Conformal Map wanted I am in search of a conformal map that will stretch the rectangle $P_1 =\lbrace(x,y) : (-W < x < W , -L < y < L )\rbrace$ to the entire real plane $P_2 = \lbrace(u,v) : u,v \in \mathbb{R}\rbrace$, where the sides of $P_1$ are mapped to infinity. For example, this transformation:
$$ u = \frac{xy}{W^2-x^2},\quad v = \frac{xy}{L^2-y^2} $$ 
satisfies the stretching, however it is not conformal. I have almost no experience in these type of problems, so any help will be very helpful. For instance, a good tip may even be the existence of a solution to this problem.
Thanks!  

Unfortunately, no such map exists: If there were a conformal bijection from your rectangle to the entire plane, the inverse mapping (or its complex conjugate) would be a bounded, entire function. But a bounded, entire function is constant by Liouville's theorem from elementary complex analysis.
Generally, no proper open subset of the plane is conformally equivalent to the plane. This is an immediate consequence of the Koebe-Poincaré uniformization theorem.
