Due October 31, 2003

- In class we found that the function
has a pole at
every integer with residue 1. One natural approach to constructing
such a function is to consider a sum over the integers of a function
with a pole at : consider

- Show that this approach does not work, by showing that the sum
above diverges for all
.
(Recall that a sum converges if and only if exists, which in turn is true if and only if there is a value so that for every , there are and so that, for and , is within of the limiting value .)

- Show that the sum in (2) converges, and that the result is an analytic function with a pole at every integer with residue 1 which satisfies .
- Evaluate the sum to show that, in fact, .

- Show that this approach does not work, by showing that the sum
above diverges for all
.
- Show that has a pole at every integer, with a residue at of .
- Bak and Newman, Chapt. 11, p. 147, problem 6.
- Bak and Newman, Chapt. 11, p. 147, problem 5.
- Bak and Newman, Chapt. 11, p. 147, problem 1(d).
- Needham, p. 446, problem 3.
- Needham, p. 446, problem 1.
- (Optional) Needham, p. 447, problem 4.

Dylan Thurston 2003-10-27