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32 1. Differential Equations and Their Solutions z0 is evidently z(t) = eitz0, and we recover the usual solution of the harmonic oscillator by taking the real part of this complex solution. But if you have had a standard course on complex function theory, then you can probably guess what the really important generalization should be. First of all, we should replace the time t by a complex variable τ, demand that the vector field V that occurs on the right- hand side of our equation dz dτ = V (z) is an analytic function of z, and look for analytic solutions z(τ). To simplify the notation, we will consider the case of a single equation, but everything works equally well for a system of equations dzi dτ = Vi(z1,... , zn). We shall also assume that V is an entire function (i.e., defined and analytic on all of C), but the generalization to the case that V is only defined in some simply connected region Ω ⊂ C presents little extra difficulty. Let us write H(Br, C) for the space of continuous, complex- valued functions defined on Br (the closed disk of radius r in C) that are analytic in the interior. Just as in the real case, we can define the map F = F V,z0 of H(Br, C) into itself by F (ζ)(τ) = z0 + τ 0 V (ζ(σ)) dσ. Note that by Cauchy’s Theorem the integral is well- defined, independent of the path joining 0 to τ, and since the indefinite integral of an analytic function is again analytic, F does indeed map H(Br, C) to itself. Clearly F (ζ)(0) = z0 and d dτ F (ζ)(τ) = V (ζ(τ)), so ζ ∈ H(Br, C) satisfies the initial value problem dz dτ = V (z), z(0) = z0 if and only if it is a fixed point of F V,z0 . The fact that a uniform limit of a sequence of analytic functions is again analytic implies that H(Br, C) is a complete metric space in the metric ρ(ζ1,ζ2) = ζ1 − ζ2 ∞ given by the “sup” norm, ζ ∞ = supτ∈B r |ζ(τ)|. We now have all the ingredients required to extend to this new setting the same Banach Contraction Principle argument used in Appendix B to prove the existence and uniqueness theorem in the real case. It follows that given z ∈ C, there is a neighborhood O of z and a posi- tive such that for each z0 ∈ O there is a unique ζz 0 ∈ H(B , C) that solves the initial value problem dz dτ = V (z), z(0) = z0. And the proof in Appendix F that solutions vary smoothly with the initial condition generalizes to show that ζz 0 is holomorphic in the initial condition z0.