I'm trying to expand cosine function by using the Mittag-Leffler theorem which is introduced in Arfken 7ed.:
$$f(z)=f(0)\exp\left(\frac{zf'(0)}{f(0)}\right)\prod_{n=1}^{\infty}\left(1-\frac{z}{z_n}e^{z/z_n}\right) \tag{11.88}$$
Where $z_n$ is simple point.
First I found $z_n=\frac{2n-1}{2}\pi$.
And the result on the book was
$cosz=\prod_{n=1}^{\infty}(1-\frac{z^2}{(n-1/2)^2\pi^2})$
So I could guess $e^{z/z_n}$ was deformed as $1+\frac{z}{z_n}$.
But if we expand $e^{z/z_n}$, it becomes $1+\frac{z}{z_n}+\frac{1}{2!}(\frac{z}{z_n})^2+\frac{1}{3!}(\frac{z}{z_n})^3+\cdots$
Should I study the gamma function? I didn't study it yet.
I wonder why the higher order disappeared. Can you give me some advice?