1. ## complex power expansion

find the complex power expansion for $\displaystyle \frac{z^2}{(1-z^2)^3}$ about 0 and determine the radius of convergence.

i can see that the radius of convergence is 1 since when z=1 the function is undefined.
help me on finding the power expansion. please

2. Originally Posted by PRLM
find the complex power expansion for $\displaystyle \frac{z^2}{(1-z^2)^3}$ about 0 and determine the radius of convergence.

i can see that the radius of convergence is 1 since when z=1 the function is undefined.
help me on finding the power expansion. please
Consider the power series

$\displaystyle f(z)=\frac{1}{1-z}=\sum_{n=0}^{\infty}z^n$

Now if we take two derivatives we get

$\displaystyle f'(z)=\frac{1}{(1-z)^2}=\sum_{n=1}^{\infty}nz^{n-1}$

Now taking one more gives

$\displaystyle f''(z)=\frac{1}{(1-z)^3}=\sum_{n=2}^{\infty}n(n-1)z^{n-2}$

Now consider

$\displaystyle g(z)=z^2(f''(z^3))=\frac{z^2}{(1-z^2)^3}= z^2\sum_{n=2}^{\infty}n(n-1)(z^3)^{n-2}=\sum_{n=2}^{\infty}n(n-1)z^{3n-4}$