# Series Representation

• Apr 18th 2010, 08:46 AM
villa223
Series Representation
Hi, I was wondering if anyone might know how to solve this problem that I've been working on.

Find the Laurent Series for f centered at z0 = 0. Also, determine the punctured disk of maximum radius on which the series converges.

f(z) = z^3 cosh(1/z^2)

Thanks
• Apr 18th 2010, 09:36 AM
tonio
Quote:

Originally Posted by villa223
Hi, I was wondering if anyone might know how to solve this problem that I've been working on.

Find the Laurent Series for f centered at z0 = 0. Also, determine the punctured disk of maximum radius on which the series converges.

f(z) = z^3 cosh(1/z^2)

Thanks

$\cosh\left(\frac{1}{z^2}\right)=\frac{e^{1/ z^2}+e^{-1/z^2}}{2}=$ $\frac{1}{2}\left[ \sum^\infty_{n=0}\frac{1}{n!z^{2n}}+\sum^\infty_{n =0}\frac{(-1)^n}{n!z^{2n}}\right]$ $=\sum^\infty_{n=0}\frac{1}{(2n)!z^{4n}}$ $=1+\frac{1}{2!z^4}+\frac{1}{4!z^8}+\frac{1}{6!z^{1 2}}+\ldots\Longrightarrow$

$z^3\cosh\left(\frac{1}{z^2}\right)=z^3+\frac{1}{2! z}+\frac{1}{4!z^5}+\ldots$ $=\sum^\infty_{n=0}\frac{1}{(2n)!z^{4n-3}}$

Tonio