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I've recently discovered the Gamma Function. Upon graphing it in Desmos, I found something interesting: the minimum isn't at $x=0$, it's at $x\approx 1.4616$, with the value of $\Gamma(x)\approx 0.8856$. After some research here and on a forum from 2012 here, I've discovered that these numbers may or may not be irrational. However, something I can't seem to find is if either of these constant shows up anywhere else, like the golden ratio and $\pi$ do.

So what I'm asking is:

  • Why is the minimum of the gamma function this pair of numbers, and

  • do they appear anywhere else?

projectilemotion
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Feathercrown
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    As far as I know, the local minimum (or the corresponding zero of the digamma function) has no special significance outside this context. – pisco Apr 14 '17 at 12:20
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    see (https://math.stackexchange.com/q/1790371) and also (https://math.stackexchange.com/q/3245) – Jean Marie Apr 14 '17 at 12:32
  • @JeanMarie Funnily enough, I came across both of those trying to find an answer. – Feathercrown Apr 14 '17 at 13:06
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    Putting $1.461632144968362341$ into the ISC we get a match! But the description is simply "minimal x of GAMMA(x)" ... so even they do not know another description for it. (ISC = http://isc.carma.newcastle.edu.au) – GEdgar Apr 14 '17 at 13:19
  • @GEdgar Huh... maybe it really is unconnected from pretty much everything else. Note, however, that fibonacci spirals often appear on things like pinecones, but "pinecone spirals" isn't a mathematical relationship or even a specific value, so it wouldn't show up there. – Feathercrown Apr 14 '17 at 13:34
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    Yes, it looks de-cone-cted. – Jean Marie Apr 14 '17 at 14:35

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