Timeline for Who achieved the analytic continuation of the Gamma function?
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Mar 16, 2019 at 2:31 | comment | added | KCd | It is not really that hard. Using integration by parts, for ${\rm Re}(x) > 0$ we have $\Gamma(x+1) = x\Gamma(x)$, so $\Gamma(x) = \Gamma(x+1)/x$. The last equation serves to define $\Gamma(x)$ when ${\rm Re}(x) > -1$ except at $x = 0$. Then that same equation serves to define $\Gamma(x)$ when ${\rm Re}(x) > -2$ except at $x = -1$, and so on. We get analytic continuation of $\Gamma(x)$ to $\mathbf C$ except at $x = 0, -1, -2, -3, \ldots$ where there are simple poles. This approach to analytic (really, meromorphic) continuation is in many complex analysis books. | |
| Mar 13, 2019 at 22:56 | vote | accept | John | ||
| Mar 13, 2019 at 14:09 | answer | added | Alexandre Eremenko | timeline score: 9 | |
| Mar 13, 2019 at 13:27 | history | asked | John | CC BY-SA 4.0 |