Hi Mitchell,
Here are some connections. Connections to other fields may require some background for a full understanding, such as a course in the field. First a definition:
Def:
$f(z)$ has a pole of integer order
$k>0$ at
$c$ if
$(z-c)^kf(z)$ is holomorphic at
$c$ but is not holomorphic for smaller powers
$k$. Example: The following has poles of orders
$1,2,3$ at
$z=10,20,30$ respectively:
$$f(z)={(z-40)^4 \over (z-10)(z-20)^2(z-30)^3} \,.$$
A number
$c$ is a pole of order
$k$ of
$f(z)$ if it is a zero of multiplicity
$k$ of
$1/f(z)$. If we express the example above as a product, zeros and poles are distinguished by the sign of the exponent; otherwise, they are algebraically similar:
$$f(z)={(z-40)^4 (z-10)^{-1}(z-20)^{-2}(z-30)^{-3}} \,.$$
Generalized mathematical significance (hope this isn't too abstract): Suppose we have a continuous object, such as a function
$f_a(z)$ that varies continuously with one parameter/coefficient (or more) represented by
$a$. And suppose there is a discrete quantity that is calculated from
$f_a(z)$, such as the number of zeros or number of poles. Finally suppose that the discrete quantity can be proved to vary continuously with
$a$. Then the discrete quantity is an invariant of
$f_a(z)$. That is, it's constant.
Contrariwise, if the discrete quantity changes (is discontinuous), something interesting is probably happening with the functions
$f_a(z)$ at the value for
$a$ at which there is a discontinuity.
For instance, a meromorphic function on the Riemann sphere has a finite number of zeros and the same number of zeros (
$Z$) as poles (
$P$) counted with multiplicity and including poles/zeros at infinity. Thus
$Z-P=0$. It is constant. You cannot change the number of zeros of
$f_a(z)$ without also changing the number of poles in the same way (assuming
$f_a(z)$ is meromorphic for all
$a$). Example:
$f_a(z)=z^2+a$ has two zeros because it has a pole of order
$2$ at infinity (and vice versa).
More examples:
I suppose it's clear that solving equations has many applications, and the number of solutions has implications for the problem in which the equation arises. A solution to
$f(z)=c$ is a zero of
$g(z)=f(c)-c$.
I suppose it's also clear from the complex analysis course that the poles themselves are important in integration. Perhaps the number of them is not so significant, but it might sometimes be helpful to know how many there are.
An application: I suppose periodicity has been an important topic in mathematical physics and therefore in mathematics since humans noticed things like days, months, and years recur periodically. A periodic function has infinitely solutions to
$f(z)=c$ if it has one. If
$f(z)$ has infinitely many zeros, then it must have an essential singularity on the Riemann sphere (examples:
$e^z$,
$\sin z$,
$\sin(1/z)$). Periodic functions must have this feature.
In control theory, the zeros and poles of the transfer function determine the behavior of a linear system. See
MIT notes
(read
$2{du\over dt} + u$ for the RHS of the example at the top of page 2) or
CalTech notes
(read
$e^{st}$ in the RHS of the equation between (6.6) and (6.7) on page 142).
