1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

90 CHAPTER 2 • COMPLEX FUNCTIONS

i'

II

"

-·~ I

II

"

y

II

"

• ···· -b= I

..... ·~ .... b=~

........................... ~ .. b=-~

······,···············b=-1

w= l.

t

---

... ~=-! ..

: ·,

b = - 1 '•

·.

Figure 2.25 The images of horizontal and vertical lines under the reciprocal trans-

formation.

which is the equation of a circle in the w plane with center w 0 = .J,, and radius

I~ I· The point at infinity is mapped to ( u, v) = (0, 0).

Similarly, the horizontal line y = b is mapped onto the circle

221 1 2 1 1

( )

2 ( )2
u + v + bv + 4b2 = u + v + 2b = 2b ,

which has center wo = -~ and radius I~ I·

Figure 2.25 illustrates the images of several lines.

-------... EXERCISES FOR SECTION 2.5

For Exercises 1- 8, find the image of the given circle or line under the reciprocal

transformation w = ~.

1. T he horizontal line Im (z) = t.

2. The circle c! (-~) = {z: lz +~I= u.


3. The vertical line Re z = - 3.


4. The circle C 1 (-2) = {z : lz + 21 = 1}.


5. The line 2x + 2y = 1.

6. The circle C1 (t) = {z: lz -~I= 1}.

7. The circle C1 (~) = {z: lz-~I= l}.


8. The circle C2 ( - 1 + i) = { z : lz + 1 -i i = 2}.