Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

y

θ 2 θ 1

C 1

C 2

z 1

z 2

z 0

w=g(z)

s

r

C 1 ′

C′ 2

φ 2 φ 1

w 0

w 1

w 2

x

Figure 24.3 Two curvesC 1 andC 2 in thez-plane, which are mapped onto

C 1 ′andC′ 2 in thew-plane.

important properties are that, except at points at whichg′(z), and henceh′(z), is

zero or infinite:

(i) continuous lines in thez-plane transform into continuous lines in the

w-plane;

(ii) the angle between two intersecting curves in thez-plane equals the angle

between the corresponding curves in thew-plane;

(iii) the magnification, as between thez-plane and thew-plane, of a small line

element in the neighbourhood of any particular point is independent of

the direction of the element;

(iv) any analytic function ofztransforms to an analytic function ofwand

vice versa.

Result (i) is immediate, and results (ii) and (iii) can be justified by the following

argument. Let two curvesC 1 andC 2 pass through the pointz 0 in thez-plane

and letz 1 andz 2 be two points on their respective tangents atz 0 , each a distance

ρfromz 0. The same prescription withwreplacingzdescribes the transformed

situation; however, the transformed tangents may not be straight lines and the

distances ofw 1 andw 2 fromw 0 have not yet been shown to be equal. This

situation is illustrated in figure 24.3.

In thez-planez 1 andz 2 are given by

z 1 −z 0 =ρexpiθ 1 and z 2 −z 0 =ρexpiθ 2.

The corresponding descriptions in thew-plane are

w 1 −w 0 =ρ 1 expiφ 1 and w 2 −w 0 =ρ 2 expiφ 2.

The anglesθiandφiare clear from figure 24.3. The transformed anglesφiare

those made with ther-axis by the tangents to the transformed curves at their