Re
1
z
Im
1
z
Figure 7.1.
Real and Imaginary Parts of
f
(
z
) =
1
z
.
Therefore, if
f
(
z
) is any complex function, we can write it as a complex combination
f
(
z
) =
f
(
x
+ i
y
) =
u
(
x, y
) + i
v
(
x, y
)
,
of two interrelated real harmonic functions:
u
(
x, y
) = Re
f
(
z
) and
v
(
x, y
) = Im
f
(
z
).
Before delving into the many remarkable properties of complex functions, let us look
at some of the most basic examples. In each case, the reader can directly check that the
harmonic functions provided by the real and imaginary parts of the complex function are
indeed solutions to the Laplace equation.
Examples of Complex Functions
(
a
)
Harmonic Polynomials
: As noted above, any complex polynomial is a linear com
bination, as in (7.2), of the basic complex monomials
z
n
= (
x
+ i
y
)
n
=
u
n
(
x, y
) + i
v
n
(
x, y
)
.
(7
.
7)
Their real and imaginary parts,
u
n
, v
n
, are the
harmonic polynomials
that we previously
constructed by applying separation of variables to the polar coordinate form of the Laplace
equation (4.94). The general formula can be found in (4.110).
(
b
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Olver
 Differential Equations, Exponential Function, Equations, Partial Differential Equations, Complex number, imaginary parts, Harmonic functions, Peter J. Olver

Click to edit the document details