selected titles in this series - ams
TRANSCRIPT
Selected Title s i n Thi s Serie s
17 Jacque s Hadamar d (Jerem y J . Gra y an d A b e Shenitzer , Editors) , Non-Euclidea n
geometry i n th e theor y o f automorphi c functions , 199 9
16 P . G . L . Dirichle t (wit h Supplement s b y R . Dedekind) , Lecture s o n numbe r theory ,
1999
15 Charle s W . Curtis , Pioneer s o f representatio n theory : Probenius , Burnside , Schur , an d
Brauer, 199 9
14 Vladimi r Maz'y a an d Tatyan a Shaposhnikova , Jacque s Hadamard , a universa l
mathematician, 199 8
13 Lar s Garding , Mathematic s an d mathematicians : Mathematic s i n Swede n befor e 1950 ,
1998
12 Walte r Rudin , Th e wa y I remembe r it , 199 7
11 Jun e Barrow-Green , Poincar e an d th e thre e bod y problem , 199 7
10 Joh n Stillwell , Source s o f hyperboli c geometry , 199 6
9 Bruc e C . Bernd t an d Rober t A . Rankin , Ramanujan : Letter s an d commentary , 199 5
8 Kare n Hunge r Parshal l an d Davi d E . Rowe , Th e emergenc e o f th e America n
mathematical researc h community , 1876-1900 : J . J . Sylvester , Feli x Klein , an d E . H . Moore ,
1994
7 Hen k J . M . Bos , Lecture s i n th e histor y o f mathematics , 199 3
6 Smilk a Zdravkovsk a an d Pete r L . Duren , Editors , Golde n year s o f Mosco w
mathematics, 199 3
5 Georg e W . Mackey , Th e scop e an d histor y o f commutativ e an d noncommutativ e
harmonic analysis , 199 2
4 Charle s W . McArthur , Operation s analysi s i n th e U.S . Arm y Eight h Ai r Forc e i n Worl d
War II , 199 0
3 Pete r L . Dure n e t al. , Editors , A centur y o f mathematic s i n America , par t III , 198 9
2 Pete r L . Dure n e t al. , Editors , A centur y o f mathematic s i n America , par t II , 198 9
1 Pete r L . Dure n e t al. , Editors , A centur y o f mathematic s i n America , par t I , 198 8
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Ill
the Theor y o f
c
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e^wcvwc^ •
S O U R C E S
Non^Euclidean in
the Theor y o f AiULtonnorphic Functions
Jeremy J. Gra y and Abe Shenitzer , Editors
Translated by Abe Shenitze r
With Historical Introduction by Jeremy J . Gra y
7
American Mathematica l Societ y London Mathematica l Societ y
https://doi.org/10.1090/hmath/017
Editor ia l Boar d
American Mathematica l Societ y Londo n Mathematica l Societ y George E . Andrew s Davi d Fowler , Chai r Bruce Chandle r Jerem y J . Gra y Karen Parshall , Chai r S . J . Patterso n George B . Seligma n
Neevklidova geometriy a v teori i avtomorfnyk h funktsi i
by Jacque s Hadamar d
Originally publishe d i n Russia n b y Gostekhizdat , M-L , 1951 .
Original Russia n tex t translate d b y Ab e Shenitzer .
1991 Mathematics Subject Classification. Primar y 01-XX , 01A55 , 01A60 ; Secondary 30-03 , 30F35 , 34A20 , 51-03 .
Library o f Congres s Cataloging-in-Publicatio n Dat a
Hadamard, Jacques , 1865-1963 . Non-Euclidean geometr y i n th e theor y o f automorphi c function s / Jacque s Hadamar d ;
Jeremy J . Gray , Ab e Shenitzer, editor s ; translated b y Abe Shenitzer ; with historica l introductio n by Jerem y J . Gray .
p. cm . — (Histor y o f mathematics ; v. 17) Includes bibliographica l references . ISBN 0-8218-2030- 3 (softcove r : alk . paper ) 1. Automorphi c functions . 2 . Geometry , Non-Euclidean . I . Gray , Jeremy , 1947 - .
II. Shenitzer , Abe . III . Title. IV . Series. QA353.A9H33 199 9 515'.9—dc21 99-3170 9
CIP
Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make fai r us e of the material, suc h a s to copy a chapte r fo r use in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given.
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© 199 9 b y the American Mathematica l Society . Al l rights reserved . Printed i n the United State s o f America .
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10 9 8 7 6 5 4 3 2 1 0 4 03 02 01 00 9 9
Contents
Acknowledgments i x
Introduction b y th e publisher s o f the Russia n translatio n x i
Historical introductio n JEREMY GRA Y 1
A brief histor y o f automorphi c functio n theory , 1880-193 0 JEREMY GRA Y 3
Chapter I . Th e grou p o f motion s o f th e hyperboli c plan e an d it s properl y
discontinuous subgroup s 1 7
Chapter II . Discontinuou s group s i n thre e geometries . Fuchsia n function s 3 7
Chapter III . Fuchsia n function s 5 7
Chapter IV . Kleinia n group s an d function s 7 1
Chapter V . Algebrai c function s an d linea r algebrai c differentia l equation s 7 9
Chapter VI . Fuchsia n group s an d geodesie s 8 7
References 9 3
Additional reference s 9 5
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Acknowledgments
The editor s wis h to thank Sara h Shenitze r fo r carefull y readin g th e translatio n and eliminatin g a number o f infelicities an d outrigh t errors . W e also wish to than k Deb Smit h fo r he r car e i n th e preparatio n o f th e final versio n o f th e manuscrip t and fo r th e patienc e wit h whic h sh e handle d score s o f last-minute corrections .
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Introduction b y th e Publisher s of th e Russia n Translatio n
In the Editors ' Introductio n t o B.A . Fuks ' book Non-euclidean geometry in the theory of conformal and pseudoconformal mappings, volum e V in thi s series , i t wa s mentioned tha t th e eminent Frenc h mathematician J . Hadamar d wrot e in the 1920 s a monograph i n connection with the preparation o f an edition of the collected work s of N.I . Lobachevski . Th e author' s manuscrip t wa s translated b y A.V . Vasil'ev an d edited b y B.A . Fuks .
Hadamard's monograp h i s a survey , an d mos t o f it s proposition s ar e state d without proof . Thi s calle d fo r a numbe r o f notes . Th e note s wer e writte n b y B.A . Fuks.
Fuks' boo k ca n serv e a s a n introductio n t o th e Hadamar d monograph .
The possibilit y o f establishin g a Lobachevskia n metri c i n a simpl y connecte d region o f the comple x plan e provide d i n th e pas t th e stimulu s fo r th e discover y o f automorphic functions . Th e metri c i n questio n playe d a n essential , an d a t time s crucial, rol e i n al l stage s o f the constructio n o f the gran d edific e o f thes e function s which have such important application s to many problems of mathematical analysis .
The basi c ai m o f Hadamard' s smal l boo k i s t o demonstrat e th e fundamenta l importance o f the Lobachevskia n metri c fo r th e theor y o f automorphi c functions .
While th e autho r usuall y omit s proofs , i n mos t case s h e provide s thei r under -lying idea s o r outlines . Whe n h e doe s this , h e tend s t o emphasiz e th e significanc e of the relevan t proposition s o r fact s o f Lobachevskian geometr y fo r eac h argument .
By no w i t i s clea r tha t Hadamard' s boo k canno t serv e a s a textboo k o n th e theory o f automorphi c function s an d tha t i t ca n onl y b e recommende d t o reader s who hav e a certai n amoun t o f knowledge o f thi s theory . A goo d sourc e fo r th e re -quired background knowledg e is chapters I I and II I of the recently published secon d edition o f V.V. Golubev' s Lectures on the analytic theory of differential equations, Gostekhizdat, M.-L. , 1950 . Chapter s I I and II I of B.A. Fuks ' Non-euclidean geome-try in the theory of conformal and pseudoconformal mappings (Gostekhizda t 1951) , volume V i n thi s series , ar e a goo d sourc e o f indispensabl e informatio n abou t th e Lobachevskian metri c i n a simpl y connecte d domai n o f th e comple x plane , abou t the group of motions generated b y this metric, and about it s properly discontinuou s subgroups.
Chapter I o f Hadamard' s book , "Th e grou p o f motion s o f th e Lobachevskia n plane an d it s properl y discontinuou s subgroups" , i s o f a n introductor y nature . I t describes th e subsequentl y use d realization s o f th e Lobachevskia n plane , bring s together th e mos t importan t propertie s o f the properl y discontinuou s subgroup s o f its group of motions, and partl y explains the characteristi c singula r feature s o f their fundamental domains .
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xii INTRODUCTIO N
Chapter II , "Properl y discontinuou s group s o f 3 geometries. Fuchsia n groups" , is devote d t o th e stud y o f th e properl y discontinuou s subgroup s o f th e group s o f motions o f th e geometrie s o f Riemann , Euclid , an d Lobachevski . (I n particular , i t deals wit h th e condition s unde r whic h a polygo n i n th e Lobachevskia n plan e ca n serve a s a fundamenta l regio n o f a properly discontinuou s subgrou p o f it s grou p o f motions.)
Chapter II I deal s with Fuchsia n function s an d Chapte r I V with Kleinia n func -tions; i n particular , thes e chapter s includ e th e theor y o f Poincare' s thet a series .
Chapter V is devoted t o application s o f the automorphi c function s constructe d in th e earlie r chapter s t o th e proble m o f uniformizatio n o f algebrai c curve s an d t o the solutio n o f ordinary linea r differentia l equation s wit h algebrai c coefficients .
The las t chapter , Chapte r VI , is titled "Fuchsia n group s an d geodesi c lines". I t is relatively shor t an d i s in the nature o f an appendix . Th e book includes reference s to work s tha t contai n comprehensiv e account s o f touched-o n issues . Additiona l references ar e foun d i n the editor' s notes .
REMARK. I n th e Englis h translation , th e Russia n footnote s appea r a s Note s at th e en d o f eac h o f the si x chapters . (Eds. )
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