Thread: Conformal equivalence between rectangles

For 0<a<1, let R_a be the rectangle whose side lengths are a and 1. Show that every rectangle is conformally equivalent to one and only one of this rectangles.


Hint: use the schwarz reflection principle and the fact that every ring in C (subset whose complement has two components) is conformally equivalent to one and only one o: C/{0}, D/{0} or Ar={z in C : 1 < |z| < r}

Thanks

Let me be more precise:

R_a={x+iy in C: 0<x<a, 0<y<pi} (not 1 anymore).

I need help mainly in which is the conformal map that takes an arbitrary rectangle to one of these R_a.... I´ve thought of traslating first... Is it confotmal to flatten or lengthen each side? It appears it is not because apparently the R_a are not conformally equivalent among themselves...