<s>[INST] Partial fraction expansion two variables How to expand
$$\frac{y}{(x-y)(y-1)}$$
by partial fraction expansion.      
 [/INST] $$\text{Plot3D}\left[\left\{\frac{y^4}{(x-1) (y-1)}-\frac{y \left(x^2 y^2+x^2 y+x^2+x y^2+x y+y^2\right)}{x^3},\frac{y}{(y-1) (x-y)},x \left(y+\frac{1}{y}+1\right)+y+1\right\},\{x,-2,2\},\{y,-2,2\},\text{PlotLegends}\to \text{Automatic}\right]$$
use this
$$\frac{y^4}{(x-1) (y-1)}-\frac{y \left(x^2 y^2+x^2 y+x^2+x y^2+x y+y^2\right)}{x^3}$$

as you can see the orange (above  ) and the blue (your fraction ) are very near is the generalitated laurent series
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