Department of
Mathematics
and Statistics
Summer Project Descriptions
Sarah
Gourlie, Geometric
patterns in complex power series, Summer 04
Faculty Advisor: Dan Kalman
One way to visualize the convergence of a complex power series is
to graph the successive terms vector fashion in the complex
plane. So, for
the series
a0 + a1 + a2
+ ...
draw a0 as a vector starting at the origin,
then draw a1 as a vector starting from the end
of a0, and so on. When this is done for
the power series 1 + z + z2
+ ... , the terms all depend on the selected value of z. This can
be viewed on a computer screen, with z varying as a mouse is dragged
across the complex plane. The results are visually quite
stunning. See some samples below.
The idea of this research project is to identify and classify various
types of geometric patterns that arise in this fashion, and to
determine the values of z that produce them.
Sarah developed computer activities to explore this topic using
software
called Mathwright. She identified and classified several
different types
of patterns, and then determined which values of z correspond to which
patterns.
Continuing her research in a capstone project for the honors program,
Sarah
extended her investigation to other infinite series. Sarah
presented the
results of her summer research at a regional meeting of the
Mathematical
Association of American, where she won an award for an outstanding
student
paper.