What is a residue?

Solution 1:

The holomorphic functions have an extraordinary property: if you compute an integral along a path, the value of the integral does not depend on the path !

More precisely, if the function is holomorphic everywhere inside a closed path, the integral is just zero. But if the function has poles (zeroes at the denominator, $c_{-k}$ terms in the Laurent series), every pole brings a nonzero contribution called its residue. You can shrink the path as much as you want, even turning it to infinitesimal circles around every pole, provided you keep the poles in.

So the residues are what is left (as regards integration) after you removed all the holomorphic parts of the domain.

Solution 2:

A function $f$ analytic in a full disk $D_r$ around $a$ can be written as derivative of some other function $F$ in $D$:$$f(z)=F'(z)\quad(z\in D_r)\ .$$ If $f$ has an isolated singularity at $a$ one may still ask whether $f$ has a primitive $F$ in the punctured disk $\dot D_r:=D_r\setminus\{a\}$, in other words: whether the ODE $y'=f(z)$ has a solution in $\dot D_r$. It turns out that the sole obstruction to the solvability of this problem is the residue $${\rm res\,}_a(f):={1\over2\pi i}\int_{\partial D_\rho} f(z)\>dz,\qquad\rho<r\ .$$ If this residue is $\ne0$ no solution exists.

In my history of math book (by Moritz Kline) I read that the name of "residue" has been introduced by Cauchy in his Exercices de mathématique (1826–30) in connection with an integral similar to the above, but around a rectangle.