An Exact Perturbative Existence and Uniqueness Theorem

We investigate singularly perturbed nonlinear complex differential systems of the form $\mathrm{\hslash }{\mathrm{\partial }}_{x}f=F(x,\mathrm{\hslash },f)$ where $\mathrm{\hslash }$ is a small complex perturbation parameter. Under a geometric assumption on the eigenvalues of the Jacobian matrix of $F$, we prove an Existence and Uniqueness Theorem for exact perturbative solutions; i.e., holomorphic solutions with prescribed perturbative expansions in $\mathrm{\hslash }$. In fact, these solutions are the Borel resummation of the formal perturbative solutions.