Find a function given its poles, residues, limit at infinity, and additional constraints So what is given is that the function f(z) holomorphic is on the Complex plane provided:


*

*$f(z)$ has a first order pole in $z = 1$

*$f(z)$ has a second order pole in $z = 0$ with residue $0$

*$\lim\limits_{z\to\infty} f(z)= -2$  

*$\displaystyle \int_{|z| = 2}zf(z) = 0$

*$f(-1) = 0$


Determine $f(z)$
I tried it and I only got:
Based on 1. and 2. we find that the denominator is equal to $(z-1)z^2$.
Based on the order of the denominator and 3. we find that $(az^3 + bz^2 + cz + d)/((z-1)z^2)$ where $a$, $b$, $c$ and $d$ are constants to be determined.
Also from 3. we find that $a = -2$.
If we use 5. then we fill in $z = -1$ and set the whole equation equal to 0 to find another constant.
But I don't know if the above steps are good and I definitely don't know how to use number 4. I think I have to make use of the residue theorem and since there are two poles inside $|z| = 2$ namely $z = 1$ and $z = 0$ we get something like $2\pi(1+0)$ I think?
Please help me out. Thanks ! 

Q)  the only singularity of a single valued function f(z) are poles of order 2 and 1 at Z=1 and Z=2 with residues of these poles 1 and 3 respectively.if f(0)=3/2 and f(-1)=1,
determine the function f(z)
