## Analytic Continuation

We start with a

Definition Letandbe analytic functions whose domains are the regionsand respectively.andare direct analytic continuations of each other if there is a regionsuch thatfor

Example:is only defined forbut we may writeon the region defined bybutis defined on the wider regionis an analytic continuation of

Example:is normally defined on the regionso that

We in fact only need Log to be one to one on the complex plane so that the inverse is defined. We can instead define Log-1 z on so thatforand we have extendedto be defined on