**
1.
**
Rouche's Theorem with Applications
**
Rouche's Theorem.
**Let *f* and *g*
be holomorphic functions in a domain *D* in** **
C
containing the closed disk
. If
for
, then *f* and
*g* have the same number of zeros in
(note that the
hypothesis implies that *f* and *g* have no
zeros on the boundary of
).
For applications
the following variant of Rouche's Theorem is useful:
**
A variant of
Rouche's Theorem.
**Let *f* and
*h *be holomorphic functions in a domain *D* in**
**
C
containing the closed disk
. If
for
, then *f* and
*f+h* have the same number of zeros in
**
Example. **
Show that the equation
has exactly three distinct
solutions in
First, we apply Rouche's
Theorem on
Let
and
When
while
By Rouche's Theorem,
and
have the same number of zeros in
That is,
has 4 zeros
in
.
Next, we apply Rouche's
Theorem on
Let
and
When
while
By Rouche's Theorem,
and
have the same number of zeros
in
. That is,
has one zero in
.
Now we show that previously mentioned zeros lie on
If
then
On
, this inequality is impossible.
Combining we see that
has 3 zeros in
To show that
these zeros are distinct (i.e, that each zero in
is
simple), suppose
and
. Then
or
implying
that
is a simple zero. |