**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

- Maps of the form z → 1 / z