# Schwarz function

Riemann–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 $\pi / \nu _ {1}$, $\pi / \nu _ {2}$, $\pi / \nu _ {3}$, where $\nu _ {1}$, $\nu _ {2}$ and $\nu _ {3}$ are some specially-chosen natural numbers.

If $1/ \nu _ {1} + 1/ \nu _ {2} + 1/ \nu _ {3} = 1$, one obtains rectilinear triangles for which the only possibilities are: $\nu _ {1} = \nu _ {2} = 2$, $\nu _ {3} = \infty$( a semi-strip); $\nu _ {1} = 2$, $\nu _ {2} = 3$, $\nu _ {3} = 6$; $\nu _ {1} = 2$, $\nu _ {2} = \nu _ {3} = 4$; $\nu _ {1} = \nu _ {2} = \nu _ {3} = 3$. 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 $1/ \nu _ {1} + 1/ \nu _ {2} + 1/ \nu _ {3} > 1$, there are the following possibilities: $\nu _ {1} = \nu _ {2} = 2$, $\nu _ {3}$ arbitrary; $\nu _ {1} = 2$, $\nu _ {2} = \nu _ {3} = 3$; $\nu _ {1} = 2$, $\nu _ {2} = 3$, $\nu _ {3} = 4$; $\nu _ {1} = 2$, $\nu _ {2} = 3$, $\nu _ {3} = 5$. 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 $1/ \nu _ {1} + 1/ \nu _ {2} + 1/ \nu _ {3} < 1$, then infinitely-many different triangles are possible, since $\nu _ {1} , \nu _ {2} , \nu _ {3}$ may increase indefinitely. Here, the Schwarz functions are automorphic functions with a continuous singular curve (circle or straight line). In particular, the cases of $\nu _ {1} = 2$, $\nu _ {2} = 3$, $\nu _ {3} = \infty$ and $\nu _ {1} = \nu _ {2} = \nu _ {3} = \infty$( a circular triangle with zero angles) lead to the modular functions (cf. Modular function) $J( z)$ and $\lambda ( z)$, respectively. The Schwarz functions were studied by H.A. Schwarz .

How to Cite This Entry:
Schwarz function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_function&oldid=48631
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article