Exercise 1.  Find      for the following functions:

1 (i).   .

Solution 1 (i).

See text and/or instructor's solution manual.

Answer.   

Solution.   Use the fact that   and the substitution   to express this function in the following form

                      

The function    has an essential singularity at  ,  and must be expanded in a Laurent series.  

Here we have    and according to Definition 8.1

the residue is seen to be  .  

We are done.   

Aside.  We can let Mathematica double check our work.

Maple can check our work too!

     > series( exp(1+1/z), z=infinity,6 );

               

We are really done.   

Aside.   is capable of finding residues.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell