A question on the laplace transform.

Question

please tell me what is a zeros at infinity? That is the Laplace transform of a function which has a zeros is $\infty.$

For example: Ploles and zeros of a function are given, find the function. Simple poles: 0, -2, poles of order: -3, zeros: -1, $\infty$

I can't understand what a zeros at $\infty$ is? please show me!

Answer 1

Assume a rational Laplace transform $H(s)$. There are cases that when the degree of the denomerator is higher than the numerator, $H(s)$ tends to zero by tending $s$ to infinity. Hence, it is said that $H(s)$ has a zero at $\infty$. For example in $$H(s)=\frac{s+1}{s^2-3s+2}$$ zeros are $z_1=-1$, $z_2=\infty$, and poles are $p_1=1,p_2=2$.

So by taking into account the poles and zeros at infinity, the number of poles and zeros in rational functions are always equal.