# 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 [1].

#### References

 [1] H.A. Schwarz, "Ueber diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt (nebst zwei Figurtafeln)" J. Reine Angew. Math. , 75 (1873) pp. 292–335 [2] V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) [3] L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951)

In the case of a rectilinear triangle the functions are also called Schwarz triangle functions (cf. [a1], [a2]) or Schwarzian $s$- functions [a3]. These functions can be written as the quotient of two independent solutions of a hypergeometric equation (with explicit expressions for its coefficients in terms of the triangle angles), cf. [a3], p. 206. Conformal mapping of circular polygons to a half-plane is also discussed in [a3] (Chapt. V, Sect. 7).