be holomorphic on some open set
of
.
Let
such that
. Show that if
is a
circle of center
and small enough radius then:
is a continuous function on
such that, for each
,
the function
is holomorphic. Show that the following
function is holomorphic:
then
be holomorphic on some open set
containing the
closed disk
of center 0 and radius
. Let
be the zeroes of
in the open disc, each zero being
repeated according to its multiplicity. Prove that
Hint: apply the maximum principle to the function
be the set of complex numbers
with
real and
. Suppose that
is holomorphic on
and
such that
for all
. Show that
has an expansion of the form
where
for any
. Hint: regard
as a function of
.
Now suppose that there is
such that for
the
function
is uniformly bounded. Show that
for
.
and the Bernouilli numbers
by
Thus
,
and
for
.
Express the coefficients of the power series expansion (at
) of
the function
in terms of the Bernouilli numbers. On the other hand, recall the formula
Use the logarithmic derivative of this identity to obtain another
power series expansion for
and conclude that
