- #1

Redwaves

- 134

- 7

- Homework Statement
- Find the residue $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$

- Relevant Equations
- $$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$

Hi,

I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$

Since I have 2 singularities which are double poles.

I'm using this formula

$$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$

then,

$$\lim_{z\to\ \pm ia} \frac{d}{dz}(\frac{z^2}{z^2 + a^2})$$

At this point, I don't get the correct answer which is $$\pm \frac{1}{4ai}$$

I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$

Since I have 2 singularities which are double poles.

I'm using this formula

$$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$

then,

$$\lim_{z\to\ \pm ia} \frac{d}{dz}(\frac{z^2}{z^2 + a^2})$$

At this point, I don't get the correct answer which is $$\pm \frac{1}{4ai}$$