This is an exercise problem.
Does there exist a conformal mapping from an equilateral triangle onto an isosceles right triangle such that, under correspondence of boundary, vertices are mapped to vertices?
I am a bit confused about the problem. Shouldn't a conformal mapping preserve the angles and thus we can conclude that there does not exist such a function? However, this problem is not trivial and I wonder what's wrong with this idea.
I am also thinking about Schwarz Christoffel transformation but that seems to require the construction of a function from a triangle to upper half plane, i.e., an inverse SC mapping. I am not sure if this is the right way to go or there are some other better method.
Any help ..???