# example of analytic continuation

The function defined by

 $f(z):=\sum_{n=0}^{\infty}z^{n}=1+z+z^{2}+\ldots$

is, as a sum of power series, analytic in the disc of convergence  $D=\{z\in\mathbb{C}\,\vdots\;\;|z|<1\}$.  The function

 $g(z):=\frac{1}{1-i}\sum_{n=0}^{\infty}\left(\frac{z-i}{1-i}\right)^{n}=\frac{1% }{1-i}+\frac{z-i}{(1-i)^{2}}+\frac{(z-i)^{2}}{(1-i)^{3}}+\ldots$

similarly is analytic in the bigger disc  $E=\{z\in\mathbb{C}\,\vdots\;\;|z-i|<\sqrt{2}\}$.  But we have

 $f(z)=\frac{1}{1-z}\quad\mathrm{in}\;D,$
 $g(z)=\frac{1}{1-i}\cdot\frac{1}{1-\frac{z-i}{1-i}}=\frac{1}{1-z}\quad\mathrm{% in}\;E;$

thus $f(z)$ and $g(z)$ coincide in the intersection domain $D\cap E$.  So we can say that $g(z)$ is the analytic continuation of $f(z)$ to the domain  $E\!\smallsetminus\!D$.  It is clear that $\frac{1}{1-z}$ is the analytic continuation of $f(z)$ to the domain $\mathbb{C}\!\smallsetminus\!\{1\}$.

Title example of analytic continuation ExampleOfAnalyticContinuation 2013-03-22 16:52:06 2013-03-22 16:52:06 pahio (2872) pahio (2872) 5 pahio (2872) Example msc 30B40 msc 30A99 RadiusOfConvergence GeometricSeries SetDifference