1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

396 CHAPTER 10 • CONFORMAL MAPPING

Let C: z (t) = x (t) + iy (t), - 1 :St :S 1 d enote a smooth curve that passes

through the point z(O) = z 0. A vector T tangent to Cat the point zo is given

by

T = z' (O),

where the complex number z' (0) is expressed as a vector.

The angle of inclination of T with respect to the positive x-axis is

/3 = Argz' (0).

The image of C under the mapping w = f (z) is the curve K given by the formula

K: w(t) = u(x(t),y(t)) +iv(x(t),y(t)). We can use the chain rule to show

that a vector T• tangent to Kat the point wo = f (zo) is given by

T " = w' (0) = f' (zo) z' (0).

The angle of inclination of T• with respect to the positive u-axis is

'Y = Arg f' (zo) + Argz' (0) = a+ /3, (10-2)

where a = Arg f' ( z 0 ). Therefore, the effect of the transformation w = f ( z) is

to rotate the angle of inclination of the tangent vector T at zo through the angle

a = Arg f' (zo) to obtain the angle of inclinat ion of the tangent vector T * at

wo. This situation is illustrated in Figure 10.L

A mapping w = f (z) is said to be angle preserving, or conformal at zo, if it

preserves angles between oriented curves in magnitude as well as in orientation.

T heorem 10 .1 shows where a mapping by an analytic function is conformal

w =f(z)

~

v

-o-----------11

Figure 10.1 The tangents at the points Zo and wo, where J is an analytic function and

f I (zo) =/: 0.