Math 411: Honours Complex Variables - University of Alberta

96 CHAPTER 14. HARMONIC FUNCTIONS
where φ will be specified later. First, note that
� y
∂v ∂
(x,y) =
∂x y0
2u ∂x2(x,t)dt+φ′ (x), by Lemma 5.3,
� y
∂
= −
2u ∂y2(x,t)dt+φ′ (x)
y0
= − ∂u ∂u
(x,y)+
∂y ∂y (x,y0)+φ ′ (x).
Hence, if we want the Cauchy–Riemann differential equations to hold for u+iv, we
require that φ ′ (x) = − ∂u
∂y (x,y0). We thus set
Then (∗) holds, so that
i.e. v is harmonic.
Example. Let
� y
∂u
v(x,y) =
y0 ∂x (x,t)dt−
� x
∂u
x0 ∂y (s,y0)ds.
∂2v ∂ ∂v
=
∂x2 ∂x ∂x
∂ ∂u
= −
∂x ∂y
∂ ∂u
= −
∂y ∂x = −∂2 v
∂y2, u: R 2 → R, (x,y) ↦→ xy.
Then u is harmonic and
� y � x
v(x,y) = tdt−
is a harmonic conjugate for u.
0
0
sds = y2
2
− x2
2
Corollary 14.1.1. Let D ⊂ C be open, and let u: D → R be harmonic. Then, for
each z0 ∈ D, there is a neighbourhood U ⊂ D of z0 such that u|U has a harmonic
conjugate.
Corollary 14.1.2. Let D ⊂ C be open, and let u: D → R be harmonic. Then u is
infinitely often partially differentiable.
Corollary 14.1.3. Let D ⊂ C be open and connected, and let u : D → R be
harmonic. Then the following are equivalent:
(i) u ≡ 0;
(ii) there exists a nonempty open set U ⊂ D with u|U≡ 0.