Difference between revisions of "Analytic continuation"

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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
 
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>
+
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>.

Revision as of 12:33, 11 April 2016

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 .