1.5. Space of harmonic functions 35

then f is harmonic. So is e−rx1 sin(tx2), and hence, since linear

combinations of harmonic functions are harmonic, so is

sinh(rx1) sin(tx2).

If A is a finite subset of

Zd,

then the space of functions on A

that are harmonic on A has dimension #(∂A). In fact, as we have

seen, there is a linear isomorphism between this space and the set of

all functions on ∂A. In Section 1.2.2, we discussed one basis for the

space of harmonic functions, the Poisson kernel,

HA,y(x) = HA(x, y) = P{STA = y | S0 = x}.

Every harmonic function f can be written as

f(x) =

y∈∂A

f(y) HA,y(x).

The Poisson kernel is often hard to find explicitly. For some sets A, we

can find other bases that are more explicit. We will illustrate this for

the square, where we use the functions (1.24) which have “separated

variables”, i.e., are products of functions of x1 and functions of x2.

Example 1.13. Let A be the square in

Z2,

A = {(x1,x2) : xj = 1,...,N − 1}.

We write ∂A = ∂1,0 ∪ ∂1,N ∪ ∂2,0 ∪ ∂2,N where ∂1,0 = {(0,x2) : x2 =

1,...,N − 1}, etc. Consider the function

hj(x) = hj,1,N (x) = sinh

βj x1

N

sin

jπx2

N

.

Since cosh(0) = 1 and cosh(x) increases to infinity for 0 ≤ x ∞,

there is a unique positive number which we call βj such that

cosh

βj

N

+ cos

jπ

N

= 2,

When we choose this βj, hj is a harmonic function. Note that hj

vanishes on three of the four parts of the boundary and

hj(N, y) = sinh(βj) sin

jπy

N

.