# Finding Laurent Series

• Jan 4th 2010, 04:11 AM
thelostchild
Finding Laurent Series
Hi sorry to trouble you but I've been to find the singular terms in the Laurent expansion of

$\displaystyle f(z) = \exp(-\frac{1}{z})$ at z=0

Am I correct in saying that because there is an essential singularity here it's undefined or am I completely wrong?
• Jan 4th 2010, 05:25 AM
shawsend
Why not just use the Taylor series for $\displaystyle e^w$ and then substitute $\displaystyle w=-1/z$.
• Jan 4th 2010, 06:02 AM
thelostchild
Quote:

Originally Posted by shawsend
Why not just use the Taylor series for $\displaystyle e^w$ and then substitute $\displaystyle w=-1/z$.

That would work if I were taking the series about the point $\displaystyle z= \infty$

but the series is about z=0 which was the issue :)
• Jan 4th 2010, 01:10 PM
mr fantastic
Quote:

Originally Posted by thelostchild
Hi sorry to trouble you but I've been to find the singular terms in the Laurent expansion of

$\displaystyle f(z) = \exp(-\frac{1}{z})$ at z=0

Am I correct in saying that because there is an essential singularity here it's undefined or am I completely wrong?

There is an essential singularity but all that means is that the principal part of the Laurent series in an infinite series. The suggestion given in post #2 is what you should do to get this series.