Scenario

Sessions

 
 

Step 1 :

Power series in a complex variable and their calculus

We introduce the definition of  a power series and, as usual, define operations like sum, product (Cauchy product), composition without issues of convergence yet.

Step 2:

Convergence of power series in a complex variable

Here we turn to the issues of convergence.  We define the radius of convergence which results in a disk of convergence.  It is noteworthy for the student to know the critical difference from real analysis regarding the interval of convergence which, in complex variables, is very hard to handle.

Step 3:

Development of tests for the radius of convergence

We introduce the Cauchy-Hadamard Rule for determining the radius of convergence of a complex power series and then look at some of the delicate care that we should take in this context.  An outcome of this examination requires studying the Cauchy root test and d’Alembert ratio test in complex form.