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)

Comments

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).

References

[a1]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241
[a2]	C. Carathéodory, "Theory of functions" , 2 , Chelsea, reprint (1981) pp. Part 7, Chapts. 2–3
[a3]	Z. Nehari, "Conformal mapping" , Dover, reprint (1975) pp. 2