27. Trigonometric representation of Gamma function

Figure 1. Graph (z-n)! when n=1(red), 2(blue)

 

- This article was written long time ago. For more accurate article, refer to article 30 -

 

$\gamma(z)$ can be defined like this.

 

(1)   $\Gamma(z)=(z-1)!=\gamma(z)+m, m$: integer.

        $\gamma(z)=(z-1)!-m$

 

The reason that $m$ is added is to divide into integer part and $\gamma(z)$ whose real part is less than $1$. 

 

Using the relation $z-n=(-1)^{n+1}\cos^{n+1}\frac{\pi (z-n+1)}{2((z-n)!+1)}$, which is previously assumed,

 

(2)   $(z-n)!=\frac{\pi (z-n+1)}{2\cos^{-1}(-(z-n)^{1/(n+1)})}-1 , |\Re(z-n)|<1, n$: natural number

 

        $z!=z, \Re(z) \leq 1$

 

It is possible to define $(z-n)!$ in order that $\gamma(z)=(z-n+1)/((z-n)!+1)$ or the other form. However, it is necessary to look more deeply at how it is related to the $\zeta(s)$ or $\Gamma(s)$. As you see figure 1, when the order of the exponent of the $\cos$ function is odd, it must be negative to create a smooth curve at the point where the real axis is a natural number.

 

 

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