# Difference between revisions of "Analytic continuation"

In complex analysis we may consider extending the domain of a given function ${\displaystyle f(z)}$ which is analytic in a region ${\displaystyle {\mathcal {R}}}$ by finding another function ${\displaystyle g(z)}$ analytic in a region ${\displaystyle {\mathcal {R}}_{2}}$. If a region ${\displaystyle {\mathcal {R}}_{3}}$ exists such that ${\displaystyle {\mathcal {R}}_{3}={\mathcal {R}}\cap {\mathcal {R}}_{2}}$ and if ${\displaystyle f(z)=g(z)}$ for all ${\displaystyle z\in {\mathcal {R}}_{3}}$ then we say that ${\displaystyle g(z)}$ is the analytic continuation of ${\displaystyle f(z)}$ into ${\displaystyle {\mathcal {R}}_{2}}$.