Timeline for Asymptotic approximation of $x^\alpha$ by entire functions
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 28, 2010 at 17:45 | answer | added | Pietro Majer | timeline score: 25 | |
| May 28, 2010 at 14:23 | vote | accept | Roland Bacher | ||
| May 28, 2010 at 12:59 | vote | accept | Roland Bacher | ||
| May 28, 2010 at 14:09 | |||||
| May 28, 2010 at 12:59 | vote | accept | Roland Bacher | ||
| May 28, 2010 at 12:59 | |||||
| May 28, 2010 at 10:28 | comment | added | Wadim Zudilin | Casorati-Weierstrass: If $f$ has an essential singularity at $a$, then the image under $f$ of any punctured disk around $a$ is dense in $\mathbb C$. Use for $a=\infty$. Take an entire function $f(z)$ and consider the entire $g(z)=zf(z)^2$; there exists a direction $\lambda$ along which $g(z)\to C\ne0$ as $z\to\infty$. Then $f(x)=c_0\sqrt{x}g(x/\lambda)$ will give an example with $\alpha=-1/2$. | |
| May 28, 2010 at 10:11 | comment | added | Wadim Zudilin | Looking at the example of $1/\Gamma(1+x)\sim (e/x)^x(2\pi x)^{-1/2}$ as $x\to+\infty$, I would say "yes" but definitely a maitre in complex analysis is wanted. :) | |
| May 28, 2010 at 9:22 | answer | added | coudy | timeline score: 16 | |
| May 28, 2010 at 9:20 | comment | added | Roland Bacher | Existence, yes or no, and if yes, an example, of an entire function $h$ such that $x^{-\alpha}h(x)\rightarrow 1$ for $x\rightarrow+\infty$ with $x$ real. | |
| May 28, 2010 at 9:05 | comment | added | Per Alexandersson | What is the question? | |
| May 28, 2010 at 8:56 | history | asked | Roland Bacher | CC BY-SA 2.5 |