complex power expansion

**PRLM** · April 18th 2010, 12:18 PM

find the complex power expansion for $\frac{z^2}{(1-z^2)^3}$ 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

**TheEmptySet** · April 18th 2010, 02:17 PM

Originally Posted by PRLM

find the complex power expansion for $\frac{z^2}{(1-z^2)^3}$ 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

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

Now if we take two derivatives we get

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

Now taking one more gives

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

Now consider

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

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