kill(all)$

van=facebook.com/LearningCalculus$

w(u,v):= u+%i*v;

z(u,v):=w(u,v)+1/w(u,v);

z(u,v);

x(u,v):=realpart(z(u,v));

y(u,v):=imagpart(z(u,v));

[[x(-1,0),y(-1,0)],[x(1,0),y(1,0)]];

x(u,v):=''(fullratsimp(realpart(z(u,v))));

y(u,v):=''(fullratsimp(imagpart(z(u,v))));

solve([X=x(u,v),Y=y(u,v)],[X,Y]);

solve([X=x(u,v),Y=y(u,v)],[u,v]);