Cauchy formula problem: undefined function

Question

I have this problem where it is supposed to use the Cauchy integral formula: $$\oint_{C} \frac{f(z)}{z(2z+1)^2}dz.$$ The poles clearly are at $z=0 ,z=\frac{-1}{2}. $ The contour is the unitary circle centered at the origin. I tryed to factorize de denominators, but for that i need $f(z)$ to find the coeficientes. How can one manage to evaluate this problem? Can i evaluate the formula over one pole first then the other or should i work with the contour?

Answer 1

Hint: Separate the expression using partial fractions. That is, use the fact that $$ \frac 1{z(2z+1)^2} = \frac 1z - \frac 2{2z + 1} - \frac 2{(2z + 1)^2}. $$ It should now be a bit clearer how Cauchy's integral formula applies to this integral.