Download PDF by Lester R Ford: Automorphic Functions

By Lester R Ford

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Example text

With pI as cluster point. The points of the latter set, with the possible exception of one point at 00, are centers of isometric circles. Hence, pI is a limit point. Furthermore, no point which is not a limit point is carried by 8 into a limit point, since otherwise 8- 1 would carry a limit point into a point not a limit point. -1f the set of limit points contains more than two points, it is a perfect set. A set is perfect, by definition, if it has the following two properties: (1) each cluster point of the set belongs to the set; that is, the set is closed; and (2) each point of the set is a cluster point of points of the set; that is, the set is dense in itself.

S~c. -A point on the boundary of R is a point. P not belonging to R but such that in any circle with P as cent,er there are points of R. P may be an ordinary point or a limit point. Obviously, P cannot lie within an isometric circle. If P is an ordinary' point, it lies on one or more isometric circles. cles other than those which pass through P. In the most general case, a boundary point p. belongs to one of the following three categories: (a) P is a limit point of the group; ((3) P i~ an. ordinary point and lies on a single isometric circle; , (,,) P is an ordinary point and lies on two or more isometric circles.

20. -If we apply to R the various transformations of the group, there results a set of congruent regions no two of which overlap (Theorem 1). -R and the regions congruent to R form a set oj regions which extend into the neighborhood of every point of the plane. Suppose, on the contrary, that there is a point Zo enclosed by a circle Q with Zo as center and of radius r sufficiently small that Q contains neither points of R nor points congruent to point~ of R. Then, all transforms of Q contain neither points of R nor points congruent to points of R.

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Automorphic Functions by Lester R Ford


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