Finding a Laurent Series

Complex Analysis · Axiom Academy

EXAMPLE Finding a Laurent Series Expanding a rational function in an annulus using partial fractions and geometric series Find the Laurent series of valid in the annulus (centered at z_0 = 0 ). Nice work. You built a Laurent series the way it is almost always done in practice: split, then expand each piece for the region you care about. Region picks the form. For , use for a Taylor (positive-power) expansion in z/a , and for a Laurent (negative-power) expansion in a/z . One function, many series. The same f has a different expansion in each annulus around the same point — here uses but . The principal part lives here. The negative powers came entirely from the term, since z=1 sits inside this annulus. The coefficient of in a Laurent series — here a_ -1 = -1 — is exactly the residue, the single number that drives contour integration.

This is the written version of the interactive lesson above. See the full Complex Analysis course.