# Math 113 Problem Set 7

Due October 31, 2003

1. 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 (1)

1. 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 .)

We can be a little more clever, by taking both endpoints of the sum simultaneously to infinity: (2)

1. 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 .
2. Evaluate the sum to show that, in fact, .
In summary, although is not the only function with a pole of residue 1 at each integer, it is in some sense the most natural one.
2. Show that has a pole at every integer, with a residue at of .
3. Bak and Newman, Chapt. 11, p. 147, problem 6.
4. Bak and Newman, Chapt. 11, p. 147, problem 5.
5. Bak and Newman, Chapt. 11, p. 147, problem 1(d).
6. Needham, p. 446, problem 3.
7. Needham, p. 446, problem 1.
8. (Optional) Needham, p. 447, problem 4.

Dylan Thurston 2003-10-27