### Author Topic: analytic extension  (Read 562 times)

#### Jingxuan Zhang

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##### analytic extension
« 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?

#### Victor Ivrii

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 $\mathbb{R}\ni x\to -\infty$.