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>, if a region <math> {\mathcal R}_3 </math> exists such
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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>. We say that <math> g(z) </math> is ''the analytic continuation'' of <math> f(z) </math>
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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 ==
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{{reflist}}

Latest revision as of 12:01, 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]

  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.