10.8.21 Series with complex terms
condition: Find all expansions of a given function in powers of a given difference \( (z-a) \). Indicate the areas of suitability of each of the decompositions. \[ f(z)=\frac{z+2}{\left(z^{2}+2 z+5\right)^{2}} \text { by powers }(z+1) \]
Expansion of complex functions in Laurent and Taylor series. Both single-valued and multi-valued functions are considered, indicating the branches along which the multi-valued functions are taken.