Natural boundaries of solutions to a class of functional differential equations
by
Jonathan Marshall
Institute of Fundamental Sciences, Massey University

The functional differential equation y'+ay+by( \alphaz)=0 has been studied extensively by many people, and is known to possess entire solutions if | \alpha| < 1, and have no solutions holomorphic at the origin if |\alpha| > 1. In this talk we generalise this situation to functional differential equations of the form

y''+ay'+by=cy(g(z)),

where a, b, c are constants, c =/= 0, and g(z) is a non-constant polynomial. We find that similar results can be found as to the existence of holomorphic solutions about a fixed point, but that these solutions are no longer entire, and in fact they have natural boundaries that are dependent only on the functional argument g.

Date received: October 18, 2000