Schwarz function

An analytic function realizing a conformal mapping of a triangle bounded by arcs of circles onto the upper half-plane (or unit disc) that remains single-valued under unrestricted analytic continuation. A Schwarz function is an automorphic function. The corresponding group depends on the form of the mapped triangle. The requirement of single-valuedness is satisfied only in the case when the angles of the triangle are , , , where , and are some specially-chosen natural numbers.
If , one obtains rectilinear triangles for which the only possibilities are: , (a semi-strip); , , ; , ; . In all these cases the Schwarz functions are represented by trigonometric functions or Weierstrass elliptic functions and are automorphic; their group is the group of motions of the Euclidean plane.
If , there are the following possibilities: , arbitrary; , ; , , ; , , . In all these cases the Schwarz functions are rational automorphic functions; their group is a finite group of motions of a sphere. As a result of the relationship between this group and regular polygons, such Schwarz functions are also called polyhedral functions.
Finally, if , then infinitely-many different triangles are possible, since may increase indefinitely. Here, the Schwarz functions are automorphic functions with a continuous singular curve (circle or straight line). In particular, the cases of , , and (a circular triangle with zero angles) lead to the modular functions (cf. Modular function) and , respectively. The Schwarz functions were studied by H.A. Schwarz .