# Difference between revisions of "Analytic continuation"

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+ | Here we follow Spiegel (1964) <ref> Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).</ref> or Levinson and Redheffer (1970). <ref> Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.</ref> | ||

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In [[Dictionary:analytic Function|complex analysis]] we may consider extending the domain of a given function <math> f(z) </math> which is [[Dictionary:analytic Function|analytic]] in a region <math> \mathcal R </math> | In [[Dictionary:analytic Function|complex analysis]] we may consider extending the domain of a given function <math> f(z) </math> which is [[Dictionary:analytic Function|analytic]] in a region <math> \mathcal R </math> | ||

− | by finding another function <math> g(z) </math> analytic in a region <math> {\mathcal R}_2 </math> | + | by finding another function <math> g(z) </math> analytic in a region <math> {\mathcal R}_2 </math>. If a region <math> {\mathcal R}_3 </math> exists such |

− | that <math> {\mathcal R}_3 = {\mathcal R} \cap {\mathcal R}_2 </math> and if <math> f(z) = g(z) </math> for all <math> z \in {\mathcal R}_3 </math> then we say that <math> g(z) </math> is ''the analytic continuation'' of <math> f(z) </math | + | that <math> {\mathcal R}_3 = {\mathcal R} \cap {\mathcal R}_2 </math> and if <math> f(z) = g(z) </math> for all <math> z \in {\mathcal R}_3 </math> then we say that <math> g(z) </math> is ''the analytic continuation'' of <math> f(z) </math> |

into <math> {\mathcal R}_2 </math>. | into <math> {\mathcal R}_2 </math>. | ||

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+ | == References == | ||

+ | {{reflist}} |

## Latest revision as of 13:01, 11 April 2016

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 which is analytic in a region
by finding another function analytic in a region . If a region exists such
that and if for all then we say that is *the analytic continuation* of
into .

## References

^{[3]}

^{[4]}

^{[5]}

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