Complex Analysis - Maths KU

y
a
b
φ = k 1
x
φ = k 0
Figure 3.14 Figure for Problem 14
3.4 Applications 171
(b) Find the complex potential Ω(z).
(c) Sketch the equipotential curves and the lines offorce.
14. The steady-state temperature φ(r) between the two concentric circular conducting
cylinders shown in Figure 3.14 satisfies Laplace’s equation in polar
coordinates in the form
r 2 d 2 φ dφ
+ r
dr2 dr =0.
(a) Show that a solution ofthe differential equation subject to the boundary
conditions φ(a) =k0 and φ(b) =k1, where k0 and k1 are constant
potentials, is given by φ(r) =A log e r + B, where
A = k0 − k1
loge (a/b) and B = −k0 loge b + k1 loge a
loge (a/b)
[Hint: The differential equation is known as a Cauchy-Euler equation.]
(b) Find the complex potential Ω(z).
(c) Sketch the isotherms and the lines ofheat flux.
Focus on Concepts
15. Consider the function f(z) =z + 1
. Describe the level curve v(x, y) =0.
z
16. The level curves of u(x, y) =x 2 − y 2 and v(x, y) =2xy discussed in Example
1 intersect at z = 0. Sketch the level curves that intersect at z = 0. Explain
why these level curves are not orthogonal.
17. Reread the discussion on orthogonal families on page 165 that includes the proof
that the tangent lines L1 and L2 are orthogonal. In the proofthat concludes
with (4), explain where the assumption f ′ (z0) �= 0 is used.
18. Suppose the two families of curves u(x, y) =c1 and v(x, y) =c2, are orthogonal
trajectories in a domain D. Discuss: Is the function f(z) =u(x, y)+iv(x, y)
necessarily analytic in D?
Computer Lab Assignments
In Problems 19 and 20, use a CAS or graphing software to plot some representative
curves in each ofthe orthogonal families u(x, y) =c1 and v(x, y) =c2 defined by
the given analytic function first on different coordinate axes and then on the same
set ofcoordinate axes.
z − 1
19. f(z) =
z +1
21. The function Ω(z) = A
dimensional fluid flow.
20. f(z) = √ �
r cos θ
2
�
θ
+ i sin ,r>0, −π