Complex Analysis - Maths KU

172 Chapter 3 Analytic Functions
(c) Use a CAS or graphing software to plot representative curves from each
ofthe orthogonal families φ(r, θ) = c1 and ψ(r, θ) = c2 on the same
coordinate axes.
� � � �
�
22. The function Ω(z) = log � z +1�
� z +1
e � z − 1 � + iArg is a complex potential ofa
z − 1
two-dimensional electrostatic field.
(a) Show that the equipotential curves φ(x, y) =c1 and the lines offorce
ψ(x, y) =c1 are, respectively
(x − coth c1) 2 + y 2 = csch 2 c1 and x 2 +(y + cot c2) 2 = csc 2 c2.
Observe that the equipotential curves and the lines of force are both families
ofcircles.
(b) The centers ofthe equipotential curves in part (a) are (coth c1, 0). Approximately,
where are the centers located when c1 →∞? When c1 →−∞?
Where are the centers located when c1 → 0 + ? When c1 → 0 − ?
(c) Verify that each circular line of force passes through z = 1 and through
z = −1.
(d) Use a CAS or graphing software to plot representative circles from each
family on the same coordinate axes. If you use a CAS do not use the
contour plot application.
Answers to selected odd-numbered problems begin
CHAPTER 3 REVIEW QUIZ
on page ANS-13.
In Problems 1–12, 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. Ifa complex function f is differentiable at point z, then f is analytic at z.
y
2. The function is f(z) =
x2 x
+ i
+ y2 x2 differentiable for all z �= 0.
+ y2 3. The function f(z) =z 2 + z is nowhere analytic.
4. The function f(z) = cos y − i sin y is nowhere differentiable.
5. There does not exist an analytic function f(z) =u (x, y)+iv (x, y) for which
u (x, y) =y 3 +5x.
6. The function u(x, y) =e 4x cos 2y is the real part ofan analytic function.
7. If f(z) =e x cos y + ie x sin y, then f ′ (z) =f(z).
8. If u(x, y)
�
and v(x,
�
y) are
�
harmonic
�
functions in a domain D, then the function
∂u ∂v ∂u ∂v
f(z) = − + i + is analytic in D.
∂y ∂x ∂x ∂y
9. If g is an entire function, then f(z) = � iz 2 + z � g(z) is necessarily an entire
function.
10. The Cauchy-Riemann equations are necessary conditions for differentiability.