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

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
.
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