Laurent series
A series \sum_{n=-\infty}^\infty a_n(z-z_0)^n with both positive and negative powers, representing f on an annulus.
A Laurent series centered at z_0 is a two-sided series \sum_{n=-\infty}^\infty a_n(z-z_0)^n, allowing negative as well as non-negative powers of (z-z_0). Any function analytic throughout an annulus r<|z-z_0|<R has a unique Laurent series representation there, convergent on the whole annulus; it generalizes the Taylor series, which is the special case where every negative-power coefficient is zero and the annulus is really a punctured disk with r=0 filled back in.
Examples
e^{1/z} = \sum_{n=0}^\infty \frac{1}{n!\,z^n} = 1+\frac1z+\frac1{2z^2}+\cdots is the Laurent series of e^{1/z} on the punctured plane z\ne 0.
\frac{1}{z(z-1)} has a different Laurent series on 0<|z|<1 than it does on |z|>1, since it is analytic on each annulus but not across the circle |z|=1.