**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

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.**