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. |