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 ! 
 A: 
Based on 1. and 2. we find that the denominator is equal to ... 

True, but it is a bad idea to write $f$ in the form of a single quotient. Partial fractions are easier to handle when integrals get involved (which they do in 4). On the basis of 1 and 2 you can  write 
$$
f(z) = \frac{A}{z-1}+\frac{B}{z^2} + g(z) \tag{*}
$$
where $g$ has no poles (i.e., is an entire function). There is  no $1/z$ because the residue at $0$ is $0$; this is   some information you were not using. Suggestions for the rest:


*This implies $\lim_{z\to\infty} g(z)=-2$. Use Liouville's theorem to find what $g $ is.

*Plug $(*)$ into the integral. Then express the integral in terms of $A$ and $B$, using the residue theorem. Now you have a linear equation for $A$ and $B$.

*Plug $z=-1$ into $(*)$. Now you have a second linear equation for $A$ and $B$. Find $A,B$ and you are done.
A: 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)
