1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

400 CHAPTER 10 • CONFORMAL MAPPING

v

y w=~

Z3 = i Z2 = I + i

x u

Zo = 0 z, = I

Figure 10.4 The mapping w = z^2.

From Equation (10-9), the distance lw 1 - wol between the images of the points
z1 and ZQ is given approximately by I f / (zo)l lz1 -zol· Therefore, we say that
the transformation w = f (z) changes small distances near z 0 by the scale factor

I f'(zo)I. For example, the scale factor of the transformation w = f(z) = z^2

near the point zo = 1 + i is I f' (1 + i)I = 12 (1 + i)I = 2/2.

We also need to say a few things about the inverse transformation z = g ( w)

of a conformal mapping w = f (z) near a point zo, where f' (zo) =/: O. A complete
justification of the following assertions relies on theorems studied in advanced

calculus.^1 We express the mapping w = f (z) in the coordinate form

u = u(x,y) and v = v(x,y). (10-10)

The mapping in Equations (10-10) represents a transformation from the xy

plane into the uv plane, and the Jacobian determinant, J (x, y), is d efined by

J (x ' y) =I u. 11. (x,11) (x,y) "v(x,y)vv(x,y) I • (10-11)

The transformation in Equations ( 10 - 10) has a local inverse, provided J (x, y) =/:

0. Expanding Equation ( 10 - 11) and using the Cauchy-Riemann equations, we
   obtain

J (xo, Yo) = u., (xo, Yo) vv (xo, Yo) - v,, (xo, Yo) Uy (xo, Yo)

= u; (xo,yo) +v~ (xo,yo) = I!' (zo)l

2

"=/: 0.

(10-12)

Consequently, Equations (10-11) and (10-12) imply that a local inverse z = g (w)

exists in a neighborhood of the point w 0. The derivative of g at w 0 is given by

(^1) See, for instance, R. Creighton Buck, Advanced Calculus, 3rd ed. (New York, McGraw-
Hill), pp. 358 - 361, 1978.