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
u x y
iv x y
=
+
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 x
iy
f x
iy
x
iy
∂
+
∂
+
=
∂
∂
which we can write in terms of
u
,
v
:
( ,
)
( ,
)
( ,
)
( ,
.
(
)
(
)
u x y
v x y
u x y
v x y
i
i
)
x
x
iy
iy
∂
∂
∂
∂
+
=
+
∂
∂
∂
∂
Equating real and imaginary parts of this equation we find:
,
.
u
v
v
u
x
y
x
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

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