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Friday, October 14, 2011

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

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