# Finding the residue of the following function

• Apr 22nd 2013, 09:47 AM
director
Finding the residue of the following function
Hi all, I have the following function $\displaystyle f(z) = \frac{1}{(z^2 + z + 1)^2}$ and I need to find the residue at the double pole $\displaystyle z = (-1)^{\frac{2}{3}}$. Any tips? I'm hoping to not have to use the Laurent expansion for this. I know that $\displaystyle (z^2 + z + 1) = \frac{(1-z)}{(1-z^3)}$. I'm aware of a limit formula for higher order poles but I don't know how to manipulate the function to get it in that form. Maybe that's not even needed. Thanks!
• Apr 24th 2013, 06:08 AM
majamin
Re: Finding the residue of the following function
Quote:

Originally Posted by director
Hi all, I have the following function $\displaystyle f(z) = \frac{1}{(z^2 + z + 1)^2}$ and I need to find the residue at the double pole $\displaystyle z = (-1)^{\frac{2}{3}}$. Any tips? I'm hoping to not have to use the Laurent expansion for this. I know that $\displaystyle (z^2 + z + 1) = \frac{(1-z)}{(1-z^3)}$. I'm aware of a limit formula for higher order poles but I don't know how to manipulate the function to get it in that form. Maybe that's not even needed. Thanks!

Note that the (double) roots of f(z) are $\displaystyle \frac{-1\pm i\sqrt{3}}{2}$ (which are equivalent to $\displaystyle (-1)^{2/3})$. Set $\displaystyle z_1=\frac{-1+ i\sqrt{3}}{2}$ and $\displaystyle z_2=\frac{-1- i\sqrt{3}}{2}$, then $\displaystyle Res(f,z_1) = \frac{1}{(n-1)!} \lim_{z->z_1}\frac{d^{n-1}}{dz^{n-1}}(z-z_1)^2 f(z)$ where n=2. Similarily for $\displaystyle z_2$.
• Apr 26th 2013, 07:21 PM
director
Re: Finding the residue of the following function