An Introduction to Harmonic Theory
Henry Jones
March 2023
1 Introduction
Interesting in their own right in theoretical realms, harmonic functions play a fundamental role
in many physical applications. By definition, a twice differentiable function u : A → R defined on
an open set A is called a real-valued harmonic function if u satisfies Laplace’s equation,
∆u =
∂
2
u
∂x
2
+
∂
2
u
∂y
2
= 0.
Of great importance to the work in this project is the linking of the properties of analytic functions
to harmonic functions, a link which is first demonstrated in the definition of harmonic conjugates.
Definition 1. If u and v are real-valued functions defined on an open subset A ⊂ C such that the
complex-valued function f = u + iv is analytic on A, we say u and v are harmonic conjugates on
A.
Having introduced the connections between analytic and harmonic functions, we introduce our
final introductory definition.
Definition 2. A complex-valued function f(z) = f(x + iy) = u(x, y) + iv(x, y) is harmonic if u
and v are harmonic.
Note that the definition of a complex-valued harmonic function does not require its real and
imaginary components u and v to be harmonic conjugates. Therefore while all analytic functions
are harmonic, complex-valued harmonic functions need not be analytic.
The focus of this paper will be the latter parts of section 2.5 in the Marsden and Hoffman text,
which ultimately leverages the properties of analytic functions to deduce results for their real and
imaginary harmonic components.
2 Foundations of Harmonic Theory
Our results in this paper heavily depend upon some preliminary results I proved in-class for analytic
function, most importantly the Mean Value Property (Theorem 2.5.2) and The Global Maximum
Modulus Principle (Theorem 2.5.6) for analytic functions. Of particular importance is Theorem
2.5.8:
1
Theorem 3. Let A be a region in C and let u be a twice continuously differentiable harmonic
function on A. Then u is C
∞
, and in a neighborhood of each point z
0
∈ A, u is the real part of
some analytic function. If A is simply connected, there is an analytic function f on A such that
u = ℜ(f).
This key result allows us conclude that harmonic functions defined on A are the real part of
analytic functions on neighborhoods of points in A. Having established these results, we can begin
to make stronger and more general statements about harmonic functions themselves.
2.1 Maximum Principles for Harmonic Functions
After establishing the Mean Value Property for Harmonic Functions (2.5.9) by using the
corresponding Theorem 2.5.2 through Theorem 2.5.8, we are in a position to deduce the Local
Maximum Principle for Harmonic Functions (2.5.10). The approach is analogous to the method by
which Mean Value Property for Analytic Functions (2.5.2) was used to deduce the local Maximum
Modulus Principle (2.5.1.)
Theorem 4. Local Maximum Principle for Harmonic Functions (2.5.10)
Let u be harmonic on a region A. Suppose that u has a relative maximum at z
0
∈ A,. That is,
u(z) ≤ u(z
0
) for all in some disk of radius ρ centered about z
0
. Then u is constant in a neighborhood
of z
0
.
Note that this theorem similarly applies to local minimums of harmonic functions. Instead of
a direct argument, we can use Theorem 2.5.1 to provide a simple proof.
Proof. By Proposition 2.5.8, we can say that because u is a twice-continuously differentiable har-
monic function, u is the real part of an analytic function f on a neighborhood of every point in
A. Since e
z
is entire, e
f(z)
is analytic and |e
f(z)
| = e
u(z)
. Because e
x
is strictly increasing in x for
all x ∈ R, the maxima of u are the same as those of |e
f(z)
|. But by the Local Maximum Modulus
Principle for analytic functions, e
f(z)
is constant is a neighborhood of z
0
. But because e
x
is strictly
increasing for x ∈ R, e
u
and therefore u are also constant in a neighborhood of z
0
.
Then, just as the Global Maximum Modulus Principle (2.5.6) was deduced from the local
version, we may deduce the global maximum principle for harmonic functions.
Theorem 5. Global Maximum Principle for Harmonic Functions (2.5.11)
Suppose that A ⊂ C is an open, connected, and bounded set. Let u = cl(A) → R be continuous
and harmonic on A and let M be the maximum of u on the boundary of A, denoted ∂A. Then
(i) u(x, y ≤ M for all (x, y) ∈ A.
(ii) If u(x, y) = M for some (x, y) ∈ A, then u is constant on A.
There is an intuitive, corresponding result for the minimum. Let m denote the minimum of u
on ∂A. Then
(i) u(x, y ≥ m for all (x, y) ∈ A.
(ii) If u(x, y) = m for some (x, y) ∈ A, then u is constant on A.
2
The proof of this theorem is identical to the argument for maximum modulus principle for
analytic functions which I have previously proved.
3 The Dirichlet Problem
A classical problem in both theoretical and applied settings is known as the Dirichlet Problem.
The problem is posed as follows:
Let A be an open, bounded region and let u
0
be a given continuous function on the boundary of
A denoted ∂A. Find a real-valued function u on the closure of A that is continuous on this closure,
is harmonic on A, and equals u
0
on ∂A.
There are several difficult proofs for the existence of solutions for domain boundaries ∂A which
are “sufficiently smooth,” but the proof for uniqueness of solutions is much more accessible to our
investigation.
Theorem 6. Uniqueness for the Dirichlet Problem (2.5.12) The solutions to the Dirichlet Problem
is unique assuming one exists.
Proof. Let u and ˜u be two solutions to the Dirichlet problem. Then consider ϕ(u − ˜u). Since u and
˜u are both harmonic on A, ϕ is harmonic and ϕ = 0 on ∂A.
Then, by the maximum principle for harmonic functions, ϕ(x, y) ≤ 0 inside A. And by the
minimum principle we must have ϕ(x, y) ≥ 0 on A. Thus we have that ϕ = 0, and solutions to the
Dirichlet problem are unique.
Note that the Dirichlet problem attempts to solve Laplace’s conditions ∆u = 0, which can be
thought of a the steady state solution to the classical heat equation u
t
= −k∆u where k is the
diffusion or ‘conduction’ constant.
With uniqueness established, our motivation in deriving the following Poisson Integral Formula
is to find a solution to the Dirichlet Problem on an open disk. To this end, we will derive a formula
that explicitly expresses the values of the solution in terms of its boundary values.
4 The Poisson Integral Formula
Theorem 7. The Poisson Integral Formula
Assume that u is defined and continuous on the closed disk {z : |z| ≤ r} and is harmonic on the
open disk D(0; r) = {z : |z| < r}. Then for any ρ < r, we have the real form of Poisson’s Formula
u(ρe
iϕ
) =
r
2
− ρ
2
2π
Z
2π
0
u(re
iθ
)
r
2
− 2rρ cos(ϕ − θ) + ρ
2
dθ, (1)
which is equivalent to the complex form of Poisson’s formula
u(z) =
1
2π
Z
2π
0
u(re
iθ
)
r
2
− |z|
2
|re
iθ
− z|
2
dθ. (2)
This function is the unique harmonic function in disk of radius r that satisfies the prescribed
boundary condition
lim
ρ→r
u(ρe
iϕ
) = u(re
iϕ
). (3)
3
Proof. We note first that because u is harmonic on D(0; r), and all such disks are simply connected,
Proposition 2.5.8 allows us to conclude that there is an analytic function f defined on D(0; r) such
that u = ℜf. Next we let 0 < s < r and let γ
n
be the circle |z| = s. Then, by Cauchy’s Integral
Formula 2.4.4, we can write
f(z) =
1
2πi
Z
γ
n
f(ζ)
ζ − z
dζ (4)
for all z such that |z| < s. Note that we have used the fact that f is analytic on D(0; r) which is
simply connected to say γ
n
is homotopic to a point as required by Cauchy’s Integral Formula.
We are motivated to manipulate this expression for f(z) to take the real parts in order to give
a formula for u. To this end, we let ˜z = s
2
/¯z, which is called the reflection of z in the circle |ζ| = s.
Thus if z lies inside the circle |z| = s, then ˜z lies outside the circle, and therefore
1
2πi
Z
γ
n
f(ζ)
ζ − ˜z
dζ = 0 (5)
for |z| < s. If we subtract (5) from (4) we obtain
f(z) =
1
2πi
Z
γ
n
f(ζ)
1
ζ − z
−
1
ζ − ˜z
dζ.
If we observe that |ζ| = s, we can simplify part of our integrand as
1
ζ − z
−
1
ζ − ˜z
=
1
ζ − z
−
1
ζ − |ζ|
2
¯z
=
1
ζ − z
−
¯z
ζ(¯z − ζ
¯
ζ
=
−ζ ¯z + |ζ|
2
+ ζ ¯z − |z|
2
ζ|ζ − z|
2
=
|ζ|
2
− |z|
2
ζ|ζ − z|
2
.
Thus we can write
f(z) =
1
2πi
Z
γ
n
f(ζ)(|ζ|
2
− |z|
2
)
ζ|ζ − z|
2
dζ.
Recall that our path has explicit parametrization γ
n
(θ) = se
iθ
for 0 ≤ θ ≤ 2π, so letting z = ρe
iϕ
(where ρ < s) we can write
f(ρe
iϕ
) =
1
2π
Z
2π
0
f(se
iθ
)(s
2
− ρ
2
)
|se
iθ
− ρe
iϕ
|
2
dθ. (6)
If we expand our denominator we see that
|se
iθ
− ρe
iϕ
|
2
=
s cos(θ) − ρ cos(ϕ) + i((s) sin(θ) − ρ sin(ϕ))
s cos(θ) − ρ cos(ϕ) − i((s) sin(θ) − ρ sin(ϕ))
= (s cos(θ) − ρ cos(ϕ))
2
+ (s sin(θ) − ρ sin(ϕ))
2
= s
2
+ ρ
2
− 2sρ cos(ϕ − θ).
Then taking equating the real parts of both sides of (6), we obtain
u(ρe
iϕ
) =
1
2π
Z
2π
0
u(se
iθ
)(s
2
− ρ
2
)dθ
s
2
+ ρ
2
− 2sρ cos(ϕ − θ)
. (7)
Holding ρ and ϕ fixed, we can not that this formula holds for any s such that ρ < s < r. Since u is
continuous on the closure of D(0; r), and since the function s
2
+ ρ
2
− 2sρ cos(ϕ − θ) is never zero
whenever s > ρ, we can conclude that the integrand
u(se
iθ
)(s
2
− ρ
2
)dθ
s
2
+ ρ
2
− 2sρ cos(ϕ − θ)
4
is a continuous function of s and θ for s > ρ. Hence, with ρ, ϕ fixed, this function is uniformly
continuous on the compact set {0 ≤ θ ≤ 2π,
r+ρ
2
≤ s ≤ r}.
Consequently the limit
lim
s→r
u(se
iθ
)(s
2
− ρ
2
)
s
2
+ ρ
2
− 2sρ cos(ϕ − θ)
=
u(re
iθ
)(r
2
− ρ
2
)
r
2
+ ρ
2
− 2rρ cos(ϕ − θ)
converges uniformly in θ, which allows us to conclude that
lim
s→r
1
2π
Z
2π
0
u(se
iθ
)(s
2
− ρ
2
)
s
2
+ ρ
2
− 2sρ cos(ϕ − θ)
=
1
2π
Z
2π
0
u(re
iθ
)(r
2
− ρ
2
)
r
2
+ ρ
2
− 2rρ cos(ϕ − θ)
.
Thus, the Poisson Integral Formula gives a harmonic function which uniquely satisfies the
particular boundary condition which is ‘encoded’ in its construction.
5 Conclusion
This project has given an overview of introductory concepts and result in Harmonic Theory before
moving onto a direct proof of the Poisson Integral Formula. Throughout our investigation, the
relationship between harmonic and analytic functions, and the various results proved earlier for
analytic functions allow for more general statements to be made about harmonic functions. The
classical Dirichlet problem is evidence for the precedent of harmonic function theory in applied
settings and motivates a derivation of the Poisson Integral Formula.
6 Reflection
This project was rewarding in the sense that I felt that I had proved a large amount of the
preliminary results in section 2.5. I felt I had a solid foundation upon which to embark upon this
project. In hindsight, I would have liked a project that could have led to some more interesting
applications or examples, or even afforded opportunity to write some code, but it was rewarding
to lean into some theoretical bulk. When I first encountered harmonic functions in my PDE class
last year, I did not come away with much understanding of them and they were only relevant to
serving as the basis for many of our PDE boundary value problems. Therefore this project was a
nice theoretical underpinning to some of the PDE work I did last year and which appears in some
of my thesis.
7 References
Basic Complex Analysis, Marsden, J.E. and Marsden, U.J.E. and Hoffman, M.J. 1999, ISBN
9780716728771. https://books.google.com/books
Explorations in Complex Analysis, Brilleslyper, M.A., Dorff, M.J., McDougall, J.M., Rolf, J.S.
and Schaubroeck, L.E., 2012. Mathematical Association of America ISBN 9781614441083.
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