# Analytic continuation

Here we follow Spiegel (1964) [1] or Levinson and Redheffer (1970). [2]

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 ${\displaystyle {\mathcal {R}}_{2}}$.

## References

1. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
2. Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.