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16.1 Harmonic Functions

The document discusses harmonic functions, which are functions whose second partial derivatives sum to zero. It defines harmonic functions and provides examples. Harmonic functions arise in applications like heat distributions and electrostatics. If a function is analytic, its real and imaginary parts are harmonic. The document also defines harmonic conjugates as functions that satisfy the Cauchy-Riemann equations along with another harmonic function.

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Mosa A Shorman
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87 views3 pages

16.1 Harmonic Functions

The document discusses harmonic functions, which are functions whose second partial derivatives sum to zero. It defines harmonic functions and provides examples. Harmonic functions arise in applications like heat distributions and electrostatics. If a function is analytic, its real and imaginary parts are harmonic. The document also defines harmonic conjugates as functions that satisfy the Cauchy-Riemann equations along with another harmonic function.

Uploaded by

Mosa A Shorman
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Lecture 16:

Harmonic Functions

Dan Sloughter
Furman University
Mathematics 39

March 31, 2004

16.1 Harmonic functions


Suppose f is analytic in a domain D,

f (x + iy) = u(x, y) + iv(x, y),

and u and v have continuous partial derivatives of all orders. From the
Cauchy-Riemann equations we know that

ux (x, y) = vy (x, y) and uy (x, y) = −vx (x, y)

for all x + iy ∈ D. Differentiating with respect to x, we have

uxx (x, y) = vyx (x, y) and uyx (x, y) = −vxx (x, y);

differentiating with respect to y, we have

uxy (x, y) = vyy (x, y) and uyy (x, y) = −vxy (x, y).

Hence

uxx (x, y) + uyy (x, y) = vyx (x, y) − vxy (x, y) = vxy (x, y) − vxy (x, y) = 0

for all x + iy ∈ D and

vxx (x, y) + vyy (x, y) = −uyx (x, y) + uxy (x, y) = −uxy (x, y) + uxy (x, y) = 0

for all x + iy ∈ D.

1
Definition 16.1. Suppose H : R2 → R has continuous second partial deriva-
tives on a domain D. We say H is harmonic in D if for all (x, y) ∈ D,

Hxx (x, y) + Hyy (x, y) = 0.

Harmonic functions arise frequently in applications, such as in the study


of heat distributions and electrostatic potentials.
Theorem 16.1. If f is analytic in a domain D and

f (x + iy) = u(x, y) + iv(x, y),

then u and v are harmonic in D.


Proof. The result follows from the discussion above combined with a result
we will prove later: if f is analytic at z0 = x0 + iy0 , then u and v have
continuous partial derivatives of all orders at (x0 , y0 ).
Example 16.1. We know that f (z) = ez is entire. Since

f (x + iy) = ex cos(y) + iex sin(y),

it follows that u(x, y) = ex cos(y) and v(x, y) = ex sin(y) are both harmonic
in C (which is also easily checked directly).
Example 16.2. We know that
1
f (z) =
z2
is analytic in {z ∈ C : z 6= 0}. Now

1 1 z̄ 2 x2 − y 2 − 2xyi
= = ,
z2 z 2 z̄ 2 (x2 + y 2 )2
so
x2 − y 2
u(x, y) =
(x2 + y 2 )2
and
2xy
v(x, y) = −
(x2 + y 2 )2
are harmonic in {(x, y) ∈ R2 : (x, y) 6= (0, 0)}.

2
Definition 16.2. If u and v are harmonic in a domain D and satisfy the
Cauchy-Riemann equations, then we say v is a harmonic conjugate of u.

Example 16.3. It is easy to check that the function

u(x, y) = x3 − 3xy 2

is harmonic. To find a harmonic conjugate v of u, we must have

ux (x, y) = vy (x, y)

and
uy (x, y) = −vx (x, y).
From the first we have
vy (x, y) = 3x2 − 3y 2 ,
from which it follows that

v(x, y) = 3x2 y − y 3 + ϕ(x)

for some function ϕ of x. It now follows from the second equation that

−6xy = −vx (x, y) = −(6xy + ϕ0 (x)),

and so ϕ0 (x) = 0. Hence for any real number c, the function

v(x, y) = 3x2 y − y 3 + c

is a harmonic conjugate of u.

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