I am working on the following problem, suppose that $f$ has a simple pole at $-1$ with $Res(f,-1) = 1$. A double pole at $2$ with $Res(f, 2) = 2$. Also $f(0) = 7/4$ and $f(1) = 5/2$. I am supposed to find the Laurent series for $f$ in $ann(0;1,2)$.

By definition the coefficients of the Laurent series in the desired annulus are $a_n = \frac{1}{2\pi i} \int_{\gamma} {\frac{f(z)}{z^{n+1}}}$. Where $\gamma$ is a circle centered at zero with radius bigger then $1$ and less then $2$. I was able to apply the residue theorem to calculate $a_{-1}$ and I know that $a_n = 0$ for $n < -2$. I am unsure how to calculate $a_{-2}$. I believe that the positive coefficients just correspond to the Taylor series coefficients correct? Is there an explicit way to calculate these with such minimal information?

Any help is greatly appreciated, thanks


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