Problems and Theorems in Analysis II

Transcription

1 George P61ya Gabor Szego s and Theorems in Analysis II Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry Reprint of the 1976 Edition Springer

2 George P6lya t Gabor Szeglf t Tram/ator: C. E. Billigheimer McMaster University Mathematics Hamilton, Ontario Canada Originally published as Vol. 216 of the Grundlehren der mathematischen Wissenschaften Mathematics Subject Classification (1991): 10-01,15-01, 15AI5,30-01,30A06, 30A08 CIP data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme P61ya, George: s and theorems in analysis I George P61ya; Gabor Szego.- [Nachdr.].- Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara, Singapore; Tokyo: Springer (Classics in mathematics) 2. Theory of functions, zeros, polynomials, determinants, number theory,geometry.-reprint [der Ausg.] Berlin, Springer, ISBN-13: DOl: / e-isbn-13: This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted onjyunder the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Springer-Verlag Berlin Heidelberg 1998 Softcover reprint of the hardcover 1 st edition 1998 The use of general descriptive names, registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. SPIN Printed on acid-free paper

3 G. P61ya G. Szego s and Theorems in Analysis Volume II Theory of Functions. Zeros. Polynomials Determinants Number Theory Geometry Translation by C. E. Billigheimer Springer-Verlag Berlin Heidelberg New York 1976

4 George P6lya Gabor Szegt;. Stanford University, Stanford, California, USA Claude Elias Billigheimer McMaster University, Hamilton, Ontario, Canada and Maimonides College, Toronto, Ontario, Canada Revised and enlarged translation of Aufgaben und Lehrsatze Qua tier Analysis II, 4th edition, 1971; Heidelberger Taschenbiicher, Bd. 74 AMS Subject Oassifications (1970): 10-01, 15-01,15 A IS, 30-01,30 A 06, 30A 08 ISBN Springer-Verlag Berlin Heidelberg New York ISBN Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are ~d, whether the whole or part of the material is concerned, specifically those of trans1ation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storl!lle in data banks. Under S4 of the German Copyright Law where copies are made for other tiiiin 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 Printed in Germany. Typesettina: William Oowes & Sons Ltd., London, Beccles and Colchester. Printing and bookbinding: Konrad Triltsch, Wilrzbura. Library of Congress Cataloging in Publication Data. P61ya, George, s and theorems in analysis. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 193, 216.) Vol. 2 tranalated by C. E. Billigheimer. Rev. and enl. translation of Aufgaben und Lehrsitze aus der Analysis, 4th ed., Contents: v.i. Series, integral calculus, theory of functions. v. 2. Theory of functions, zeros, polynomials, determinants, number theory, geometry. 1. Mathematical analysis - s, exercises, etc. I. Szeg6, Gabor, 189S.. joint author. n. Title. m. Series. QA301. P SIS'

5 Contents Part Four. Functions of One Complex Variable. Special Part Chapter 1. Maximum Term and Central Index, Maximum Modulus and Number of Zeros Prob- Solu- Numben lern Pqe tion Paae 1 (1-40) Analogy between p(r) and M(r), v(r) and N (r) (41-47) Further Results on p(r) and v(r) (48-66) Connection between p(r), v(r), M(r) and N(r) (67-76) p.(r) and M(r) under Special Regularity Assumptions Chapter 2. Schlicht Mappings 1 (77-83) Introductory Material (84-87) Uniqueness Theorems (88-96) Existence of the Mapping Function (97-120) The Inner and the Outer Radius. The Normed Mapping Function ( ) Relations between the Mappings of Different Domains ( ) The Koebe Distortion Theorem and Related Topics Chapter 3. Miscellaneous s 1 ( ) Various Propositions ( ) A Method of E. Landau ( ) Rectilinear Approach to an Essential Singularity ( ) Asymptotic Values of Entire Functions ( ) Further Applications of the Phragmen-Lindelof Method (*206-*212) Supplementary s

6 vi Part Five. '!be Location of Zeros Chapter 1. RoUe's Theorem IDd Descartes' Rule of Sigas Numbers (1-21) (22-27) (28-41) (42-52) (53-76) (77-86) (87-91) (92-100) Zeros of Functions, Changes of Sign of Sequences Reversals of Sign of a Function First Proof of Descartes' Rule of Signs Applications of Descartes' Rule of Signs. Applications of Rolle's Theorem. Laguerre's Proof of Descartes' Rule of Signs What is the Basis of Descartes' Rule of Signs? Generalizations of Rolle's Theorem Contents Solu tion Chapter 2. The Geometry of the Complex PllDe IDd the Zeros of Polynomials 1 ( ) Center of Gravity of a System of Points with respect to a Point ( ) Center of Gravity of a Polynomial with respect to a Point. A Theorem of Laguerre ( ) Derivative of a Polynomial with respect to a Point. A Theorem of Grace Chapter 3. MiscelllDeous s 1 ( ) Approximation of the Zeros of Transcendental Functions by the Zeros of Rational Functions ( ) Precise Determination of the Number of Zeros by Descartes' Rule of Signs ( ) Additional s on the Zeros of Polynomials Part Six. Polynomials and Trigonometric Polynomials 1 (1-7) Tchebychev Polynomials (8-15) General s on Trigonometric Polynomials (16-28) Some Special Trigonometric Polynomials (29-38) Some s on Fourier Series (39-43) Real Non-negative Trigonometric Polynomials (44-49) Real Non-negative Polynomials (50-61) Maximum-Minimum s on Trigonometric Polynomials (62-66) Maximum-Minimum s on Polynomials (67-76) The Lagrange Interpolation Formula (77-83) The Theorems of S. Bernstein and A. Markov (84-102) Legendre Polynomials and Related Topics ( ) Further Maximum-Minimum s on Polynomials

7 Contents Part Seven. Determinants and Quadratic Forms Numbers 1 (1-16) Evaluation of Determinants. Solution of Linear (17-34) ( ) ( ) (55-72) Equations Power Series Expansion of Rational Functions Generation of Positive Quadratic Forms. Miscellaneous s Determinants of Systems of Functions vii Solution Part Eight. Number Theory Chapter 1. Arithmetical Functions (1-11) (12-20) ( ) (28-37) (38-42) (43-64) (65-78) (79-83) s on the Integral Parts of Numbers. Counting Lattice Points The Principle of Inclusion and Exclusion Parts and Divisors Arithmetical Functions, Power Series, Dirichlet Series. Multiplicative Arithmetical Functions Lambert Series and Related Topics Further s on Counting Lattice Points Chapter 2. Polynomials with Integral Coefficients and Integral-Valued Functions 1 (84-93) Integral Coefficients and Integral-Valued Polynomials (94-115) Integral-Valued Functions and their Prime Divisors ( ) Irreducibility of Polynomials Chapter 3. Arithmetical Aspects of Power Series 1 ( ) Preparatory s on Binomial Coefficients ( ) On Eisenstein's Theorem ( ) On the Proof of Eisenstein's Theorem ( ) Power Series with Integral Coefficients Associated with Rational Functions ( ) Function-Theoretic Aspects of Power Series with Integral Coefficients ( ) Power Series with Integral Coefficients in the Sense of Hurwitz ( ) The Values at the Integers of Power Series that Converge about z=oo

8 viii Chapter 4. Some s on Algebraic Integers Numbers 1 ( ) Algebraic Integers. Fields. 2 ( ) Greatest Common Divisor. 3 ( ) Congruences. 4 ( ) Arithmetical Aspects of Power Series Contents Solution Chapter 5. Miscellaneous s 1 ( ) Lattice Points in Two and Three Dimensions 2 ( ) Miscellaneous s Part Nine. Geometric s 1 (1-25) Some Geometric s Appendix 1 Additional s to Part One. New s in English Edition Author Index Subject Index Topics Errata

9 Notation and Abbreviations We have attempted to be as consistent as possible in regard to notation and abbreviations and to denote quantities of the same nature by the same symbol, at least within the same part. A particular notation may be specified for a few sections. Otherwise the meaning of every letter is explained anew in every problem except when we refer to a previous problem. A problem that is closely related to the preceding one is introduced by the remark "continued"; if it is related to some other problem the relevant number is mentioned, e.g. "continuation of 136". We denote parts by roman numerals, chapters (where necessary) by arabic numerals. The problems are numbered in bold-face. Within the same part only the number of the problem is given; if, however, we refer to another part its number is also indicated. For example if we refer to problem (or solution) 123 of Part IV in a problem (or solution) of Part IV we write "123"; if we refer to it in a problem (or solution) of any other part we write "IV 123". Remarks in square brackets [ ] in a problem are hints, while in a solution (particularly at the beginning of the solution) they are citations or references to other problems that are used in various steps of the proof. All other remarks appear in ordinary parentheses. A reference to a problem number indicates in general that one should consult both problem and solution, unless the opposite is explicitly stated, e.g. "solution 75". Almost always references to the sources are given only in the solution. If a problem has already appeared in print, this fact is indicated in the citations. If the author but no bibliography is mentioned, the problem has been communicated to us as a new problem. s whose number is preceded by the symbol * (as in *206 of PartlY) or contains a decimal point (asin174.1 of Part IY) are new, that is they are either not contained in the original German edition, or else are contained there but are essentially modified in the present English version. If the problem is the same as in the original edition but the solution has some essentially new feature, the symbol is used only in the solution. The abbreviations of the names of journals are taken from the index of Mathematical Reviews and, if not listed there, from the World List of Scientific Periodicals Published , Peter Brown, British Museum, Washington, Butterworths, The most frequently quoted journals are: Abh. Akad. Wiss. St. Petersburg=Akademie der Wissenschaften, St. Petersburg Acta Math. = Acta Mathematica, Stockholm Acta Soc. Sc. Fennicae = Acta Societatis Scientiae Fennicae Amer. Math. Monthly =American Mathematician Monthly Arch. Math. Phys. = Archiv der Mathematik und Physik Atti Acad. Naz. Lincei Rend. = Atti dell' Accademia Nazionale dei Lincei Rendiconti. Cl. Sci. Fis. Mat. Natur. Classe di Scienze Fisiche, Matematiche e Naturali, Roma Berlin. Ber. = Berliner Berichte C.R. Acad. Sci. (Paris) Ser. A-B=Comptes rendus hebdomadaires des seances de l'academie des Sciences, Paris, Series A et B Giorn. Mat. Battaglini = Giornale di Matematiche di Battaglini Jber. deutsch. Math. Verein. =Jahresbericht der deutschen Mathematiker-Vereinigung J. Math. spec. =Journal de Mathematiques Specie1es, Paris J. reine angew. Math. = Journal fiir die reine und angewandte Mathematik Math. Ann. = Mathematische Annalen Math. es term ert = Matematikai es termeszettudomanyi ertesito Math. Z. = Mathematische Zeitschrift Miinchner Ber. = Miinchner Berichte

10 x Notation and Abbreviations Nachr. Akad. Wiss. Gottingen Nouv. AnnIs Math. Nyt. Tidsskr. Proc. Amer. Math. Soc. Proc. Lond. Math. Soc. Trans. Amer. Math. Soc. = Nachrichten der Gesellschaft der Wissenschaften Gottingen = Nouvelles Annales de mathematiques = Nyt tidsskrift for matematik = Proceedings of the American Mathematical Society = Proceedings of the London Mathematical Society = Transactions of the American Mathematical Society The following textbooks are quoted repeatedly and are usually cited by the name of the author only or by a suitable abbreviation (e.g. Hurwitz-Courant; MPR.): G.H. Hardy and E.M. Wright: An Introduction to the Theory of Numbers, 4th Ed. Oxford: Oxford University Press E. Hecke: Vorlesungen tiber die Theorie der algebraischen Zahlen, New York: Chelsea Publishing Co E. Hille: Analytic Function Theory, Vol. I: Boston - New York - Chicago - Atlanta - Dallas - Palo Alto - Toronto - London: Ginn & Co. 1959; Vol. II: Waltham/Mass. - Toronto London: Blaisdell Publishing Co A. Hurwitz - R. Courant: Vorlesungen tiber allgemeine Funktionentheorie und elliptische Funktionen, 4th Ed. Berlin - Gottingen - Heidelberg - New York: Springer K. Knopp: Theory and Applications of Infinite Series, 2nd Ed. London - Glasgow: Blackie & Son G. Kowalewski: EinfUhrung in die Determinantentheorie, 4th Ed. Berlin: Walter de Gruyter G. P6lya: How to Solve It, 2nd Ed. Princeton: Princeton University Press Quoted as HSI. G. P6lya: Mathematics and Plausible Reasoning, Vols. 1 and 2, 2nd Ed. Princeton: Princeton University Press Quoted as MPR.. G. P6lya: Mathematical Discovery, Vols. 1 and 2, Cor. Ed. New York: John Wiley & Sons Quoted as MD. G. Szego: Orthogonal Polynomials, American Mathematical Society Colloquium Publications Vol. XXIII, 3rd Ed. New York: American Mathematical Society E. C. Titchmarsh: The Theory of Functions, 2nd Ed. Oxford - London - Glasgow - New York - Melbourne - Toronto: Oxford University Press E. T. Whittaker and G.N. Watson: A Course of Modem Analysis, 4th Ed. London: Cambridge University Press The following notation and abbreviations are used throughout the book: a,.-+a means "a,. tends to a as " aa~b,. (read "a,. is asymptotically equal to b,.") means "b,.;60 for sufficiently large nand ~ -+1 as ". O(a,.), with a,. > 0, denotes a quantity that, divided by a,., remains bounded, o(a,.) a quantity that, divided by a,., tends to 0 as 1t-+oo. Such notation is used analogously in limit processes other than for 1t-+OO. x-+a+o means "x converges to a from the right"; x->-a-o means "x converges to a from the left". exp (x) = e", where e is the base of natural logarithms. Given n real numbers a10 a2,..., an, max (ai, a2,..., a,.) denotes the largest (or one of the largest) and min (ai, a2,..., a,.) the smallest (or one of the smallest) of the numbers al, a2,..., a,.. Max f(x) and min f(x) have an analogous meaning for a real function defined on an interval a, b, providedf(x) assumes a maximum or a minimum on a, b. Otherwise we retain the same notation for the least upper bound and the greatest lower bound, respectively. Analogous notation is used in the case of functions of a complex variable. sgn x denotes the signum (Kronecker) function: +1 for x>o sgn x= { 0 for x=o -1 for x<o.

11 Notation and Abbreviations [x] denotes the greatest integer that is not greater than x (x-l < [x]~x). Square brackets are, however, also used instead of ordinary parentheses where there is no danger of confusion. (They are also used in a very special sense restricted to Part I, Chap. 1, 5.) z is the conjugate to the complex number z. For the determinant with general term aa., '\, p.= 1,2,..., n, we use the abbreviated notation laa.i~ or laa.ia. = 1... " or laah aa.,... aa"i~. A non-empty connected open set (containing only interior points) is called a region. The closure of a region (the union of the open set and of its boundary) is called a domain. As this terminology is not the one most frequently used, we shall sometimes emphasize it by speaking of an "open region" and a "closed domain". A continuous curve is defined as a single-valued continuous image of the interval 0 ~, ~ 1, i.e. as the set of points z=x+iy, where x=",(t), Y='/J(/), with ",(I) and ';'(t) both continuous functions on the interval O~ t~ 1. The curve is closed if ",(0) = ",(1), ';'(0)= ';'(1), and is without double points if ",(t1) = rp(t.), ';'(t1) = ';'(t2), tl < '2, imply 'I =0, t2 = 1. A curve without double points is also called a simple curve. A simple, continuous curve that is not closed is often referred to as a simple arc. A simple closed continuous curve (a Jordan curve) in the plane determines two regions of which it forms the common boundary. The paths of integration of line integrals or complex integrals are assumed to be continuous and rectifiable. (a, b) denotes the open interval a<x<b, [a, b) the half-open interval a~x<b, (a, b] the half-open interval a<x~b, [a, b] the closed interval a~x~b. When we do not need to distinguish between these four cases we use the term "interval a, b". "Iff" is used occasionally as an abbreviation for "if and only if". xi