This isn’t a analysis stage query. However as a consequence of some purpose I can not ask this query on Math Stack Change. So, I’m asking this query right here.

By definition we all know that we are able to measure the multiplicity of a root of a operate $f(z)$ as comply with.

If the a root of some operate $f(z)$ is $alpha_i$, then the multiplicity of that root is $a_i$ for which $lim limits_{x to alpha_i} |frac{f(z)}{(z-alpha_i)^{a_i}}|$ has a finite worth.

So, following this we are able to see that every one the roots ($(2n+1)ipi$,
) of the polynomial $e^z+1$ has multiplicity $1$.

So, $$e^z+1= 2prod_{n=-infty}^{infty} left(1-frac{z}{(2n+1)ipi}proper)$$…..(1)

Which additional could be written as

$$e^z+1= 2prod_{n=0}^{infty} left(1+frac{z^2}{((2n+1)ipi)^2}proper)$$

However this solely consists of $z^{2k}$ not the odd powers.

If we immediately examine this from (1) we see the coefficients of $z^3$, we see they’re $0$ which could be immediately concluded from (1).

However really $e^z+1=2+z+frac{z^2}{2!}+….$. So, what am I doing unsuitable?