1. ## Laurent Series

I found this exercise can someone explain what I have to do it!?

Find the expansion of f(z)=1/(z(z-1)) about the point z=1 and say for what region of the Argand plane your expansion is valid for. What therefore is the residue of this unctions at the point z=1?

Thanks!!

2. What have you tried ?

Do you know how to get the expansion of a function at some point ?
Do you know what a residue is ?

3. I just have some idea if u can explain me that is very helpful

4. Originally Posted by NaNa
I found this exercise can someone explain what I have to do it!?

Find the expansion of f(z)=1/(z(z-1)) about the point z=1 and say for what region of the Argand plane your expansion is valid for. What therefore is the residue of this unctions at the point z=1?

Thanks!!

take u = z-1

your expression reduces to 1 / [(u+1) u]

use partial fraction to break it into 1/u - 1/(u+1)

= 1/u - (u+1)^(-1)

apply binomial series for the second term
and change back to u = z-1
it will be valid in |u| <1 or 0 < |z-1| < 1

the residue at z=1 will be the coefficient of 1 / (z-1) in the series