be
a complex parameter. Exercise 1. Let
be
a complex parameter.
i) Show that the Laurent series
expansion of the function exp(
)
at the origin 
has
the form
exp()=
for
with
for
Similarly, show that the
Laurent series expansion of the function exp(
)
at the origin has
the form
exp()=
for
with
for
(Notice that if
’=-
then
exp(
)=exp()
for 
ii) Let
and
be
two nonnegative integers. Show that
if
with
being
a nonnegative integer and 
otherwise,
and deduce the expansions in power series of
and
as
functions of the parameter ( ,
as a function of 
carries
the name of a Bessel function of the first kind).
Exercise 2. Let
be
a meromorphic function in a neighborhood of the origin
and
having a simple pole at the origin. Let
be
an arbitrary complex number. Show that the Laurent expansion of the function
has
the form
-
where
is
a polynomial in of
degree 
(One
can make an identification by using the Taylor expansion of the function
Exercise 3. Let
be
a holomorphic function in the upper half plane
defined
by 
suppose
for
Show
that there exists a holomorphic function
on
the punctured disk 
such
that 
for
Deduce
that has
the expansion
with
for
some 
.
Show that this series
converges uniformly on each compact subset of
Show
that if there exists a constant 
and
an integer 
such
that
for
all sufficiently large
,
and uniformly in then
Exercise 4. i) Show
that the function 
is
meromorphic on the whole complex plane and has the points
as
simple poles. Find the Laurent expansion at the point
If
(
)
denote the coefficients of the expansion for
show
that 
for
an integer 
and
if we put
=(
)
(
)
! 
for
show
that we have the recurrence formula:
-
+
for
(by
identification of the coefficients on the two sides of the relation
).
ii) For
put
Let
be
the perimeter of a square having vertices
Show
that if is
on ,
then 
Integrating
along
the contour in
the direct sense and making 
deduce
that
(N.B. The numbers
are
called the Bernoulli numbers)
Exercise 5.
Find the Laurent series of
in
a)
b)
Exercise 6.
Find the Laurent series of 
in
a)
b)
c)
d)
