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In my studymaterial I see different definitions of the Gamma function In Mathematica it is for example:

Gamma[z] == 1/z Product[(1 + 1/k)^z/(1 + z/k),
                        {k, 1, Infinity}]/; Not[Element[-z, Integers]\[And] -z >= 0]

( z={0,1,2,..} Is there an easy way to get (derive) some other forms of the Gamma function here defined ? Hoping that one of the forms looks the same as in my studymaterial.

Example: $Pi(s) = \prod _{n=1}^{\infty } \frac{n^{1-s} (n+1)^s}{n+s}$

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2 Answers 2

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There is MathematicalFunctionData which can be exploited to get various representations of the Euler Gamma function. As a starting point one can evaluate e.g.

MathematicalFunctionData["Properties"]

to get a perspective what kind of properties one can search for, in case of Gamma it works like e.g.

MathematicalFunctionData["Gamma", "AlternativeRepresentations"]

or even more informative

MathematicalFunctionData["Gamma", "NamedIdentities"][[19 ;; 23]] //  
TraditionalForm

enter image description here

A kind of convenient browser of mathematical properties one can get evaluating

Manipulate[ Entity["MathematicalFunction", "Gamma"][z],
            {z, Entity["MathematicalFunction", "Gamma"]["Properties"]}]
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  • $\begingroup$ Thanks, getting information in the traditional form can make searching for my formula easier. $\endgroup$
    – janhardo
    Commented Feb 19, 2022 at 19:24
  • $\begingroup$ @janhardo You are welcome! You have also link to Wolfram functions MathematicalFunctionData["Gamma", "WolframFunctionsSiteLink"]. Nontheless I find the browser with Manipulate a more convenient way of exploiting built-in mathematical data . $\endgroup$
    – Artes
    Commented Feb 19, 2022 at 19:31
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Use Entity["MathematicalFunction", "Gamma"]

(prodRep = #[z] & /@
     Most[Entity["MathematicalFunction", "Gamma"][
       "ProductRepresentations"]]) // Column[#, Frame -> All] & //
 TraditionalForm

enter image description here

Also look at Entity["MathematicalFunction", "Gamma"]["Properties"]

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  • $\begingroup$ Thanks, Can also use this to perform a targeted search and put the result in a box in the desired output..handy $\endgroup$
    – janhardo
    Commented Feb 19, 2022 at 19:40

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