Timeline for Example of continuous function that is analytic on the interior but cannot be analytically continued?
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Jul 30, 2014 at 0:07 | comment | added | XL _At_Here_There | @engelbrekt，you mean $h(z)$ is without unit circle as natural boundary? | |
| Jan 6, 2010 at 3:51 | comment | added | engelbrekt | The advantage of this example is the ease of showing rigorously that the function blows up at the roots of unity of form $\exp(2{\pi}ik/2^m)$, due to the functional equation. It is not obvious that this happens at other roots of unity; you would have to establish an estimate to exclude that there is not some weird kind of cancellation happening. And in fact, weird cancellation can happen! The function $h(z) = (1 - z)^{-1}(1 - z^2)^{-1}(1 - z^3)^{-1}{\cdots}$ blows up (radial approach) very fast at every root of unity. At most other points of the unit circle, it actually tends to zero. | |
| Jan 5, 2010 at 21:20 | comment | added | Anweshi | Why isn't the following used: $g(z)$ has the unit circle as natural boundary, because its derivative $f(z)$ blows up at all roots of unity and has the unit circle as the natural boundary. | |
| Jan 5, 2010 at 21:11 | history | answered | engelbrekt | CC BY-SA 2.5 |