Holomorphic dependence of local ODE solutions

Statement

If $F(z,y,\lambda)$ is jointly holomorphic near $(z_0,y_0,\lambda_0)$ , the equation

\partial_z y=F(z,y,\lambda),\qquad y(z_0)=y_0

has a unique local solution jointly holomorphic in $z$ , initial state and $\lambda$ , on sufficiently small common neighborhoods.

Picard argument

Choose a complex product neighborhood with uniform bounds on $F$ and its state derivative. On a small time disc the Picard map preserves a ball and contracts it uniformly for all parameters in the chosen neighborhood. Its iterates are holomorphic and converge uniformly on smaller compact sets, so their limit is holomorphic. A version with real time, continuous time dependence and holomorphic state/parameter dependence follows by the same integral iteration. Extending a common parameter neighborhood along a longer trajectory requires uniform domain and coefficient control.

References

Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Theorem 4.2.