# Toronto Math Forum

## APM346--2019 => APM346--Misc => Topic started by: Jingxuan Zhang on January 26, 2019, 10:29:57 AM

Title: analytic extension
Post by: Jingxuan Zhang on January 26, 2019, 10:29:57 AM
Suppose I have a absolutely convergent series solution of an ODE on the real line and I want to extend this to the whole plane. Naïvely I will just write the same formula, replaced with a complex variable. But to what extent is this justifiable?
Title: Re: analytic extension
Post by: Victor Ivrii on January 26, 2019, 11:31:38 AM
If series has an infinite radius of convergence then it converges on the whole plane. If the radius of convergence is finite ...

However, even if the radius of convergence is infinite, it does not answer to many questions. F.e. from the decomposition of $e^z$ one cannot derive that $e^x$ fast rapidly as $\mathbb{R}\ni x\to +\infty$ and rapidly decays as $\mathbb{R}\ni x\to -\infty$.