  For a power series which converges in some neighborhood of the origin in the
complex plane, it turns out that the zeros of its partial sums---its
sections---often behave in a controlled manner, producing intricate patterns as
they converge and disperse. We open this thesis with an overview of some of the
major results in the study of this phenomenon in the past century, focusing on
recent developments which build on the theme of asymptotic analysis. Inspired
by this work, we derive results concerning the asymptotic behavior of the zeros
of partial sums of power series for entire functions defined by exponential
integrals of a certain type. Most of the zeros of the n'th partial sum travel
outwards from the origin at a rate comparable to n, so we rescale the variable
by n and explicitly calculate the limit curves of these normalized zeros. We
discover that the zeros' asymptotic behavior depends on the order of the
critical points of the integrand in the aforementioned exponential integral.
  Special cases of the exponential integral functions we study include classes
of confluent hypergeometric functions and Bessel functions. Prior to this
thesis, the latter have not been specifically studied in this context.
