# What is the general form of a transfer function

I repeatedly see two representations of the general transfer function in the literature. The first is the following which is factorization of the numerator and denominator polynomials:

$$\textbf{G}(s) = k \frac{ (s - a_{1})(s - a_{2})(s - a_{3})\ldots }{ (s - b_{1})(s - b_{2})(s - b_{3})(s - b_{4})\ldots }$$

The alternative, below, eludes me and I'll checked many sources but none seem to justify it. The difference is a lowly $s$ term in the denominator raised to a power. Is this general form related the open-loop transfer function?

$$\textbf{G}(s) = k \frac{ (s - a_{1})(s - a_{2})(s - a_{3})\ldots }{s^m (s - b_{1})(s - b_{2})(s - b_{3})(s - b_{4})\ldots }$$

My question is what is the difference, if any in these two forms?

You may write the second equation on the first form by using $b_i = 0$ corresponding to those poles appearing at $s = 0$. Hence, the only difference is that in the second form you know that there are $m$ poles at $s=0$ while in the first form they may still be there, but one will have to check the values of $b_i$ to determine if they are there.
Note that a system with poles at $s=0$ is a marginally stable system and that the poles are typically introduced by integrators, which have a transfer function $1/s$.