## Holomorphic solutions to linear first-order functional differential equations

### Summary

Summary: 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.

### Mathematics Subject Classification

30D05, 34M05, 37F10, 37F50

### Keywords/Phrases

complex functional differential equations, pantograph equation