition of implicit

formula. Then we

have shown mappings involving elemantary functions

and mappings of trigonometric functions with some

examples. Moreover, we have shown on figures how

functions are mapped.