Conformal mappings, w = z+k, w = kz,

CONFORMAL MAPPINGS, w = z+k, w = kz,

W= 1/z  w = z²

Introduction :

Consider the transformation w = f (z), where f (z) is a single valued function of z, under this transformation, a point z₀ and any two curves C₁ and C₂ passing through z₀ in the z plane, will be mapped onto a point w₀ and two curves C₁' and C₂' in the w plane.

If the angle between C₁ and C₂ at z₀ is the same as the angle between C₁' and C₂' at wo, both in magnitude and sense, then the transformation W = f(z) is said to be conformal at the point z₀.

Conformal mapping [A.U D15/J16 R-08]

A transformation that preserves angles between every pair of curves through a point, both in magnitude and sense, is said to be conformal at that point.

Isogonal

A transformation under which angles between every pair of curves through a point are preserved in magnitude, but altered in sense is said to be an isogonal at that point.

Note: 3.4 (i) A mapping w = f (z) is said to be conformal at

z = z₀, if f ' (z₀) ≠ 0.

Note:  (ii) The point, at which the mapping w = f (z) is not conformal, i.e., f'(z) = 0 is called a critical point of the mapping.

If the transformation w = f (z) is conformal at a point, the inverse transformation z = f^-1 (w) is also conformal at the corresponding point.

The critical points of z = f^-1 (w) are given by re given by dz/dw = 0 Hence the crirical points of the transformation w = f (z) are given by dw/dz = 0 and dz/dw = 0

Note : (iii) Fixed points of mapping.

Fixed or invariant point of a mapping w = f(z) are points that are mapped onto themselves, are "Kept fixed" under the mapping. Thus they are obtained from w = f (z) = z.

The identity mapping w = z has every point as a fixed point. The mapping w =  has infinitely many fixed points.

w = 1/z has two fixed points, a rotation has one and a translation has none in the complex plane.

PROBLEMS BASED ON CONFORMAL MAPPING

Example 3.4.1. When do you say w= f(z) is a conformal mapping.

Solution: A mapping w = f(z) is said to be conformal at  z = z₀, if f ' (z₀) ≠ 0

Example 3.4.2 Define an isogonal transformation.

Solution: See Page No. 3.65, 3.4 (b)

Example 3.4.3 What is an invariant point in mapping ?

Solution: See Page No. 3.66, Note 3.4 (iii)