Complex Analysis - Maths KU

158 Chapter 3 Analytic Functions
26. Show that |f ′ (z)| 2 = u 2 x + v 2 x = u 2 y + v 2 y.
Focus on Concepts
27. Suppose u(x, y) and v(x, y) are the real and imaginary parts ofan analytic
function f. Can g(z) =v(x, y) +iu(x, y) be an analytic function? Discuss
and defend your answer with sound mathematics.
28. Suppose f(z) is analytic. Can g(z) =f(z) be analytic? Discuss and defend
your answer with sound mathematics.
29. In this problem you are guided through the start ofthe proofofthe proposition:
If f is analytic in a domain D, and |f(z)| = c, where c is constant,
then f is constant throughout D.
Proof We begin with the hypothesis that |f(z)| = c. If f(z) =u(x, y)+iv(x, y)
then |f(z)| 2 = c 2 is the same as u 2 + v 2 = c 2 . The partial derivatives ofthe
last expression with respect to x and y are respectively
2uux +2vvx = 0 and 2uuy +2vvy =0.
Complete the proofby using the Cauchy-Riemann equations to replace vx and
vy in the last pair ofequations. Then solve for ux and uy and draw a conclusion.
Use the Cauchy-Riemann equations again and solve for vx and vy.
30. In this problem you are guided through the start ofthe proofofthe proposition:
If f is analytic in a domain D, and f ′ (z) =0, then f is constant
throughout D.
Proof We begin with the hypothesis that f is analytic in D and hence it is
differentiable throughout D. Hence by (9) ofthis section and the assumption
that f ′ (z) =0inD, wehavef ′ (z) = ∂u ∂v
+ i = 0. Now complete the proof.
∂x ∂x
31. Use the proposition in Problem 30 to show that if f and g are analytic and
f ′ (z) = g ′ (z), then f(z) = g(z) +c, where c is a constant. [Hint: Form
h(z) =f(z) − g(z).]
32. If f(z) and f (z) are both analytic in a domain D, then what can be said about
f throughout D?
33. Suppose x = r cos θ, y = r sin θ, and f(z) =u(x, y)+iv(x, y). Show that
and
∂u ∂u ∂u
= cos θ + sin θ,
∂r ∂x ∂y
∂v ∂v ∂v
= cos θ + sin θ,
∂r ∂x ∂y
∂u ∂u ∂u
= − r sin θ + r cos θ
∂θ ∂x ∂y
(12)
∂v ∂v ∂v
= − r sin θ + r cos θ.
∂θ ∂x ∂y
(13)
Now use (1) in the foregoing expressions for vr and vθ. By comparing your results
with the expressions for ur and uθ, deduce the Cauchy-Riemann equations
in polar coordinates given in (10).
34. Suppose the function f(z) =u(r, θ) +iv(r, θ) is differentiable at a point z
whose polar coordinates are (r, θ). Solve the two equations in (12) for ux and