Factorials are the number of arrangements of things. Is there a similar interpretation for the Gamma function or is it just an extension of factorials? Does $\left(\frac{1}{2}\right)!$ mean anything?? I looked up on YouTube and browsed for a while, but all I see is a derivation of gamma and its relation to factorials. Thanks in advance.
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See for example this thread and "Is the Gamma function mis-defined?" – Raymond Manzoni Dec 02 '24 at 17:13
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@RaymondManzoni I'm sorry but I'm asking if there's any geometrical interpretation or a physical meaning for gamma. Like how 3 factorial means numbers of way 3 items can be arranged . – Pranay Varanasi Dec 02 '24 at 17:21
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There is a graphical interpretation and it’s related to the volume of an Nth ball https://en.m.wikipedia.org/wiki/Volume_of_an_n-ball – Aderinsola Joshua Dec 02 '24 at 17:33
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There isn't a physical interpretation of the generalization. But that is common in Mathematics. For example, what is the interpretation of $x^{1/2}$? What do you mean by "Multiply $x$ by itself one-half times"? Does that also bothers you? – jjagmath Dec 02 '24 at 17:33
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The Gamma function is not just an extension of factorials. Perhaps the answers to this quesiton on construction of the gamma function will help you to more fully appreciate it. – Steven Clark Dec 02 '24 at 20:42