Recall the definition of the complex exponential. For z = x + iy,
z x+iy x x
e = e = e cosy+ e sin y
This is an analytic function because the Cauchy Riemann equations hold and also the partial derivatives of
the real and imaginary parts are continuous. Thus it has all derivatives. Recall that the derivative of
f
(z)
= u
(x,y)
+ iv
(x,y)
is u_{x}
(x,y)
+ iv_{x}
(x,y)
. In this case this gives
z
(e )
^{′} = e^{z} and so the usual rules
of differentiation hold. Indeed, this was how this function was considered in the first place.
Now consider the winding number. It was discussed earlier but here is the formal definition
again.
Definition 16.3.1Let γ :
[a,b]
→ ℂ and suppose z
∕∈
γ^{∗}. The winding number, n
(γ,z)
is definedby
1 ∫ dw
n (γ,z) ≡ 2πi w−-z-.
γ
The main interest is in the case where γ is closed curve. However, the same notation will be used for
any such curve.
Theorem 16.3.2Let γ :
[a,b]
→ ℂ be continuous and have bounded variation with γ
(a)
= γ
(b)
. Alsosuppose that z
∕∈
γ^{∗}. Define
∫
n (γ,z) ≡ -1- -dw--. (16.6)
2πi γ w− z
(16.6)
Then n
(γ,⋅)
is continuous and integer valued. Furthermore, there exists a sequence, η_{k} :
[a,b]
→ ℂ suchthat η_{k}is C^{1}
([a,b])
,
||ηk − γ || < 1-,ηk (a) = ηk(b) = γ(a) = γ (b),
k
and n
(ηk,z)
= n
(γ,z)
for all k large enough. Also n
(γ,⋅)
is constant on every connected component ofℂ∖γ^{∗}and equals zero on the unbounded component ofℂ∖γ^{∗}.
Proof: First consider the assertion about continuity.
|∫ ( ) |
|| --1-- --1--- ||
|n(γ,z)− n(γ,z1)| ≤ C | γ w − z − w − z1 dw|
^
≤ C (Length of γ )|z1 − z|
whenever z_{1} is close enough to z. This proves the continuity assertion. Note this did not depend on γ being
closed.
Next it is shown that for a closed curve the winding number equals an integer. To do so, use Theorem
4.2.3 to obtain η_{k}, a function in C^{1}
equals a constant. In particular, using the fact that η
(a)
= η
(b)
,
−g(b) −g(a)
e (η(b)− z) = e (η(a)− z) = (η(a)− z) = (η (b)− z)
and so e^{−g}
(b)
= 1. This happens if and only if −g
(b)
= 2mπi for some integer m. Therefore, 16.7
implies
∫ b ′ ∫
2m πi = -η(s)ds-= -dw--.
a η (s)− z η w− z
Therefore,
12πi-
∫_{ηk}
wdw−z-
is a sequence of integers converging to
21πi
∫_{γ}
wdw−z-
≡ n
(γ,z)
and so n
(γ,z)
must
also be an integer and n
(ηk,z)
= n
(γ,z)
for all k large enough.
Since n
(γ,⋅)
is continuous and integer valued, it follows from Corollary 2.7.15 on Page 201 that it must
be constant on every connected component of ℂ∖γ^{∗}. It is clear that n
(γ,z)
equals zero on the unbounded
component because from the formula,
( 1 )
lim |n (γ,z)| ≤ lim V (γ,[a,b]) ------
z→∞ |z|→ ∞ |z|− c
where c ≥ max
{|w| : w ∈ γ∗}
. ■
Corollary 16.3.3Suppose γ :
[a,b]
→ ℂ is a continuous bounded variation curve and n
(γ,z)
isan integer where z
∕∈
γ^{∗}. Then γ
(a)
= γ
(b)
. Also z → n
(γ,z)
for z
∕∈
γ^{∗}is continuous.
Proof: Letting η be a C^{1} curve for which η
(a)
= γ
(a)
and η
(b)
= γ
(b)
and which is close enough to γ
that n
(η,z)
= n
(γ,z)
, the argument is similar to the above. Let
∫ tη′(s)ds
g(t) ≡ η(s)−-z. (16.8)
a
(16.8)
Then
( ) ′
e−g(t)(η(t)− z) = e− g(t)η′(t) − e− g(t)g′(t)(η (t)− z)
− g(t) ′ − g(t)′
= e η (t) − e η (t) = 0.