Mathematical Tools for Physics

14—Complex Variables 425

The structure of a Laurent series is such that it will converge in an annulus. Examine the absolute
convergence of such a series.
∑∞

−∞

∣

∣akzk

∣

∣=

∑−^1

−∞

∣

∣akzk

∣

∣+

∑∞

0

∣

∣akzk

∣

∣

The ratio test on the second sum is

if for large enoughk,

|ak+1||z|k+1

|ak||z|k

=

|ak+1|

|ak|

|z|≤x < 1 then the series converges.

This defines the largest value of|z|for which the sum of positive powers converges.

If|ak+1|/|ak|has a limit then |z|max= lim

|ak|

|ak+1|

Do the same analysis for the series of negative powers, applying the ratio test.

if for large enoughnegativek,

|ak− 1 ||z|k−^1

|ak||z|k

=

|ak− 1 |

|ak|

1

|z|

≤x < 1 then the series converges.

This defines thesmallestvalue of|z|for which the sum of negative powers converges.

If|ak− 1 |/|ak|has a limit ask→−∞then |z|min= lim

|ak− 1 |

|ak|

If|z|min < |z|max then there is a range ofz for which the series converges absolutely (and so of course it
converges).

|z|min<|z|<|z|max an annulus

|z|min

|z|max

If either of these series of positive or negative powers is finite, terminating in a polynomial, then respectively
|z|max=∞or|z|min= 0.