1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

r 10

mapping

Overview
The terminology "conformal mapping" should have a familiar sound. In 1569 the
Flemish cartographer Gerardus Mercator (1512-1594) devised a cylindrical map
projection that preserves angles. T he Mercator projection is still used today
for world maps. Another map projection known to the ancient Greeks is the
stereographic projection. It is also conformal (i.e., angle preserving), and we
introduced it in Chapter 2 when we defined the Riemann sphere. In complex
analysis a funct ion preserves angles if and only if it is analytic or anti-analytic
(i.e., the conjugate of an analytic function). A significant result, known as the
Riemann mapping theorem, states that any simply connected domain (other
than t he entire complex plane) can be mapped conformally onto t he unit disk.

10.1 Basic Properties of Conformal Mappings

Let f be an analytic function in the domain D and let z 0 be a point in D. If

f' (zo) =F 0, then we can express fin the form

f (z) = f (zo) + !' (zo)(z- zo) + 17(z)(z- zo), (10-1)

where 11 ( z) -+ 0 as z -+ zo. If z is near zo, then the transformation w = f ( z)

has the linear a pproximation

S (z) = A + B (z -zo) = B z + A - Bzo,

where A = f (zo) and B = f' (zo). Because 11 (z) -> 0 when z -+ zo, for points

near zo the transformation w = f ( z) has an effect much like the linear mapping

w = S ( z). The effect of the linear mapping S is a rotation of the plane through

the angle a = Arg/ ' ( zo), followed by a magnification by the factor I f ' ( zo) I,

followed by a rigid translation by t he vector A -B zo. Consequently, the mapping

w = S (z) preserves the angles at the point zo. We now show that the mapping

w = f (z) also preserves angles at zo.

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