What is the residue at ##z=0## for ##\frac{1}{z^3}+e^{2z}##?

[kq6up]

Homework Statement

Find the residue of $\frac{1-cos2z}{z^3}$ at $z=0$

Homework Equations

$Res=\frac{1}{n!}\frac{d^n}{dz^n}[f(z)(z-z_0)^{n+1}]$ Where the order of the pole is $n+1$

The Attempt at a Solution

Differentiating $(1-cos2z)z^3$ twice, leaves me with zeros against every term giving me $0$ for a residue which is incorrect. Mathematica gives me 2 for a residue. Since $z_0=0$ I am only left multiplying the top function by z. I have no idea what I am missing here.

What am I doing incorrectly?

Thanks
Chris

 

[mfb]

What is the order of your pole?

 

[kq6up]

3rd order. I think I made a mistake that the 1/z^3 factor was not included in f(x). If I include it, it cancels out the z's outside of the cos2z function altogether. In that case, the part that gets differentiated and evaluated is just 1-cos2x.

Thanks,
Chris

 

[mfb]

3rd order.

It is not. Don't forget the numerator.

 

[kq6up]

I got it. I having trouble with a different problem now. A new thread is in order for that one.

Thanks,
Chris