= Holomorphic dependence of ordinary differential equations on parameters
{title2=$\partial_z f(z,\alpha)=A(z,f(z,\alpha),\alpha)$}
If the right-hand side and initial data are jointly <holomorphic>, the local solution is jointly <holomorphic> in its independent variable and parameters. On a smaller closed <polydisc>, bound $|A|$ by $M$ and $|\partial_w A|$ by $L$. Choose $r$ so that $rM$ stays inside the dependent-variable disc and $rL<1$. The integral <contraction mapping> has jointly <holomorphic> iterates converging uniformly, so <locally uniform convergence of holomorphic functions> proves the assertion. Shifting the dependent variable turns initial values into parameters.
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