1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2.2 • THE MAPPINGS w = z" AND w = zf. 65

y v

w=z'

x

"

Figure 2.12 T he mappings w = z^2 a nd z =wt.

Solution Using Equations (2-9), we determine that the vertical line x = a is

mapped onto the set of points given by the equations u = a^2 - y^2 and v = 2ay.

If a i= 0 , t hen y = 2 1:, and

(2-10)

Equation (2-10) represents a parabola with vertex at a^2 , oriented horizontally,

and opening to the left. If a > 0 , the set { (u, v) : u = a^2 - y^2 , v = 2ay} has

v > 0 precisely when y > 0, so the part of the line x = a lying above the x-axis

is mapped to the top half of the parabola.

The horizontal line y = bis mapped onto the parabola given by the equations

u = x^2 - b^2 and v = 2xb. If b i= 0, then as before we get

vz

u = -b 2 + 402· (2-11)

Equation (2-11) represents a parabola with vertex at - b^2 , oriented horizon-

tally and opening to the right. If b > 0, the part of the line y = b to the

right of the y-axis is mapped to the top half of t he parabola because the set

{ (u, v): u = x^2 - b^2 , v = 2bx} has v > 0 precisely when x > O.

Quadrant I is mapped onto quadrants I and II by w = z^2 , so the rectangle

0 < x < a, 0 < y < b is mapped onto the region bounded by the top halves of

the parabolas given by Equations (2-10) and (2- 11 ) and t he u-axis. The vertices

0, a , a + ib, and ib of the rectangle are mapped onto the four points 0 , a^2 ,

a^2 - b^2 + i2ab, and - b2, respectively, as indicated in Figure 2.13.

Finally, we can easily verify that the vertical line x = 0, y i= 0 is mapped

to the set { ( - y^2 , 0) : y i= 0}. This is simply the set of negative real numbers.

Similarly, the horizontal line y = 0, x i= 0 is mapped to the set { ( x^2 , 0) : x =f 0},

which is the set of positive real numb ers.