Lectures on Transcendental Numbers
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546 Kurt Mahler
Lectures on Transcendental Numbers Edited and Completed by B.
Divi~
and W. J. Le Veque
Springer-Verlag Berlin· Heidelberg· New York 1976
Author Kurt Mahler Department of Mathematics Research School of Physical Sciences Australian National University Canberra, ACT 2600/ Australia
Editors B. Divi~ t W. J. Le Veque Claremont Graduate School Claremont, California/USA
Library 01 Coalret. Cala",la, I. Pabll.atl•• Da.a
Mahler, Kurt.
Lectures on tl"8ollBcendentaJ. numbers.
(Lecture notes in ....thematic. ; 5116) Bibliography: p. Inclucles index. 1. Nwnbers, TranscendentaJ.. I. Div:l.s, B. II. LeVeque, William Judson. III. r1t1e. IV.
ries:
SeLecture notes in ....tbematics (Berlin) ; 546.
W.L2B no. 546 {QA247.51 510'.88 [512'.731 76-44348
AMS Subject Classifications (1970): 10F35 ISBN 3-540-07986-6 Springer·Veriag Berlin' Heidelberg· New York ISBN 0-387·07986-6 Springer-Verlag New York' Heidelberg· Berlin This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concerned, specifically those of translation, reo printing. re-use of illustrations, broadcasting. reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use. a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1976
After the manuscript of this book had gone to the printers, our collaborator, Professor B. Divis, died suddenly at the early age of 32, while attending a number theoretic meeting at Illinois State University. This is a great loss to his family, to mathematics, as well as to us.
K. Mahler W.J. Le Veque
PREFACE
The rather small list of modern books on transcendental numbers (Siegel 1949; Gelfond 1952;
Schneider 1957;
Lang 1962;
enriched by four new ones (Stolarsky 1974; 1975).
Ramachandra 1969) has recently been Waldschmidt 1974;
Baker 1975;
Masser
Baker's book is particularly valuable because it contains an account of his
powerful new method and a review of some of its applications. The present book is derived from lectures given by Mahler during the last twenty years at different times and places, and edited and sometimes enlarged by Divis and LeVeque.
It aims at giving an account of some old and classical results and
methods on transcendency and in particular to present in all its details the important Siegel-Shidlovski theory of the transcendency of a special class of entire functions (the Siegel
E-functions) which satisfy linear differential equations with
rational functions as coefficients.
In the earlier books this theory was only
sketched. Chapter 1 discusses the existence and gives first examples of transcendental numbers and concludes with a rather general necessary and sufficient condition for transcendency.
As it will be needed later on, Chapter 2 gives a short account of the
theory of formal Laurent and power series in one variable, and it con
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