2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, 
BC, Canada.
Electronic Journal of Differential Equations,
Conference 12, 2005, pp. 39-46. 

Title: Holomorphic solutions to linear first-order
       functional differential equations

Authors: Bruce van Brunt (Massey Univ., New Zealand)
         Jonathan C. Marshall (Massey Univ., New Zealand)
 
Abstract: 
 In this paper we study holomorphic solutions to linear first-order
 functional differential equations that have a nonlinear functional
 argument.  We focus on the existence of local solutions at a fixed
 point of the functional argument and the holomorphic continuation of
 these solutions.  We show that the Julia set for the functional
 argument dominates not only the conditions for  holomorphic
 continuation, but also the existence of local solutions.
 In particular,  nonconstant holomorphic solutions in a neighbourhood
 of a repelling or neutral fixed point are uncommon in that the
 functional argument must satisfy conditions that force it to have an
 exceptional point in the former case, and a Siegel fixed point in the
 latter case.  In contrast,  local holomorphic solutions always exist
 near attracting fixed points.  In this case  a subset of the Julia set
 forms a natural boundary for holomorphic continuation.

Published April 20, 2005.
Math Subject Classifications: 30D05, 34M05, 37F10, 37F50.
Key Words: Complex functional differential equations; 
           pantograph equation.