1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 353

Prove that the function defined by (1) maps the deleted complex

plane C - {O} onto the circle

l

w _ ac - bd I = I ad - be I

cc-dd cc-dd

provided that lei f= Id!. In the case icl = ldl, (1) maps the complex

plane, with the line cz + dz = 0 omitted, onto the straight line

(bd - ac)w + (bd - ac)w + lal^2 - lbl2 = O

*11. A nonsingular affine transformation of the z-plane onto thew-plane is
defined by a complex function f = u +iv, where

with real coefficients and D = ai b 2 - a 2 b1 f= 0, or, equivalently, by a

function of the form f(z) = Az + Bz + C, with A, B, C complex and

IAI f= IBI. Under such a transformation parallel lines are mapped into

parallel lines and circles into ellipses.

(a) Find the conditions that are to be imposed on the coefficients

in order that circles be transformed into circles. What do those

conditions imply for f?

(b) Let w = f(z) E '.D(A). Since

dw = f zdz + fzdz

at each point z E A the preceding equation defines a nonsingular

affine transformation of dz into dw, provided that lfzl f= lfzl or

Jf f= 0. Show that the circle ldzl = r maps into the ellipse

(v; + v~) du^2 - 2(u.,v., + uyvy) dudv + (u; + u;) dv^2 = r^2 J^2 (1)

( c) Let a and b (a 2:: b) be the semiaxes of the ellipse (1 ), and k = a/b.

Assuming that k f= 1, let () (0 ::::; () < 7r) be the inclination angle

of the major axis of the ellipse with respect to Ou. Prove that

equation (1) can be written in the form

where

a du^2 + 2/3 du dv + 'Y dv^2 = ab

a= k sin^2 () + ~ cos^2 (),

'Y = k cos^2 e + I sin^2 e,

f3 = (k -~) sin()cos()

a1-/3^2 = 1