# The origin of Discrete `Liouville's theorem'

It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).

If a bounded function $f : \mathbb Z^2 \rightarrow \mathbb{R}$ satisfies the following condition $$\forall (x, y) \in \mathbb{Z}^2,\quad f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, y - 1)}{4}$$ then f is constant function.

(A stronger version of discrete “Liouville's theorem” also follows from Heilbronn's proof.)

But recently I knew from Alexander Khrabrov that there is an older article of Capoulade with almost the same result (Sur quelques proprietes des fonctions harmoniques et des fonctions preharmoniques, - Mathematica (Cluj), 6 (1932), 146-151.)

Unfortunately the last article is not available for me. Does anybody have an access to this paper? Is this really the first proof discrete Liouville's theorem?

(This question is inspired by Liouville Theorem from Mathematics)

• I'll make a trip to Leiden University library today or tomorrow to obtain this elusive article by Capoulade (I hope). Dec 28, 2013 at 13:13
• no luck yet, I know it's out there in storage, and I'm confident I'll get it eventually, but if there's someone with easier access --- don't let me hold you back! Dec 30, 2013 at 7:36
• got it (first time this journal left it's offsite storage...); I'll scan it and post it as a "partial answer" shortly. Dec 31, 2013 at 11:00

• Rather off-topic, but I wonder whether it is possible to prove Liouville's theorem considering a function like $F(x,y)=\sum_{i,j} f(i,j) x^iy^j$. (As written, this does not make much sence, of course, since the series diverges.) We then have $$4F(x,y) = \sum_{i,j} \left(xf(i-1,j)x^{i-1}j+\dotsb+\frac1yf(i,j+1)x^iy^{j+1} \right)$$ whence $4F(x,y)=(x+1/x+y+1/y)F(x,y)$, implying $F(x,y)=0$.