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**chisigma** Setting $\displaystyle z-1=s$ the complex function becomes...

$\displaystyle \displaystyle f(s)= \frac{1}{s}\ \frac{3+4 s + s^{2}}{3+3 s + s^{2}} = \frac{1}{s}\ (1+ \frac{s}{3+3 s + s^{2}}) = \frac{1}{s} + \frac{1}{3+3 s + s^{2}}$ (1)

Now you have to find the Laurent expansion of (1) around $\displaystyle s=0$. The first term is the 'non analythic part' of the expansion, the second in the 'analythic part'. Are You able to proceed?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$