Analytic continuation

Here we follow Spiegel (1964) or Levinson and Redheffer (1970).

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