70 CHAP'l'ER 2 • COMPLEX FUNCTIONS
- Find the image of the sector r > 0, -7' < () <^2 ; under the following mappings.
(a) w = z~.
(b) w = z~.
(c) w = z•. l- Use your knowledge of the principal square root function to explain the fallacy in
the following logic: 1 = y'(-1) (-1) = J(-ij/(-ij = (i) (i) = -1.
2 J. 1 iA.l'K 2 (w)
10. Show that the functions f (z) = z and g (w) = w• = lwl2 e with domains
given by Equations (2-6) and (2-7), respectively, satisfy Equations (2-3). Thus,
f and g are inverses of each other that map the shaded regions in Figure 2.14
one-to-one and onto each other.- Show what happens when a = 0 and b = 0 in Example 2.13
- Establish the result referred to in Definition 2.2.
2.3 Limits and Continuity
Let u = u (x, y) be a real-valued function of the two real variables x a.nd y.
Recall that u has the limit uo as (x, y) approaches (xo, Yo) provided the value of
u(x, y) can be made to get as close as we want to the value uo by taking (x, y)
to be sufficiently close to (xo, Yo). When this happens we write
lim u (x, y) = uo.
(:r:,y)-(:r:o,vo)In more technical language, u has the limit UiJ as (x, y) approaches (xo, Yo)
iff ju (x, y) -uol can be made arbitrarily small by making both Ix - xol and
IY - Yol small. This condition is like the definition of a limit for functions of one
variable. The point (x, y) is in the xy plane, and the distance between (x , y)
and (xo, Yo) is V<x -xo)^2 + (y - vo)2. With this perspective we can now give
a precise definition of a limit.Definition 2.3: limit of u (x, y)The expression Lim u(x, y) = uo means that for each number c > 0,
(x,y)-(:i:o,Yo)
t here is a corresponding number 8 > 0 such that1 u(x, y)- uo 1 < c whenever o < v<x -xo)
2
+ <v - vo)
2
< 8. (2-15)•EXAMPLE 2.14 Show, ifu(x, y) = ( ?~3
x Y^2 ), then (x,y)-(0,0) Lim u(x, y) = 0.