Complex Analysis - Maths KU

170 Chapter 3 Analytic Functions
2
–1
y
φ = 20
φ = 10
Figure 3.12 Figure for Problem 12
y
π/4
φ = 30
φ = 0
Figure 3.13 Figure for Problem 13
x
Remarks
In Section 4.5 and in Chapter 7 we shall introduce a method which enables
us to solve Dirichlet problems using analytic mappings.
EXERCISES 3.4 Answers to selected odd-numbered problems begin on page ANS-13.
In Problems 1–4, identify the two families of level curves defined by the given analytic
function f. By hand, sketch two curves from each family on the same coordinate
axes.
1. f(z) =2iz − 3+i 2. f(z) =(z− 1) 2
3. f(z) = 1
4. f(z) =z +
z
1
z
In Problems 5–8, the given analytic function f(z) =u + iv defines two families of
level curves u(x, y) =c1 and v(x, y) =c2. First use implicit differentiation to
compute dy/dx for each family and then verify that the families are orthogonal.
5. f(z) =x − 2x 2 +2y 2 + i(y − 4xy)
6. f(z) =x 3 − 3xy 2 + i(3x 2 y − y 3 )
7. f(z) =e −x cos y − ie −x sin y
8. f(z) =x +
x
x2 + i
+ y2 �
y −
y
x2 + y2 �
In Problems 9 and 10, the given real-valued function φ is the velocity potential for
the planar flow ofan incompressible and irrotational fluid. Find the velocity field F
ofthe flow. Assume an appropriate domain D ofthe plane.
9. φ(x, y) =
x
x 2 + y 2
10. φ(x, y) = 1
2 A log e
� x 2 +(y +1) 2 � , A>0
11. (a) Find the potential φ ifthe domain D in Figure 3.10 is replaced by 0