Riemann mapping between arbitrary triangles Question---Is there nice formula for Riemann mapping between arbitrary triangles with vertices (a_1,a_2,a_3) and (b_1,b_2,b_3)? 
Comment---I look for the conformal equivalence of interiors promised by Riemann mapping theorem (which will automatically be continuous at boundary). I know that I can easily map any triangle (a) to triangle formed by (0,1,a') by complex linear map. I do not see a simple map between two triangles in this form. The map could be described using "Schwartz-Christoffel map"  and its inverse. This does not make the situation clear. 
Edit: If closed formula doesn't exist, are there nice special cases where formula does exist.
 A: In such questions everything depends on your exact definition of "closed formula".
The "closed formula" does not exist for general triangles. Let me explain.
You can map the upper half-plane onto your triangles by functions $f_1$ and $f_2$,
so that $(0,1,\infty)$ go to vertices. These maps $f_1$ and $f_2$ are expressed as integrals. These integrals are not elementary functions (except for few very special triangles) but they are special functions, namely hypergeometric.
"Special functions" essentially means that "everything in known about them", many expressions in the forms of series or integrals, their global properties, like
analytic continuation, etc. So you may consider them "explicit".
However the map you are asking is $f_1^{-1}\circ f_2$. And there is no way to obtain an explicit expression for the inverse of a hypergeometric function, again except for some very special triangles. Those special triangles for which you can obtain more or less explicit expressions have been listed. I recommend Caratheodory's book Theory of functions of a complex variable, vol. 2.
