78 CHAPTER 2 • COMPLEX FUNCTIONS
(d) z-l+i Jim z2.;!l+- 2 •-2. .;+l ti.
(e) z-1+t Jim •',; t •-l-- 2.;+ 2^3 i by factoring.
- Determine where the following functions are continuous.
(a ) z^4 - 9z^2 + iz - 2.
(b) ,21;.1,.(c) z;+6zt5 • +3•+2'(d) ~·•+1 -
(e) :'!.'f.
<r> ~r~~.
State why z -Jim z o (e"' cosy +ix^2 y) = e"'O cos yo+ ix&yo.
State why z-zo Jim (In (x^2 + y^2 ) + iy] =In (x~ + y5) + iyo, provided lzol # 0.
Show that
(a) Jim J.:!. = 0.•- 0
(b) Jim "'
2
z- o z = 0.- Let f (z) = •i:\» when z # 0, and let f (0) = 0. Show that f (z) is continuous for
all values of z. - Let f (z) = •' = x•-~•+12zv.
w ill +y2
(a) Find Jim f (z) as z - 0 along the line y = x.
- (b) Find Jim f (z) as z - 0 along the line y = 2x.
•- 0
( c) Find Jim f ( z) as z --> 0 along the parabola y = x^2.
•- 0
( d) What can you conclude about the limit of f ( z) as z -+ O? Why?
3 3
- (b) Find Jim f (z) as z - 0 along the line y = 2x.
- Let f (z) = f (x, y) = ,./f 2 va + i 5 .,t{ 112 when z # 0, and let f (0) = O.
(a) Show that Jim f (z) = f (0) = 0 if z approaches zero along any straight
•- 0
line that passes through the origin.
(b) Show that f is not continuous at the point O.- For z # 0, let f(z) =~-Does f(z) have a limit as z-> O?
- Does Jim Argz exist? Why? Hint: Use polar coordinates and let z approach - 4
.t.-- 4
from the upper and lower half-planes.