Harmonic Functions
1. Show that the following functions are harmonic, and find harmonic
conjugates:
(a) x2 – y2 (c) sinhxsiny (e) tan-ley/x), x> 0
(b) xy + 3x2y – y3 (d) ex2 – y2 cos(2xy) (f) x/(x 2 + y2)
2. Show that if v is a harmonic conjugate for u, then -u is a harmonic
conjugate for v.
3. Define u(z) = Im(I/z2 ) for z f:. 0, and set u(O) = O.
(a) Show that all partial derivatives ofu with respect to x exist at
all points of the plane C, as do all partial derivative of u with
respect to y.
cPu cPu
(b) Show that 8x 2 + 8y2 = o.
(c) Show that u is not harmonic on C.
82 u
(d) Show that 8x8y does not exist at (0,0).
4. Show that if h(z) is a complex-valued harmonic function (solution
of Laplace’s equation) such that zh(z) is also harmonic, then h(z)
is analytic.
5. Show that Laplace’s equation in polar coordinates is
82u 18u 1 82u
8r 2 + ;: 8r + r2 8()2 = O. 58 II Analytic Functions
6. Show using Laplace’s equation in polar coordinates that log \z\ is
harmonic on the punctured plane C\{O}.
7. Show that log \z\ has no conjugate harmonic function on the punc-
tured plane C\ {O}, though it does have a conjugate harmonic func-
tion on the slit plane C\( -00,0].
8. Show using Laplace’s equation in polar coordinates that u(re i8 ) =
Ologr is harmonic. Use the polar form of the Cauchy-Riemann
equations (Exercise 3.8) to find a harmonic conjugate v for u. What
is the analytic function u + iv?
