This document provides an overview of conformal mapping. It begins with an introduction to conformal mapping, which preserves angles under transformation. It then discusses different types of mappings, including linear mappings and complex functions. It defines conformal mapping as a transformation that preserves both the magnitude and direction of angles. It describes some elementary conformal transformations like translation, rotation, magnification, and inversion. It also discusses the important concept of bilinear transformations. Finally, it outlines some applications of conformal mapping in fields like complex analysis, numerical analysis, fluid flow, and scattering problems.

1. SUBMITTED BY
Name: Diganta Bhuyan
Roll No- 223203
Dipartment –Mathematics
PG 2nd semester

2.  OVERVIEW
 Introduction .
 Mapping.
 Conformal mapping.
 Elementary transformation of
conformal mapping .
 Bilinear transformation.
 Applications.
 Conclusion.

3.  INTRODUCTION
 In mathematics a Conformal mapping is function
which preserve angle. In 1569 , the flemish
cartographer Gerardus Mercator devised a cylindrical
map projection that preserve angle. Another map
projection known to the ancient Greek is the
stereographic projection and both example are
Conformal. Conformal map is very important in
complex analysis as well as physics and engineering.

4.  MAPPING
 Before we start conformal mapping we need to
understand about Mapping.
In linear it is a mathematical relation such that
each element of a given set is associated with an
element of another set .
A complex function w=f(z) can be regarded as a
mapping or transformation of the points in the z=x+iy
plane to the points of the w=u+iv plane.

5.  CONFORMAL MAPPING
If the transformation u=u(x,y) and v=v(x,y) the point (x₀,y₀) of xy-
plane is mapped into the point (u₀,v₀) of uv-plane and the
intersecting curve c₁ and c₂ are respectively maped into the curve
c₁ʹ and c₂ʹ and they intersect at (u₀,v₀).
if the transformation is such that the angle at (x₀,y₀)
between c₁ and c₂ is equal to the angle at (u₀,v₀)
between c₁ʹ and c₂ʹ is equal in magnitude as well as sense then the
transformation is called conformal maaping or conformal
transformation. In Conformal mapping the magnitude of angle
is same and the direction of angle is also same.
If the direction is opposite between the curve then it is called
Isogonal Transformation.

6.  ELEMENTARY TRANSFORMATION
 Translation: : w=z+α, α is complex number.
By this transformation every point in z-plane is
displaced in direction of α.
 Rotation :: w=ze^iα , α is real
By this figure in z-plane rotate through an angle
α in w-plane.
 Magnification:: w= αz, The figure in w-plane is
magnified α-times than the size of z-plane.
 Inversion:: w=1/z ,Reflect opposite of the point of the z-
plane.
Above transformation is important for
understanding some properties of Conformal
Mapping.

7.  BILINEAR TRANSFORMATION
 It is a Important topic in conformal mapping. It is a
combination of translation, rotation, magnification
and inversion.
The transformation of the form w=az+bz/cz+d is
called a Bilinear or Mobius Transformation.where
a,b,c,d are complex constant and ad-bc≠0

8.  APPLICATION
Conformal mapping is an important technique used
in complex analysis and has many applications I
different physical situation.
• Flows in complicated Geometries.
• Numerical Analysis and algorithm development
strategies.
• Flow of Ideal fluid.
• Conformal mapping can be used in scattering and
diffraction problems.

9.  CONCLUSION
Overall, Conformal map are an important tool in
mathematics and physics, and they have a wide range
of application in many different fields.