Confusion about Liouville's theorem for Harmonic functions

Question

The excercises in my textbook ask me to prove Liouville's Theorem for harmonic functions. Their formulations are

Lioville's Theorem: Every bounded, harmonic function on the whole complex plane must be constant

and

A real valued function on a domain $D$ of the complex plane is Harmonic if $u(x,y)\in C^2$ and $$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$$ for all $z=x+iy \in D$

Although I don't even know how to interpret this, because harmonic functions are real valued. But the theorem says "harmonic on the whole complex plane"...

By the way, I should point out that in my book thus far, I have only learned up to Cauchy's integral formula. I have not learned residues or any more complicated integration methods. Thanks!

Answer 1

There are quite a few ways to do this, but I'll give a sketch of one. Recall the mean-value property

$$u(z_0)=\frac{1}{\pi r^2}\int\limits_{D_r(z_0)} u(z)dydx=\int\limits_{0}^{2\pi} \int\limits_0^r u( z_0+se^{i\theta})s ds d\theta.$$ Now, for any $p,q\in \mathbb{C},$ we have that $$|u(p)-u(q)|\leq \frac{1}{\pi r^2}\iint\limits_{\Delta (p,q,r)}|u(z)|dxdy,$$ where $\Delta(p,q,r)=(D_r(p)\setminus D_r(q))\cup (D_r(q)\setminus D_r(p))$ is the symmetric difference. Now, if $d=|p-q|,$ the $\Delta (p,q,r)\subseteq D_{r+d}(p)\setminus D_{r-d}(p),$ which implies that $A(\Delta (p,q,r))\leq \pi ((r+d)^2-(r-d)^2)=4\pi dr.$ So, if $u$ is bounded by $M$ over $\mathbb{C},$ then $$|u(p)-u(q)|\leq\frac{4Md}{r}$$ for all $r>0$. Taking the limit as $r\rightarrow\infty$ yields the result.

This proof comes from Taylor's notes on complex analysis. For alternative proof, still without harmonic conjugates, you can reference Evans, pages 29-30.

Answer 2

Cauchy's Integral Formula states that, for f(z) a holomorphic function (except possibly at finitely many points)

$f(w) = \int_{\gamma} \frac{f(z)}{z-w} dz $ for $\gamma$ a circular contour within which $w$ resides.

Now assume f is bounded, say $|f(z)|\leq M$. Then we, for a sufficiently large radius of contour

$|f(w_1)-f(w_2)| \\= | \int_{\gamma} \frac{f(z)}{z-w_1} - \frac{f(z)}{z-w_2} dz | \\= | \int_{\gamma} \frac{(w_1-w_2)f(z)}{(z-w_1)(z-w_2)} dz| \\ \leq \text{sup}_{\gamma} |\frac{(w_1-w_2)f(z)}{(z-w_1)(z-w_2)}| \text{length}(\gamma) \\ \leq \frac{M2\pi R |w_1-w_2|}{(R-|w_1|)(R-|w_2|)}$

Now taking R to infinity gives the desired result, for holomorphic functions. I'll leave it to you to show any harmonic function may be written as a holomorphic one.