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
Exercise 6. Find the Laurent series of in