# Laurent series of a complex multivariate function

Suppose $f(x,y)$ to be a function of variables $x$, $y$ $$f(x,y)=\frac{-\left(\pi^{2}-4x^{2}\right)y\left(\pi^{2}-4y^{2}\right)\left(\pi^{2}\cos2x-4x(x-\pi\sin x)\right)}{\pi\left(\left(x^{2}+y^{2}\right)\left(64x^{2}y^{2}+4\pi^{4}\right)-64\pi^{2}x^{2}y^{2}+\pi\left(\pi^{2}-4y^{2}\right)^{2}\sin x(\pi\sin x-4x)\right)}.$$ For $(x,y)\rightarrow(0,0)$ is $f(x,y)\rightarrow-\infty$. How to find the Laurent series at $(0,0)$ for such complex form of the multivariate function? Is it possible, or, only some numeric approach is available?

There are many samples of the elementary functions but more complicated examples are missing.

Mathematica provides the following solution:

f = -((Pi^2 - 4*x^2)*y*(Pi^2 - 4*y^2)*(Pi^2*Cos[2*x] - 4*x*(x - Pi*Sin[x])))/
(Pi*(-64*Pi^2*x^2*y^2 + 4*Pi^4*(x^2 + y^2) + 64*x^2*y^2*(x^2 + y^2) +
Pi*(Pi^2 -4*y^2)^2*Sin[x]*(-4*x + Pi*Sin[x])))
Normal[Series[f, {x, 0, 3}, {y, 0, 3}]]


$$x^2 \left(\left(\frac{16}{\pi ^4}-\frac{16}{\pi ^5}-\frac{4}{\pi ^3}\right) y^3+\frac{\frac{\pi }{4}-\frac{\pi ^2}{4}+\frac{\pi ^3}{16}}{y^3}+\left(\frac{12}{\pi ^3}-\frac{8}{\pi ^2}+\frac{1}{\pi }\right) y+\frac{2-\frac{3}{\pi }-\frac{\pi }{4}}{y}\right)+\frac{y}{\pi }-\frac{\pi }{4 y} + O(4).$$