# What does zero at infinity mean?

#### dwsmith

##### Well-known member
For the following algebraic expressions for the Laplace transform of a signal, determine the number of zeros located in the finite $$s-$$plane and the number of zeros located at infinity.

Does that mean pole since at a pole the plot will diverge to $$\infty$$? Otherwise, what would be a zero at infinity?

#### ThePerfectHacker

##### Well-known member
Consider the complex-valued function of a complex variable, $f(z) = z^{-2}$. We notice that as $|z|\to \infty$ that $f(|z|) \to 0$ so we say $f(z)$ is zero at infinity. I think when someone asks for "number of zeros at infinity" they are asking for the multiplicity of $\infty$ as a zero.

In general, if $f(z)$ sends $\infty$ to zero then,
$$\lim_{z\to 0} f\left( \frac{1}{z} \right) = 0$$
We can define the function,
$$g(z) = \left\{ \begin{array}{ccc} f(z^{-1}) & \text{ if }& z\not = 0 \\ 0 & \text{ if }& z = 0 \end{array} \right.$$
Now count the multiplicity of $0$ for $g(z)$ at $0$.

In the example given above $f(z)=z^{-2}$ then $g(z) = z^2$ so $g(z)$ has multiplicity two zero. Thus, $f(z)$ has two zeros at infinity.