1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2.2 • THE MAPPI NGS w = z" AND w = zk 67

We can use knowledge of the inverse mapping z = w^2 to get further insight

into how the mapping w = z~ acts on rectangles. If we let z = x + iy "! 0, then

z = w^2 = u? -v^2 + i2uv,

and we note that the point z = x + iy in the z plane is related to the point
w = u + iv = z~ in the w plane by the system of equations

x = u^2 -v^2 and y =2uv. (2-13)

• EXAMPLE 2.13 Show that the transformation w = f (z) = z~ usually maps
  vertical and horizontal lines onto portions of hyperbolas.
  Solution Let a > 0. Equations (2-13) map the right half-plane given by
  Re (z) > a (i.e., x > a) onto the region in the right half-plane satisfying u^2 - v^2 >
  a and lying to the right of the hyperbola u^2 - v^2 =a. If b > 0, Equations (2-13)
  map the upper half-plane Im (z) > b (i.e., y > b} onto the region in quadrant

I satisfying 2uv > b and lying above the hyperbola 2uv = b. This situation

is illustrated in Figure 2.15. We leave as an exercise the investigation of what

happens when a = 0 or b = 0.

v

W=.tl '

3

y

9

2

y = b 4 t= w2

x=a --

x u

• 3 3 6 9 2 3
  Figlire 2. 15 The mapping w = z~.

We can easily extend what we've done to integer powers greater than 2. We

begin by letting n be a positive integer, considering the function w = f ( z) = zn,

for z = reifJ i= 0, and then expressing it in the polar coordinate form

(2-14)

If we use polar coordinates w = pe•4> in the w plane, the mapping defined

by Equation (2-14) can be given by the system of equations

p = r" and </> = no.

The image of the ray r > 0, (} = a is the ray p > 0, </> = na, and the angles

at the origin are increased by the factor n. The functions cos nO and sin nO are