1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

160 CHAPTER 5 • ELEMENTARY FUNCTIONS

v

w = exp(z)

y

x

"

The zplane. Thew plane.

Figure 5.3 The image of R under the transformation w =exp (z).

•EXAMPLE 5.2 Consider a rectangle R = {(x,y): a$ x $ band c $ y $ d},

where - n < c < d $ n. Show that the transformation w = e• = e"'+iy maps R

onto a portion of an annular region bounded by two rays.

Solution The image points in the w plane satisfy the following relationships

involving t he modulus and argument of w:

ea = lea+i11I $ le"+illl $ leb+iyl = eb, and

c = Arg ( ex+ic) $ Arg ( ex+iy) $ Arg ( ex+id) $ d,

which is a portion of the annulus {pei¢: e" $ p $ eb} in thew plane subtended

by the rays </> = c and ¢ = d. In Figure 5.3, we show the image of the rectangle

{

R= (x,y): -1 $ x $1 and -7[ · }

4

$ y $

3

.

-------~EXERCISES FOR SECTION 5.1

1. Using Definition 5.1, explain why exp (0) = e^0 = 1.


2. The questions for this problem relate to Figure 5.2. The shaded portion in the w
   plane indicates the image of the shaded portion in the z plane, with t he lighter
   shading indicating expansion of the area of corresponding regions.

(a) Why is there no shading inside the circle si?

(b) Explain why the images of r1, r2, and r3 appear to make, respectively,

angles of -^7 ; , i, and^3 ; radians with the positive u--axis.