Complex Analysis - Maths KU

448 Chapter 7 Conformal Mappings
In Problems 37–40, use a CAS to plot the equipotential curves for the given electrostatic
potential φ(x, y).
37. φ(x, y) is the electrostatic potential from Problem 7.
38. φ(x, y) is the electrostatic potential from Problem 8.
39. φ(x, y) is the electrostatic potential from Problem 9.
40. φ(x, y) is the electrostatic potential from Problem 10.
In Problems 41–44, use a CAS to plot the streamlines of the given flow.
41. The flow from Problem 13.
42. The flow from Problem 14.
43. The flow from Problem 15.
44. The flow from Problem 16.
CHAPTER 7 REVIEW QUIZ
Answers to selected odd-numbered problems begin
on page ANS-23.
In Problems 1–15, 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 analytic at a point z0, then the mapping w = f(z) is conformal at z0.
2. The mapping w = z 2 + iz +1 is not conformal at z = − 1
2 i.
3. The mapping w = z 2 +1 is not conformal at z = ±i.
4. The mapping w =¯z fails to be conformal at every point in the complex plane.
5. A linear fractional transformation is conformal at every point in its domain.
6. The image of a circle under a linear fractional transformation is a circle.
z − i
7. The linear fractional transformation T (z) = maps the points 0, −1, and
z +1
i onto the points −i, ∞, and 0, respectively.
8. Given any three distinct points z1, z2, and z3, there is a linear fractional transformation
that maps z1, z2, and z3 onto 0, 1, and ∞.
9. The inverse of the linear fractional transformation T (z) =(az + b)/ (cz + d) is
T −1 (z) =(cz + d)/ (az + b).
10. If f ′ (z) =A(z +1) −1/2 (z − 1) −3/4 , then w = f(z) maps the upper half-plane
onto an unbounded polygonal region.
11. If f ′ (z) =A(z +1) −1/2 z −1/2 (z − 1) −1/2 , then w = f(z) maps the upper halfplane
onto a rectangle.
12. Every Dirichlet problem in the upper half-plane can be solved using the Poisson
integral formula.