In this paper, we study a class of linear functional differential equations of which the pantograph equation is a prominent member. Specifically, we study the existence of solutions holomorphic at a fixed point of the functional argument. The local theory for equations with attracting fixed points is known [17, 13], but little is known about the case where the fixed point is repelling. We formulate an eigenvalue problem for the repelling fixed point case and show that the corresponding spectrum is discrete. Hence, that holomorphic solutions occur only as special cases. The second order pantograph equation is used to illustrate this result. A key step in this process is to reformulate the problem in terms of a compact operator. Aside from exploiting well known results for the spectra of such operators, we use results such as the Fredholm Alternative to derive existence results for the non-homogeneous problem.