By C. Caratheodory
This publication is a translation through F. Steinhardt of the final of Caratheodory's celebrated textual content books, Funktiontheorie, quantity 1.
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Additional resources for Theory of Functions of a Complex variable, Volume One
Sample text
4), which has certain advantages. 4), for which C = O. 8) l-az • W= (a= - ~, . 2) and C = 0 imply D2 = a a(A 2 + 8 2). d) -;r . 9) 52 PART ONE. III. 10) l5d-yy>O.
42 PART ONE. II. 1) By reflection in h, a point z is mapped on the point (51. 2) By reflection in g*, the point z* is mapped onto (51. 3) The composition of the last two operations yields W = e2i ('P-'I') z. (51. 4) This Moebius transformation IS a rotation through the angle () = 2 (1p - cp), and we see that cp (or 1p), hence also h (or g*), can be chosen arbitrarily. Weare led, by similar reasoning, to analogous conclusions if the pencil of invariant circles is elliptic. After transforming this pencil into the pencil of straight lines through z = 0 by means of a suitable preliminary Moebius transformation, we find in this case that any Moebius transformation leaving each of these lines invariant must be the resultant of two successive reflections in two concentric circles (51.
A Moebius transformation of this type is called loxodromic. 3) is likewise loxodromic. 55. 3), we obtain Vir = (ex + 6) -: ~~' {ex + 6)2 - 4 (ex 6 - Pr) ~~ Lf = {ex + 6)2 (-: ~ ~ r, {55. 1) and from this we obtain the rational invariant ;. = (rx + 6)1 4 (rx 6 - Py) = if! + 1~ = 1 + 4 f! (f! - 1)1 4 f! {55. 2) This invariant ;. serves very well to characterize, and distinguish between, the various cases just discussed. For if e is real, positive, and =1= 1, then). > 1. If e is unimodular, say e = ei{l, then ;.