16.3 The Complex Exponential and Winding Number
Recall the definition of the complex exponential. For z = x + iy,
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
In this case this gives
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
Definition 16.3.1 Let γ :
→ ℂ and suppose z
γ∗. The winding number, n
The main interest is in the case where γ is closed curve. However, the same notation will be used for
any such curve.
Theorem 16.3.2 Let γ :
→ ℂ be continuous and have bounded variation with γ
suppose that z
is continuous and integer valued. Furthermore, there exists a sequence, ηk
→ ℂ such
that ηk is C1
for all k large enough. Also n
is constant on every connected component of
ℂ∖γ∗ and equals zero on the unbounded component of ℂ∖γ∗.
Proof: First consider the assertion about continuity.
is close enough to z.
This proves the continuity assertion. Note this did not depend on γ
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 ηk, a function in C1
large enough, ηk
= a,b, ηk
converges uniformly to γ
It is shown that each of
is an integer. To simplify the notation, write
It follows that e−g
equals a constant. In particular, using the fact that
and so e−g
= 1. This happens if and only if −g
for some integer m.
is a sequence of integers converging to
also be an integer and
is continuous and integer valued, it follows from Corollary
on Page 201
that it must
be constant on every connected component of ℂ∖γ∗.
It is clear that n
equals zero on the unbounded
component because from the formula,
where c ≥ max
Corollary 16.3.3 Suppose γ :
→ ℂ is a continuous bounded variation curve and n
an integer where z
γ∗. Then γ
. Also z → n
γ∗ is continuous.
Proof: Letting η be a C1 curve for which η
and which is close enough to
the argument is similar to the above. Let
for some integer, m. Therefore, from 16.9
Thus c = η
and letting t
which shows η
This proves the corollary since the assertion about continuity was already