### A Note on the Alexander Theorem on the Complex Plane

We investigate the Banach manifold consisting of complex ${}^{r}$ functions on the unit disc having boundary values in a given one-dimensional submanifold of the plane. We show that ∂/∂λ̅ restricted to that submanifold is a Fredholm mapping. Moreover, for any such function we obtain a relation between its homotopy class and the Fredholm index.