# Difference between revisions of "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.
3. Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
4. Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
5. Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.