The Uniqueness Theorem

The uniqueness theorem makes functions well defined. It states

Letandbe analytic on a regionand suppose thatagrees withon a setwherehas a limit point inthenon

Ifhas a limit point inthen there are an infinite number of distinct points inwhere

Example: Ifandprove that

Letbe an analytic function onsuch that

The setand has a limit point

Furthermore ifthen

so thaton

All the conditions of the uniqueness theorem are satisfied so thaton