Timeline for smooth families of analytic functions
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
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| Aug 23, 2012 at 16:32 | answer | added | Bazin | timeline score: 1 | |
| Jun 20, 2012 at 13:27 | comment | added | Jochen Wengenroth | For each $x$, the Taylor series has infinite radius of convergence and you can extend $f$ to a function on $\mathbb R \times \mathbb C$ which is holomorphic in the second variable. If this function is differentiable w.r.t. $x$ you can use Cauchy's formula $$f(x,z) = \frac{1}{2\pi i} \int_S \frac{f(x,\xi)}{\xi-z} d\xi$$ (where $S$ is a circle around $0$ with radius $>|z|$) to differentiate with respect to $x$ under the integral. But I do not think that the extended function is always differentiable w.r.t. $x$. Thm 1.2.6 of Hörmander (Anal. part. Diff. Op. I) ponits in this direction. | |
| Jun 20, 2012 at 11:54 | history | asked | Florian | CC BY-SA 3.0 |