Conformal Mapping

Geometric interpretation of a complex function.



If D is the domain of real-valued functions and u(x,y) and v(x,y) then the system of equations u = u(x,y) and v = v(x,y) describes a transformation (or mapping) from the x y - plane into the u v -plane, also called the w-plane.

Therefore, we consider the function w= f(z) = u(x,y) + i v (x,y)
to be a transformation (or mapping) from the set D in the z-plane onto the range R in the w-plane.

Conformal Mapping:

A function f: C → C is conformal at a point z₀ if and only if it is holomorphic and its derivative is everywhere non-zero on C.


i.e., if f is analytic at z₀ and f’(z₀) ≠ 0


Isogonal Mapping:


An isogonal mapping is a transformation w = f (z) that preserves the magnitudes of local angles, but not their orientation.


Standard Transformations:


• Translation

- Maps of the form z → z + k, where k є C

• Magnification and rotation

- Maps of the form z → k z , where k є C

•  Inversion

- Maps of the form z → 1 / z

Transformation w = a z

Let w = a z, where a ≠ 0


If a = │a│ e^(i α) and, z = │z │ e^(i θ), then


w = │a│ │z│ e^i(θ + α )


The image of z is obtained by rotating the vector z through the angle α and magnifying or contracting the length of z by the factor │a│.


Thus the transformation w = a z is referred to as a rotation or magnification.



Example 1:


Find the image of the region y > 1 under the map w = ( 1 – i ) z


Solution:


Let w = u + i v ; z = x + i y


Given w = ( 1 – i ) z

i.e., z = 1/2 ( 1 + i) w      [ since ( 1 – i ) ( 1 + i) = 2]


i.e., x + i y = 1/2 ( 1 + i) (u + i v )

i.e., x = (u- v )/2 ; y = (u+v)/2


Hence the region y >1 is mapped on the region u + v > 2 in w –plane.

Example 2 :

Determine the region R of the w plane into which the triangular region D enclosed by the lines
x = 0, y = 0, x + y = 3 is transformed under the transformation w = 2z.


Solution:
Let w = u +i v; z = x + i y

Given, w =2 z

i.e., u +i v = 2 (x + i y)

i.e., u = 2 x ; v = 2 y


When x = 0, u = 0

The line x = 0 is transformed into the line u = 0 in the w – plane.


When y = 0, v = 0

The line y = 0 is transformed into the line v = 0 in the w – plane.


When x + y = 3 , we get


u/2 + v/2 = 3


i.e., u + v = 6

The line x + y = 3 is transformed into the line u + v = 6 in the w – plane.