Closed form expression for the gamma function for complex numbers $s=a+ib$ with $b≠0$

Question

Let us consider the famous gamma function.

My question is:

Is it possible to derive a closed expression for the gamma function for complex numbers $s=a+ib$ with $a≠1/2$, $b≠0$.

I know that such a closed form exists for integer values of $x=n$ for which $G(n)=(n-1)!$. Also, some non integers values have a closed form. Also if $a=1/2$ then a closed form exists. See this link:

https://mathoverflow.net/questions/112682/riemann-siegel-function-and-gamma-function

You may find expressions for the module for $a=0, a=\frac 12$ or $a=1$ here ($(6.1.29)$ and more) (btw your link shows rather a relation of the argument of zeta with $\zeta(1/2+it)$ : $\Gamma$ is the 'simpler' function!). — Raymond Manzoni, Dec 23 '12 at 09:16
You are welcome ! (sorry for the typos 'module' $\to$ modulus and the MO link gives a relation of the argument of zeta with $\Gamma(1/2+it)$ : zeta is the complicated function and the relation just shows that its argument is simple since it may be written as a function of $\Gamma$ and other elementary functions). $\Gamma$ itself has only trivial closed forms as opposed to the derivative of $\log(\Gamma)$ i.e. the digamma or $\psi$ function that admits a closed form at every rational value. — Raymond Manzoni, Dec 23 '12 at 10:14
As hinted by Nameless, a closed form containing only elementary functions is impossible. — John2000, Dec 23 '12 at 10:25
I don't know of an impossibility proof. Proofs exist that the gamma function can't be written as an elementary function but this doesn't exclude specific closed forms as at $\frac 12$. — Raymond Manzoni, Dec 23 '12 at 10:32
Can you give a link to that proof. "Proofs exist that the gamma function can't be written as an elementary function" — John2000, Dec 23 '12 at 10:35
The closest thing I could find is the Hölder theorem from the discussion here. See too the more recent paper1 in french and paper2. — Raymond Manzoni, Dec 23 '12 at 11:03

Nameless · Accepted Answer · 2012-12-23T09:34:14.990

3

The $\Gamma$ function lacks a closed form containing only elementary functions. Here are the equivalent formulas of the $\Gamma$ function however:

$$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$$ $$\Gamma(z)=\frac1z\prod_{n=1}^{\infty}\frac{(1+\frac1n)^z}{1+\frac zn}$$ $$\Gamma(z)=\frac{e^{-\gamma z}}{z}\prod_{n=1}^{\infty}\frac{e^{\frac zn}}{1+\frac zn}$$ where $\gamma$ is the Euler–Mascheroni constant. Of course we can relate the $\Gamma$ function with elementary functions via the following indentity: $$\Gamma(1-z)\Gamma(z)=\frac{\pi}{\sin \pi z}$$ We also have Riemann's functional equation $$\zeta(s)=2^s\pi^{s-1}\sin\frac{\pi s}2\Gamma(1-s)\zeta(1-s)$$ which relates the $\Gamma$ and the $\zeta$ functions.

If by closed form we mean an expression containing only elementary functions then no, $\Gamma$ has no such form. For more information read this

edited Dec 23 '12 at 09:34

answered Dec 23 '12 at 09:16

Nameless

13,766

I mean this case: $$\Gamma(s)=f(a,b)$$ such that the expression of $f$ is well know. I find one but I want to use it as contraduction in a proof. – John2000 Dec 23 '12 at 09:19
The expression of $f$ is given in term of $a$ and $b$ in an explicit manner. – John2000 Dec 23 '12 at 09:21
I find this explicit formula: $arg(Γ(α+iβ))=λ(α,β)-σ(α,β)$ with $λ(α,β)$ and $σ(α,β)$ are given in term of $cos,sinh,sin, cosh$. – John2000 Dec 23 '12 at 09:29
1

@user53124 And so? Even if $\arg \Gamma$ is elementary, $\Gamma$ is not elementary so any attempt to find such a form wil be futile. – Nameless Dec 23 '12 at 09:31
"If by closed form we mean an expression containing only elementary functions then no, $\Gamma$ has no such form" : does not exists or impossible. – John2000 Dec 23 '12 at 09:31
So please indicate to me steps of the proof. Thank you very much. – John2000 Dec 23 '12 at 09:39
You want a proof of that? I think it would be overly complicated to write it here. – Nameless Dec 23 '12 at 09:42
I can give you my email. – John2000 Dec 23 '12 at 09:43
@user53124 You won't have to. When I find a link with the proof I will post it here – Nameless Dec 23 '12 at 09:45
Ok, and thank you very much for your kind help. – John2000 Dec 23 '12 at 09:47
I think the last link is dead... :( – Hyeonseo Yang Sep 27 '18 at 16:08

Closed form expression for the gamma function for complex numbers $s=a+ib$ with $b≠0$

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