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 Incoming links: Harmonic function as the real part of a holomorphic function

Harmonic Liouville theorem Created 2026-09-24 Updated 2026-10-14

A bounded harmonic function on the complex plane is constant. More generally, an entire harmonic function bounded on either side is constant: write it as

Re f

e^{f}

e^{- f}

Suppose first that

u

is harmonic. Since

\frac{\partial}{\partial y} u_{x} = \frac{\partial}{\partial x} u_{y} and \frac{\partial}{\partial x} u_{x} = - \frac{\partial}{\partial y} u_{y},

g := u_{x} - i u_{y}

D

g

f

. If

f = p + i q

, then

p_{x} = Re f^{'} = u_{x}, p_{y} = - Im f^{'} = u_{y} .

Thus

p - u

has zero gradient and is constant on the connected disc. Subtracting that real constant from

f

gives

u = Re f .

Conversely, if

u = Re f

for a holomorphic

f = u + i v

, differentiating the Cauchy-Riemann equations gives

Δ u = u_{xx} + u_{yy} = v_{y x} - v_{x y} = 0.

Hence $u$ is harmonic. This proves the harmonic function as the real part of a holomorphic function criterion on $D$ .