Name ID

JamAbdulSattar k11-2251 (c)Sadiq Shah K11-2395 (c)Nazim Khan K11-2410 (c)Akhtar Afzal k11-2441 (A)Muhammad Ishtiaq K11-2442 (c)

Group Members:

Conformal Mapping

Mapping: To give realistic view to the physical

world. Conformal Mapping: Advanced method of forming maps

preserving angles and to give the realistic view to the image or map.

INTRODUCTION

According to many historians,

“Mapping is older than “the written word “.

Mapping is used to give life to the physical world on a piece of paper.

And to prove this statement that mapping is very old science, a lot of examples are present in the history.

History of Mapping

A map of planned town is found on the wall of 9ft in Ankara, Turkey which is 6200 B.C old.

A map of South Africa is found which was made by the Chinese Great Ming Empire in 1389.

Different maps of the world were drawn by different travelers to help others in sea journey.

EXAMPLES

All these mappings were done to help improving the life style and travelling.

But at old times there were also the quest of

improving different sciences.

In 1999, the oldest known map of the moon is discovered in Ireland, which is 3000 B.C old.

There are different drawings found in France which are about “Summer triangle” changes in weather and are 14000 B.C old.

The history of Conformal mapping is traced back to 16th century.

G-Mercator was the mathematician who contributed the first in conformal mapping.

Origination: Its origination was from the rejection of idea that

earth is flat.

Conformal Mapping

Firstly it was thought that earth is flat but this idea was rejected because there is no Isometry (distance preserving),as one position and travelled distance can’t be exactly shown, it was the origination of Conformal mapping which supports the idea that earth is sphere by preserving angles.

Theorem 

If f: A → B is analytic and f '(z) ≠ 0 for all z in A, then f is conformal.

“If a function is harmonic over a particular space where it satisfies certain boundary conditions, and it is transformed via a conformal map to another space, the transformation is also harmonic and satisfies corresponding boundary conditions”.

uxx + uyy = 0 ( Laplace equation)

It is a geometric approach to a complex analysis.

Applicable at all the points where function is analytic except the critical points.

In conformal mapping we define another plane W which is also a complex plane and then convert Z-plane to W-plane to get conformal map.

General example of Conformal mapping.

Mapping of z-plane onto W-plane

Failure of conformality points W = Sin To find the points at which the conformality fails,

we take derivative and then find the critical points which are the points of failure of conformality

So taking derivative on both sides dw = d/dz(Sin ) dw= cos -----(1)   for finding the critical points we get dw= 0 put it in eq (1) cos = 0  So Z= 1/2 , 3/2,5/2 , 7/2,…… Z=(2n-1)/2 where n E N

We can find the inverse of complex number by using Conformal Mapping.

Procedure:.Let Z=2 as in general here π/4<α<π/2 Also r=2 Required: W=1/Z ? Solution:

β is the angle of conformal mapping,

and R Is the radius of the circle in which we draw conformal map.

putting the value of Z in equation which is to be determined.

we get

W=1/(2eiα) , W=0.5e-iα (1) π/4< α< π/2

also

by comparing equation (1) and (2) we get

R=0.5, β=-α to find limit of β putting value of α

if α=π/4 β=-π/4

if α=π/2 β=-π/2 thus -π/4<β<-π/2thus we can easily determine inverse of Z by using

conformal map.

Application of conformal mapping

(1)We can find the inverse of complex number with the help of conformal mapping.

(2)Integration can also be determined by conformal mapping.

(3)Laplace equations can also be solved with the help of conformal mapping.

(4)We can also determine the conduction of heat by it.

(5)Conformal mapping has also advantages in fluid dynamics.