Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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96 CHAPTER 14. HARMONIC FUNCTIONS<br />
where φ will be specified later. First, note that<br />
� y<br />
∂v ∂<br />
(x,y) =<br />
∂x y0<br />
2u ∂x2(x,t)dt+φ′ (x), by Lemma 5.3,<br />
� y<br />
∂<br />
= −<br />
2u ∂y2(x,t)dt+φ′ (x)<br />
y0<br />
= − ∂u ∂u<br />
(x,y)+<br />
∂y ∂y (x,y0)+φ ′ (x).<br />
Hence, if we want the Cauchy–Riemann differential equations to hold for u+iv, we<br />
require that φ ′ (x) = − ∂u<br />
∂y (x,y0). We thus set<br />
Then (∗) holds, so that<br />
i.e. v is harmonic.<br />
Example. Let<br />
� y<br />
∂u<br />
v(x,y) =<br />
y0 ∂x (x,t)dt−<br />
� x<br />
∂u<br />
x0 ∂y (s,y0)ds.<br />
∂2v ∂ ∂v<br />
=<br />
∂x2 ∂x ∂x<br />
∂ ∂u<br />
= −<br />
∂x ∂y<br />
∂ ∂u<br />
= −<br />
∂y ∂x = −∂2 v<br />
∂y2, u: R 2 → R, (x,y) ↦→ xy.<br />
Then u is harmonic and<br />
� y � x<br />
v(x,y) = tdt−<br />
is a harmonic conjugate for u.<br />
0<br />
0<br />
sds = y2<br />
2<br />
− x2<br />
2<br />
Corollary 14.1.1. Let D ⊂ C be open, and let u: D → R be harmonic. Then, for<br />
each z0 ∈ D, there is a neighbourhood U ⊂ D <strong>of</strong> z0 such that u|U has a harmonic<br />
conjugate.<br />
Corollary 14.1.2. Let D ⊂ C be open, and let u: D → R be harmonic. Then u is<br />
infinitely <strong>of</strong>ten partially differentiable.<br />
Corollary 14.1.3. Let D ⊂ C be open and connected, and let u : D → R be<br />
harmonic. Then the following are equivalent:<br />
(i) u ≡ 0;<br />
(ii) there exists a nonempty open set U ⊂ D with u|U≡ 0.