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Here are closed formulas for many points of the gamma function.

Is there a formula for $\Gamma{(\frac{3}{4})}$ ?

Is it related to any known geometric concept?

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    $\begingroup$ You have $\Gamma(\frac{1}{4})\Gamma(\frac{3}{4})=\pi\csc(\frac{pi}{4})=\pi\sqrt{2}$ and there are relations between $\Gamma(\frac{1}{4})$ and the lemniscatic elliptic curve. Honestly, "closed formula" isn't really that well-defined of a term; if you allow theta functions and the like, I think you can get one for $\Gamma(\frac{3}{4})$; if you don't, you probably can't. $\endgroup$
    – W. Cadegan-Schlieper
    Jan 23, 2017 at 0:53
  • $\begingroup$ Thanks W. Schlieper. I would mark your comment as answer if you write it as answer. $\endgroup$
    – kiyopoviy
    Jan 23, 2017 at 0:59
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    $\begingroup$ Wolfram Alpha gives several expressions for $\Gamma(1/4)$ but none of them are closed.. $\endgroup$
    – Henricus V.
    Jan 23, 2017 at 1:49
  • $\begingroup$ @HenryW. I love when that happens! XD $\endgroup$
    – Simply Beautiful Art
    Jan 23, 2017 at 1:50

2 Answers 2

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First of all, "closed form" is not really a well-defined concept; the question, to some extent, is "what functions do we need to use to describe this value?" Most of the time the choice for "closed form" or "elementary function" will basically be exponentials, logarithms, and polynomials (and/or solutions); when complex numbers are allowed this includes descriptions like $\pi = 4 \arctan(1)$ which is equivalent to $\pi i = 4 \ln(1+i)$ (which, itself, is perhaps most fundamentally that $2 \pi i$ is the period of the natural exponential function)

In this particular case, such a formula for $\Gamma(\frac{3}{4})$ is unknown and unlikely but, I believe, not shown not to exist. As far as $\Gamma(s)$, formulas in general will focus on real numbers $0 < s < \frac{1}{2}$ because of the defining relation between $s$ and $s+1$ and the formula $\Gamma(s)\Gamma(1-s)=\pi \csc(\pi s)$, so someone interested in $\Gamma(\frac{3}{4})$ should really look into $\Gamma(\frac{1}{4})$, since any formula for one will produce a similar formula for the other.

The particular case $\Gamma(\frac{1}{4})$ turns out to be connected to the elliptic curve $y^2=x^3-ax$ (or equivalently, $\mathbb{C}/(\mathbb{Z}+i\mathbb{Z})$, the square torus). The algebraic-geometric mean equations in the linked article can be related to that through the elliptic integral, though my favorite is probably the one using the Dedekind eta function:

$$\Gamma\left(\frac{1}{4} \right)=2\pi^{3/4}\eta(i)$$

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  • $\begingroup$ :-( Now I have to go find a favorite form... $\endgroup$
    – Simply Beautiful Art
    Jan 23, 2017 at 1:54
  • $\begingroup$ I'd want to say that the fixation on "elementary functions/constants" and other sets is just that: a fixation, similar to the idea that the decimal expansion of a real is somehow the "true" expression thereof. There are many, many ways to represent most mathematical objects and, I think, what people really are after with these kinds of questions is "how to best understand this object and/or what forms are the most illuminating?" And the latter question requires a specification of a purpose, as what might be useful in one situation may not in another. $\endgroup$
    – The_Sympathizer
    May 10, 2019 at 8:48
  • $\begingroup$ Moreover, an expression using "non-elementary" special functions can actually in some cases, be neater and more concise, if not even more meaningful, than that using elementary functions even when the latter expressions exist. $\endgroup$
    – The_Sympathizer
    May 10, 2019 at 8:53
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If you check this answer out, you will find many different forms for $\Gamma(5/4)$. One then notes that

$$\Gamma(3/4)=\frac\pi{\Gamma(1/4)\sin(3\pi/4)}=\frac\pi{4\Gamma(5/4)\sin(3\pi/4)}$$

and just substitute any form of $\Gamma(5/4)$ you desire.

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