Functions of a Complex Variable: Contour Integration and
Steepest Descent
Michael Fowler 9/19/08
Analytic Functions
Suppose we have a complex function
f
=
u
+
iv
of a complex variable
z
=
x
+
iy
, defined in some
region of the complex plane, where
u
,
v
,
x
,
y
are real. That is to say,
()
(, )
,
f z
uxy i
vxy
=
+
with
u
(
x
,
y
) and
v
(
x
,
y
) real functions in the plane.
We now assume that in this region
f
(
z
) is differentiable, that is to say,
0
(
)
lim
z
df z
f z
z
f z
dz
z
Δ→
+
Δ−
=
Δ
is well-defined. What does this tell us about the functions
u
(
x
,
y
) and
v
(
x
,
y
), the real and
imaginary parts of
f
(
z
)?
In fact, the property of differentiability for a function of a complex variable tells us a lot! It does
not
just mean that the function is reasonably smooth.
The crucial difference from a function of a
real variable is that
Δ
z
can approach zero
from any direction
in the complex plane, and
the limit
in these different directions must be the same
.
Of course, there are only two independent
directions, so what we are really saying is
()()
,
f
xi
y
fxi
y
y
∂
+∂
+
=
∂∂
which we can write in terms of
u
,
v
:
(
,
)(
,
)
(
,
,
.
u
x
yv
x
y
u
x
x
y
ii
)
x
y
i
y
+=
+
Equating real and imaginary parts of this equation we find:
,
.
uv
v
u
x
yx
y
∂
∂
==
−
∂
∂
These are called the
Cauchy-Riemann equations
.
It immediately follows that both
u
(
x
,
y
) and
v
(
x
,
y
) must satisfy the two-dimensional Laplacian