Table Of Content

Uppsala Lectures on Calculus On Euler’s footsteps E.V. Shchepin Topology Atlas, 2003 Topology Atlas Toronto, Ontario, Canada http://at.yorku.ca/topology/ [email protected] E.V. Shchepin V.A. Steklov Institute of Mathematics Russian Academy of Sciences [email protected] Cataloguing in Publication Data Uppsala lectures on Calculus. On Euler’s footsteps / by E.V. Shchepin. viii, 129 p. ISBN 0-9730867-2-6 1. Calculus I. Shchepin, E.V. II. Title. Dewey QA303 MSC (2000) 28-01 Copyright (cid:13)c  E.V. Shchepin. All rights reserved. Individual users of this publicationarepermittedtomakefairuseofthematerialinteaching,researchand reviewing. No part of this publication may be used for commercial redistribution without the prior permission of the publisher or the copyright holder. ISBN 0-9730867-2-6 PublishedSeptember. Preliminaryversionsofthisdocumentweredistributed ontheWorldWideWeb. Thisdocumentisavailableinseveralformatsforprinting, online browsing and viewing; Please contact the publisher for details. Contents Preface v The Legend of Euler’s Series vii Chapter 1. Series 1 1.1. Autorecursion of Infinite Expressions 2 1.2. Positive Series 7 1.3. Unordered Sums 13 1.4. Infinite Products 16 1.5. Telescopic Sums 20 1.6. Complex Series 24 Chapter 2. Integrals 33 2.1. Natural Logarithm 34 2.2. Definite Integral 40 2.3. Stieltjes Integral 44 2.4. Asymptotics of Sums 47 2.5. Quadrature of Circle 52 2.6. Virtually monotone functions 57 Chapter 3. Derivatives 63 3.1. Newton-Leibniz Formula 64 3.2. Exponential Functions 69 3.3. Euler Formula 74 3.4. Abel’s Theorem 79 3.5. Residue Theory 83 3.6. Analytic Functions 89 Chapter 4. Differences 95 4.1. Newton Series 96 4.2. Bernoulli Numbers 102 4.3. Euler-Maclaurin Formula 107 4.4. Gamma Function 112 4.5. The Cotangent 118 4.6. Divergent Series 124 iii Euler’s Introductio in Analysin Infinitorum, . iv Preface This book represents an introductory course of Calculus. The course evolved from the lectures, which the author had given in the Kolmogorov School in years – for the one-year stream. The Kolmogorov School is a special physics- mathematical undergraduate school for gifted children. Most of the graduates of KolmogorovSchoolcontinuetheireducationinMoscowUniversity,wheretheyhave to learn the Calculus from the beginning. This motivates the author efforts to create a course of Calculus, which on the one hand facilitates to the students the perception of the standard one, but on the other hand misses the maximum possible of the standard material to provide the freshnessofperceptionofthecustomarycourse. Inthepresentformthecoursewas giveninUppsalaUniversityintheautumnsemesterofforagroupofadvanced first-year students. ThematerialofthecoursecoversthestandardCalculusofthefirst-year,covers the essential part of the standard course of the complex Calculus, in particular, it includes the theory of residues. Moreover it contains an essential part of the theory of finite differences. Such topics presented here as Newton interpolation formula, Bernoulli polynomials, Gamma-function and Euler-Maclaurin summation formula one usually learns only beyond the common programs of a mathematical faculty. And the last lecture of the course is devoted to divergent series—a subject unfamiliar to the most of modern mathematicians. The presence of a number of material exceeding the bounds of the standard courseisaccompaniedwiththeabsenceofsomeof“inevitable”topics1andconcepts. There is no a theory of real numbers. There is no theory of the integral neither Riemann nor Lebesgue. The present course even does not contain the Cauchy criterion of convergence. Such achievements of the ninetieth century as uniform convergence and uniform continuity are avoided. Nevertheless the level of rigor in the book is modern. In the first chapter the greek principle of exhaustion works instead of the theory of limits. “Less words, more actions” this is the motto of the present course. Under “words” we mean “concepts and definitions” and under “actions” we mean “calculations and formulas”. Every lecture gives a new recipe for the evaluation of series orintegralsandisequippedwithproblemsforindependentsolution. Moredifficult problemsaremarkedwithanasterisk. ThecoursehasalottodowiththeConcrete Mathematics of Graham, Knuth, Patashnik.2 The order of exposition in the course is far from the standard one. The stan- dardmoderncourseofCalculusstartswithsequencesandtheirlimits. Thiscourse, following to Euler’s Introductio in Analysin Infinitorum,3 starts with series. The introductionoftheconceptofthelimitsisdelayeduptotenthlecture. TheNewton- Leibniz formula appears after all elementary integrals are already evaluated. And power series for elementary functions are obtained without help of Taylor series. 1A.Ya.Hinchinwrote: “ThemoderncourseofCalculushastobeginwiththetheoryofreal numbers”. 2R.L.Graham,D.E.Knuth,O.Patashnik,Concrete Mathematics,Addison-Wesley,. 3L. Euler, Introductio in Analysin Infinitorum, . Available in Opera Omnia, Series I, Volume8,Springer,. v vi preface The course demonstrates the unity of real, complex and discrete Calculus. For ex- ample, complex numbers immediately after their introduction are applied to eval- uate a real series. Two persons play a crucial role in appearance of these lectures. These are Alexandre Rusakov and Oleg Viro. Alexandre Rusakov several years was an assis- tent of the author in the Kolmogorov School, he had written the first conspectus of the course and forced the author to publish it. Oleg Viro has invited the author to Uppsala University. Many hours the author and Oleg spent in “correcting of English” in these lectures. But his influence on this course is far more then a simple correction of English. This is Oleg who convinces the author not to construct the integral, and simply reduces it to the concept of the area. The realization of thisideaascendingtoOleg’steacherRokhlinisoneofcharacteristicfeaturesofthe course. Themainmotivationoftheauthorwastopresentthepowerandthebeautyof theCalculus. Theauthorunderstandthatthiscourseissomewheredifficult,buthe believesthatitisnowheretiresome. Thecoursegivesanewapproachtoexposition ofCalculus,whichmaybeinterestingforstudentsaswellasforteachers. Moreover it may be interesting for mathematicians as a “mathematical roman”. The Legend of Euler’s Series “One of the great mathematical challenges of the early th century was to find an expression for the sum of reciprocal squares 1 1 1 1 ((cid:63)) 1+ + + + +... 22 32 42 52 Joh. Bernoulli eagerly sought for this expression for many decades.”1 In  Jac. Bernoulli proved the convergence of the series. In – Goldbach and D. Bernoulli evaluated the series with an accuracy of 0.01. Stirling in  found eight digits of the sum. L. Euler in  calculated the first eighteen digits (!) after the decimal point ofthesum((cid:63))andrecognizedπ2/6,whichhasthesameeighteendigits. Heconjec- turedthattheinfinitesumisequaltoπ2/6. InEulerdiscoveredanexpansion of the sine function into an infinite product of polynomials: sinx x2 x2 x2 x2 ((cid:63)(cid:63)) =(cid:18)1− (cid:19)(cid:18)1− (cid:19)(cid:18)1− (cid:19)(cid:18)1− (cid:19)··· x π2 22π2 32π2 42π2 Comparing this presentation with the standard sine series expansion x3 x5 x7 x9 sinx=x− + − + −··· 3! 5! 7! 9! Eulernotonlyprovedthatthesum ((cid:63))isequaltoπ2/6, moreoverhecalculatedall sums of the type 1 1 1 1 1+ + + + +··· 2k 3k 4k 5k for even k. Putting x=π/2 in ((cid:63)(cid:63)) he got the beautiful Wallis Product π 2·24·46·68·8 = ··· 2 1·33·55·77·9 which had been known since . But Euler’s first proof of ((cid:63)(cid:63)) was not satis- factory. In , in his famous Introductio in Analysin Infinitorum, he presented a proof which was sufficiently rigorous for the th century. The series of reciprocal squares was named the Euler series. If somebody wants to understand all the details of the above legend he has to study a lot of things, up to complex contour integrals. This is why the detailed mathematical exposition of the legend of Euler’s series turns into an entire course of Calculus. The fascinating history of Euler’s series is the guiding thread of the present course, On Euler’s footsteps. 1E.Hairer,G.Wanner,Analysis by its History,Springer,. vii CHAPTER 1 Series 1 1.1. Autorecursion of Infinite Expressions On the contents of the lecture. The lecture presents a romantic style of early analytics. The motto of the lecture could be “infinity, equality and no definitions!”. Infinity is the main personage we will play with today. We demonstrate how infinite expressions (i.e., infinite sums, products, fractions) arise in solutions of simple equations, how it is possible to calculate them, and how the results of such calculations apply to finite mathematics. In particular, we will deduce the Euler-Binet formula for Fibonacci numbers, the first Euler’s formula of the course. We become acquainted with geometric series and the golden section. Achilles and the turtle. The ancient Greek philosopher Zeno claimed that Achillespursuingaturtlecouldneverpassitby,inspiteofthefactthathisvelocity was much greater than the velocity of the turtle. His arguments adopted to our purposes are the following. First Zeno proposed a pursuing algorithm for Achilles: Initialization. Assign to the variable goal the original position of the turtle. Action. Reach the goal. Correction. If the current turtle’s position is goal, then stop, else reassign to the variable goal the current position of the turtle and go to Action. Secondly, Zeno remarks that this algorithm never stops if the turtle constantly moves in one direction. And finally, he notes that Achilles has to follow his algorithm if he want pass the turtle by. He may be not aware of this algorithm, but unconsciously he must perform it. Because he cannot run the turtle down without reaching the original position of the turtle and then all positions of the turtle which the variable goal takes. Zeno’s algorithm generates a sequence of times {t }, where t is the time of k k execution of the k-th action of the algorithm. And the whole time of work of the ∞ algorithm is the infinite sum t ; and this sum expresses the time Achilles (cid:80)k=1 k needs to run the turtle down. (The corrections take zero time, because Achilles really does not think about them.) Let us name this sum the Zeno series. Assume that both Achilles and the turtle run with constant velocities v and w, respectively. Denote the initial distance between Achilles and the turtle by d . 0 Then t1 = dv0. The turtle in this time moves by the distance d1 =t1w = wvd0. By hissecond actionAchillesovercomes thisdistanceintime t2 = dv1 = wvt1, whilethe turtlemovesawaybythedistanced =t w = wd . Soweseethatthesequencesof 2 2 v 1 times{tk}anddistances{dk}satisfythefollowingrecurrencerelations: tk = wvtk−1, dk = wvdk−1. Hence {t } as well as {d } are geometric progressions with ratio w. And the k k v time t which Achilles needs to run the turtle down is t=t +t +t +···=t + wt + w2t +···=t 1+ w + w2 +··· . 1 2 3 1 v 1 v2 1 1(cid:16) v v2 (cid:17) In spite of Zeno, we know that Achilles does catch up with the turtle. And oneeasily gets thetime t heneeds todoitbythefollowingargument: thedistance between Achilles and the turtle permanently decreases with the velocity v − w. Consequently it becomes 0 in the time t = vd−0w = t1v−vw. Comparing the results we come to the following conclusion (1.1.1) v =1+ w + w2 + w3 +··· . v−w v v2 v3 2