Complex Analysis - Maths KU

Chapter 2 Review Quiz 139
10. The linear mapping w = � 1 − √ 3i � z + 2 acts by rotating through an angle
of π/3 radians clockwise about the origin, magnifying by a factor of 2, then
translating by 2.
11. There is more than one linear mapping that takes the circle |z − 1| = 1 to the
circle |z + i| =1.
12. The lines x = 3 and x = −3 are mapped onto to the same parabola by w = z 2 .
13. There are no solutions to the equation Arg(z) = Arg � z 3� .
14. If f(z) =z 1/4 is the principal fourth root function, then f(−1) = − 1
√ √
1
2+ 2i.
2 2
15. The complex number i is not in the range of the principal cube root function.
16. Under the mapping w =1/z on the extended complex plane, the domain |z| > 3
is mapped onto the domain |w| < 1
3 .
17. If f is a complex function for which lim Re(f(z)) = 4 and
z→2+i
lim Im(f(z)) = −1, then lim
z→2+i z→2+i
f(z) =4− i.
18. If f is a complex function for which lim
x→0 f(x +0i) = 0 and lim
y→0 f(0 + iy) =0,
then lim
z→0 f(z) =0.
19. If f is a complex function that is continuous at the point z =1+i, then the
function g(z) =3[f(z)] 2 − (2 + i)f(z)+i is continuous at z =1+i.
20. If f is a complex function that is continuous on the entire complex plane, then
the function g(z) =f(z) is continuous on the entire complex plane.
In Problems 21–40, try to fill in the blanks without referring back to the text.
21. If f(z) =z 2 + i¯z then the real and imaginary parts of f are u(x, y) =
and v(x, y) = .
22. If f(z) =
|z − 1|
z2 , then the natural domain of f is .
+2iz +2
23. If f(z) =z − ¯z, then the range of f is contained in the axis.
24. The exponential function e z has real and imaginary parts u(x, y) =
and v(x, y) = .
25. A parametrization of the line segment from 1 + i to 2i is z(t) = .
26. A parametrization of the circle centered at 1−i with radius 3 is z(t) = .
27. Every complex linear mapping is a composition of at most one , one
, and one .
28. The complex mapping w = iz+2 rotates and , but does not .
29. The function z 2 squares the modulus of z and its argument.
30. The image of the sector 0 ≤ arg(z) ≤ π/2 under the mapping w = z 3 is
.
31. The image of horizontal and vertical lines under the mapping w = z 2 is
.
32. The principal nth root function z 1/n maps the complex plane onto the region
.