Complex Analysis - Maths KU

138 Chapter 2 Complex Functions and Mappings
14. Let f be a complex function. Explain the relationship between the vector field
associated with f(z) and the vector field associated with g(z) =if(z). Illustrate
with sketches using a simple function for f.
15. Consider the planar flow associated with f(z) =c where c is a complex constant.
(a) Find the streamlines of this flow.
(b) Explain why this flow is called a uniform flow.
16. Consider the planar flow associated with f(z) =1− 1/z 2 .
(a) Use a CAS to plot the vector field associated with f in the region |z| > 1.
(b) Verify analytically that the unit circle x 2 + y 2 = 1 is a streamline in this
flow.
(c) Explain why f(z) =1− 1/z 2 is called a flow around the unit circle.
Computer Lab Assignments
In Problems 17–22, use a CAS to plot the vector field associated with the given
complex function f.
17. f(z) =2z− i 18. f(z) =z 3
19. f(z) =1− z 2
20. f(z) = 1
z
21. f(z) =2+i 22. f(z) =1− 1
¯z 2
CHAPTER 2 REVIEW QUIZ
Answers to selected odd-numbered problems begin
on page ANS-12.
In Problems 1–20, answer true or false. If the statement is false, justify your answer
by either explaining why it is false or giving a counterexample; if the statement is
true, justify your answer by either proving the statement or citing an appropriate
result in this chapter.
1. If f(z) is a complex function, then f(x +0i) must be a real number.
2. arg(z) is a complex function.
3. The domain of the function f(z) = 1
z2 is all complex numbers.
+ i
4. The domain of the function f(z) =e z2 −(1+i)z+2 is all complex numbers.
5. If f(z) is a complex function with u(x, y) = 0, then the range of f lies in the
imaginary axis.
6. The entire complex plane is mapped onto the real axis v =0byw = z +¯z.
7. The entire complex plane is mapped onto the unit circle |w| =1byw = z
|z| .
8. The range of the function f(z) = Arg(z) is all real numbers.
9. The image of the circle |z − z0| = ρ under a linear mapping is a circle with a
(possibly) different center, but the same radius.