Background #1
Here is a part of an answer of @Sankyu Kim in MathOverflow.
Consequently, we get the Euler-chi function $\chi(z):=\frac{\zeta(1-z)}{\zeta(z)}$.
And I want to know if Sankyu Kim's last expression is a closed form. (First assuming that its true though I failed to check the computation)
$$\Gamma(z)=\frac{(2\pi)^z}{2\cos(\frac{\pi z}2)}\chi(z)$$
This question is in context of this comment.
@ Sangkyu Kim: But the gamma function has no closed form.
If it is a closed form then he may have found the closed form of Gamma function.
Background #2
I am gathering information about this question's(about gamma function) solution.
What I know
$$\xi(z)=\xi(1-z)$$, where $$\xi(z):=π^{-\frac z2}\Gamma(\frac z2)\zeta (z)$$.
What I've tried to derive the closed form of gamma function
I used the equation above and derived the following.
$$π^\frac{2z-1}{2}\frac{\Gamma(\frac{1-z}{2})}{\Gamma(\frac z2)}\chi(z)=1$$
Further I applied $$\Gamma(z)=\frac{(2\pi)^z}{2\cos(\frac{\pi z}2)}\chi(z)$$ and it gave me function equation. It gave me:
$$\Gamma(z)=\frac{(2\pi)^z}{2\cos(\frac{\pi z}2)}\frac{\Gamma(\frac z2)}{\Gamma(\frac{1-z}2)}π^{-\frac{2z-1}2}$$
This is a functional equation about the gamma function.
I recently learned that Laplace transform could be used to solve some functional equations like fibonacci, so I tried to find the laplace transform of Gamma function.
I failed. (As it doesn't exist)
I just now googled to find the laplace transform of gamma function, but I can't find any. So I launched a new question.
If you know how to solve that functional equation, please let me know. Thanks!
Question:
Does $\chi (z)$ have a closed form?
How do I solve the functional equation above?
What I mean by closed form
a form that the value of the function can be evaluated with finite numbers of evaluation s of elementary functions
What I mean by elementary functions
Rational, exponential, logarithmic, trigonometric, etc.. (Normal(?) functions)