Complex Analysis - Maths KU

3.3 Harmonic Functions 163
In Problems 11 and 12, verify that the given function u is harmonic in an appropriate
domain D. Find its harmonic conjugate v and find analytic function f(z) =u + iv
satisfying the indicated condition.
11. u(x, y) =xy + x +2y; f(2i) =−1+5i
12. u(x, y) =4xy 3 − 4x 3 y + x; f(1 + i)=5+4i
13. (a) Show that v(x, y) =
the origin.
x
x2 is harmonic in a domain D not containing
+ y2 (b) Find a function f(z) =u(x, y)+iv(x, y) that is analytic in domain D.
(c) Express the function f found in part (b) in terms of the symbol z.
14. Suppose f(z) =u(r, θ) +iv(r, θ) is analytic in a domain D not containing
the origin. Use the Cauchy-Riemann equations (10) ofSection 3.2 in the form
rur = vθ and rvr = −uθ to show that u(r, θ) satisfies Laplace’s equation in
polar coordinates:
r 2 ∂ 2 u ∂u
+ r
∂r2 ∂r + ∂2u =0. (5)
∂θ2 In Problems 15 and 16, use (5) to verify that the given function u is harmonic in a
domain D not containing the origin.
15. u(r, θ) =r 3 cos 3θ
16. u(r, θ) = 10r2 − sin 2θ
r 2
Focus on Concepts
17. (a) Verify that u(x, y) = e x2 −y 2
domain D.
cos 2xy is harmonic in an appropriate
(b) Find its harmonic conjugate v and find analytic function f(z) =u + iv satisfying
f(0) = 1. [Hint: When integrating, think ofreversing the product
rule.]
18. Express the function f found in Problem 11 in terms of the symbol z.
19. (a) Show that φ(x, y, z) =
1
� is harmonic, that is, satisfies Laplace’s
x2 + y2 + z2 equation ∂2u ∂x2 + ∂2u ∂y2 + ∂2u = 0 in a domain D ofspace not containing the
∂z2 origin.
(b) Is the two-dimensional analogue ofthe function in part (a), φ(x, y) =
1
� , harmonic in a domain D ofthe plane not containing the origin?
x2 + y2 20. Construct an example accompanied by a briefexplanation that illustrates the
following fact:
If v is a harmonic conjugate of u in some domain D, then u is, in
general, not a harmonic conjugate of v.
21. If f(z) =u(x, y)+iv(x, y) is an analytic function in a domain D and f(z) �= 0
for all z in D, show that φ(x, y) = log e |f(z)| is harmonic in D.