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MATHEMATICAL ANALYSIS -PROBLEMS AND EXERCISES II

Series of Lecture Notes and Workbooks for TeachingUndergraduate MathematicsAlgoritmuselm´eletAlgoritmusok bonyolults´agaAnalitikus m´odszerek a p´enz¨ugyben ´es a k¨ozgazdas´agtanban

Anal´ızis feladatgyujtem´eny I

Anal´ızis feladatgyujtem´eny II

Bevezet´es az anal´ızisbe

Complexity of Algorithms

Differential Geometry

Diszkr´et matematikai feladatok

Diszkr´et optimaliz´al´as

Geometria

Igazs´agos eloszt´asok

Introductory Course in Analysis

Mathematical Analysis - Exercises I

Mathematical Analysis - Problems and Exercises II

M´ert´ekelm´elet ´es dinamikus programoz´as

Numerikus funkcion´alanal´ızis

Oper´aci´okutat´as

Oper´aci´okutat´asi p´eldat´ar

Parci´alis differenci´alegyenletek

P´eldat´ar az anal´ızishez

P´enz¨ugyi matematika

Szimmetrikus strukt´ur´ak

T¨obbv´altoz´os adatelemz´es

Vari´aci´osz´am´ıt´as ´es optim´alis ir´any´ıt´as L´aszl´o Feh´er, G´eza K´os,´Arp´ad T´oth

MATHEMATICAL

ANALYSIS -

PROBLEMS AND

EXERCISES II

E¨otv¨os Lor´and University

Faculty of Science

Typotex

2014
c?2014-2019, L´aszl´o Feh´er, G´eza K´os,´Arp´ad T´oth, E¨otv¨os Lor´and University, Faculty of Science Editors: G´eza K´os, Zolt´an Szentmikl´ossy

Reader: P´eter P´al Pach

Creative Commons NonCommercial-NoDerivs 3.0 (CC BY-NC-ND3.0) This work can be reproduced, circulated, published and performed for non- commercial purposes without restriction by indicating theauthor"s name, but it cannot be modified.

ISBN 978 963 279 420 4

Prepared under the editorship of Typotex Publishing House http://www.typotex.hu)

Responsible manager: Zsuzsa Votisky

Technical editor: J´ozsef Gerner

Made within the framework of the project Nr. T´AMOP-4.1.2-08/2/A/KMR-

2009-0045, entitled "Jegyzetek ´es p´eldat´arak a matematika egyetemi oktat´a-

s´ahoz" (Lecture Notes and Workbooks for Teaching Undergraduate Mathe- matics). KEY WORDS: Analysis, calculus, derivate, integral, multivariable, complex. SUMMARY: This problem book is for students learning mathematical calcu- lus and analysis. The main task of it to introduce the derivate and integral calculus and their applications.

Contents

I Problems11

1 Basic notions. Axioms of the real numbers13

1.0.1 Fundaments of Logic. . . . . . . . . . . . . . . . . . . 13

1.0.2 Sets, Functions, Combinatorics. . . . . . . . . . . . . 16

1.0.3 Proving Techniques: Proof by Contradiction, Induction18

Fibonacci Numbers. . . . . . . . . . . . . . . . . . . . 21

1.0.4 Solving Inequalities and Optimization Problems. . . 22

1.1 Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.1.1 Field Axioms. . . . . . . . . . . . . . . . . . . . . . . 24

1.1.2 Ordering Axioms. . . . . . . . . . . . . . . . . . . . . 25

1.1.3 The Archimedean Axiom. . . . . . . . . . . . . . . . 25

1.1.4 Cantor Axiom. . . . . . . . . . . . . . . . . . . . . . 26

1.1.5 The Real Line, Intervals. . . . . . . . . . . . . . . . . 27

1.1.6 Completeness Theorem, Connectivity. . . . . . . . . 29

1.1.7 Powers. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Convergence of Sequences31

2.1 Theoretical Exercises. . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Order of Sequences, Threshold Index. . . . . . . . . . . . . . 37

2.3 Limit Points, liminf, limsup. . . . . . . . . . . . . . . . . . . 40

2.4 Calculating the Limit of Sequences. . . . . . . . . . . . . . . 42

2.5 Recursively Defined Sequences. . . . . . . . . . . . . . . . . 46

2.6 The Numbere. . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.7 Bolzano-Weierstrass Theorem and Cauchy Criterion. . . . . 50

2.8 Infinite Sums: Introduction. . . . . . . . . . . . . . . . . . . 50

3 Limit and Continuity of Real Functions55

3.1 Global Properties of Real Functions. . . . . . . . . . . . . . 55

3.2 Continuity and Limits of Functions. . . . . . . . . . . . . . . 57

3.3 Calculating Limits of Functions. . . . . . . . . . . . . . . . . 60

3.4 Continuous Functions on a Closed Bounded Interval. . . . . 64

5

3.5 Uniformly Continuous Functions. . . . . . . . . . . . . . . . 65

3.6 Monotonity and Continuity. . . . . . . . . . . . . . . . . . . 66

3.7 Convexity and Continuity. . . . . . . . . . . . . . . . . . . . 66

3.8 Exponential, Logarithm, and Power Functions. . . . . . . . . 67

3.9 Trigonometric Functions and their Inverses. . . . . . . . . . 68

4 Differential Calculus and its Applications71

4.1 The Notion of Differentiation. . . . . . . . . . . . . . . . . . 71

4.1.1 Tangency. . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Higher Order Derivatives. . . . . . . . . . . . . . . . . . . . 77

4.3 Local Properties and the Derivative. . . . . . . . . . . . . . 78

4.4 Mean Value Theorems. . . . . . . . . . . . . . . . . . . . . . 78

4.4.1 Number of Roots. . . . . . . . . . . . . . . . . . . . . 79

4.5 Exercises for Extremal Values. . . . . . . . . . . . . . . . . . 79

4.5.1 Inequalities, Estimates. . . . . . . . . . . . . . . . . . 80

4.6 Analysis of Differentiable Functions. . . . . . . . . . . . . . . 81

4.6.1 Convexity. . . . . . . . . . . . . . . . . . . . . . . . . 82

4.7 The L"Hospital Rule. . . . . . . . . . . . . . . . . . . . . . . 82

4.8 Polynomial Approximation, Taylor Polynomial. . . . . . . . 84

5 The Riemann Integral and its Applications87

5.0.1 The Indefinite Integral. . . . . . . . . . . . . . . . . . 87

5.0.2 Properties of the Derivative. . . . . . . . . . . . . . . 89

5.1 The Definite Integral. . . . . . . . . . . . . . . . . . . . . . . 89

5.1.1 Inequalities for the Value of the Integral. . . . . . . . 91

5.2 Integral Calculus. . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.1 Connection between Integration and Differentiation. 92

5.3 Applications of the Integral Calculus. . . . . . . . . . . . . . 92

5.3.1 Calculating the Arclength. . . . . . . . . . . . . . . . 93

5.4 Functions of Bounded Variation. . . . . . . . . . . . . . . . 93

5.5 The Stieltjes integral. . . . . . . . . . . . . . . . . . . . . . . 93

5.6 The Improper Integral. . . . . . . . . . . . . . . . . . . . . . 94

6 Infinite Series97

7 Sequences and Series of Functions103

7.1 Convergence of Sequences of Functions. . . . . . . . . . . . 103

7.2 Convergence of Series of Functions. . . . . . . . . . . . . . . 105

7.3 Taylor and Power Series. . . . . . . . . . . . . . . . . . . . . 107

8 Differentiability in Higher Dimensions109

8.1 Real Valued Functions of Several Variables. . . . . . . . . . 109

8.1.1 Topology of then-dimensional Space. . . . . . . . . . 109

8.1.2 Limits and Continuity inRn. . . . . . . . . . . . . . 112

8.1.3 Differentiation inRn. . . . . . . . . . . . . . . . . . . 114

8.2 Vector Valued Functions of Several Variables. . . . . . . . . 120

8.2.1 Limit and Continuity. . . . . . . . . . . . . . . . . . 120

8.2.2 Differentiation. . . . . . . . . . . . . . . . . . . . . . 120

9 Jordan Measure, Riemann Integral in Higher Dimensions123

10 The Integral Theorems of Vector Calculus131

10.1 The Line Integral. . . . . . . . . . . . . . . . . . . . . . . . . 131

10.2 Newton-Leibniz Formula. . . . . . . . . . . . . . . . . . . . . 132

10.3 Existence of the Primitive Function. . . . . . . . . . . . . . . 133

10.4 Integral Theorems. . . . . . . . . . . . . . . . . . . . . . . . 135

11 Measure Theory139

11.1 Set Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . 139

11.2 Measures and Outer Measures. . . . . . . . . . . . . . . . . . 140

11.3 Measurable Functions. Integral. . . . . . . . . . . . . . . . . 141

11.4 Integrating Sequences and Series of Functions. . . . . . . . . 141

11.5 Fubini Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 142

11.6 Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . 143

12 Complex differentiability145

12.0.1 Complex numbers. . . . . . . . . . . . . . . . . . . . 145

12.0.2 The Riemann sphere. . . . . . . . . . . . . . . . . . . 148

12.1 Regular functions. . . . . . . . . . . . . . . . . . . . . . . . . 148

12.1.1 Complex differentiability. . . . . . . . . . . . . . . . . 148

12.1.2 The Cauchy-Riemann equations. . . . . . . . . . . . 149

12.2 Power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

12.2.1 Domain of convergence. . . . . . . . . . . . . . . . . . 149

12.2.2 Regularity of power series. . . . . . . . . . . . . . . . 150

12.2.3 Taylor series. . . . . . . . . . . . . . . . . . . . . . . 151

12.3 Elementary functions. . . . . . . . . . . . . . . . . . . . . . . 151

12.3.1 The complex exponential and trigonometric functions151

12.3.2 Complex logarithm. . . . . . . . . . . . . . . . . . . . 152

13 The Complex Line Integral and its Applications155

13.0.3 The complex line integral. . . . . . . . . . . . . . . . 155

13.0.4 Cauchy"s theorem. . . . . . . . . . . . . . . . . . . . 156

13.1 The Cauchy formula. . . . . . . . . . . . . . . . . . . . . . . 157

13.2 Power and Laurent series expansions. . . . . . . . . . . . . . 159

13.2.1 Power series expansion and Liouville"s theorem. . . . 159

13.2.2 Laurent series. . . . . . . . . . . . . . . . . . . . . . . 160

13.3 Local properties of holomorphic functions. . . . . . . . . . . 161

13.3.1 Consequences of analyticity. . . . . . . . . . . . . . . 161

13.3.2 The maximum principle. . . . . . . . . . . . . . . . . 161

13.4 Isolated singularities and residue formula. . . . . . . . . . . 162

13.4.1 Singularities. . . . . . . . . . . . . . . . . . . . . . . . 162

13.4.2 Cauchy"s theorem on residues. . . . . . . . . . . . . . 162

13.4.3 Residue calculus. . . . . . . . . . . . . . . . . . . . . 165

13.4.4 Applications. . . . . . . . . . . . . . . . . . . . . . . 166

Evaluation of series. . . . . . . . . . . . . . . . . . . . 166 Evaluation of integrals. . . . . . . . . . . . . . . . . . 167

13.4.5 The argument principle and Rouch´e"s theorem. . . . 170

14 Conformal maps173

14.1 Fractional linear transformations. . . . . . . . . . . . . . . . 173

14.2 Riemann mapping theorem. . . . . . . . . . . . . . . . . . . 175

14.3 Schwarz lemma. . . . . . . . . . . . . . . . . . . . . . . . . . 178

14.4 Caratheodory"s theorem. . . . . . . . . . . . . . . . . . . . . 179

14.5 Schwarz reflection principle. . . . . . . . . . . . . . . . . . . 180

II Solutions181

15 Hints and final results183

16 Solutions195

PrefaceThis collection contains a selection from the body of exercises that have been used in problem session classes at ELTE TTK in the past few decades. These classes include the current analysis courses in the Mathematics BSc programs as well as previous offerings of Analysis I-IV and Complex Functions. We recommend these exercises for the participants and teachers of the Mathematician, Applied Mathematician programs and for themore experi- enced participants of the Teacher of Mathematics program. All exercises are labelled by a number referring to its difficulty. This number roughly means the possible position of the problem inan exam. For the Teacher program the range is 1-7, for the Applied Mathematician program

2-8, and for the Mathematician program 3-9. (Usually the students need

to solve five problems correctly for maximum grade; the sixthand seventh problems are to challenge the best students.) Problems withdifficulty 10 are not expected to appear on an exam, they are recommended for students aspiring to become researchers. For many exercises we are not aware of the exact origin. They are passed on by "word of mouth" from teacher to teacher, or many times from the teacher of the teacher to the teacher. Many exercises may have been created several generations before. However one of the sources can be identified, it is "the mimeo", a widely circulated set of problems duplicated by a mimeograph in the70"s. The problems within "the mimeo" were mainly collected or created by Mikl´os Laczkovich, L´aszl´o Lempert and Lajos P´osa. Let us give only a (most likely not complete) list of our colleagues who were recently giving lectures or leading problem sessions at the Department of Analysis in Real and Complex Analysis:

M´aty´as Bogn´ar, Zolt´an Buczolich,

´Akos Cs´asz´ar, M´arton Elekes, Margit G´emes, G´abor Hal´asz, Tam´as Keleti, Mikl´os Laczkovich, Gy¨orgy Petruska,

Szil´ard R´ev´esz, Rich´ard Rim´anyi, Istv´an Sigray, Mikl´os Simonovics, Zolt´an

Szentmikl´ossy, R´obert Szoke, Andr´as Szucs, Vera T. S´os. 9 Some problems from the textbook Anal´ızis I. of Mikl´os Laczkovich and Vera T. S´os are reproduced in this volume with their kind permission. We are grateful for their generosity. We thank everyone whose help was invaluable in creating thisvolume, the above mentioned professors and all the students who participated in these classes. As usual when typesetting the problems we mayhave added some errors of mathematical or typographical nature; for which we take sole responsibility.

Part I

Problems

11

Chapter 1

Basic notions. Axioms of

the real numbers

1.0.1 Fundaments of Logic

1.0.1.(1)Calculate the truth table

A?(B=?A)

Answer→

1.0.2.(3)Calculate the truth tables.

1.A?B2.

A?B3.A?(B?C)

1.0.3.(2)LetP(x) mean ,,xis even" and letH(x) mean ,,xis divisible by

six". What is the meaning of the following formulas and are they true? (¬ denotes the negation.)

1.P(4)?H(12)

2.?x?P(x)?H(x)?

3.?x?H(x)? ¬P(x)?

4.?x?P(x)?H(x)?

5.?x?P(x)?H(x+ 1)?

6.?x?H(x)?P(x)?

7.?x?¬H(x)? ¬P(x)?

13

141. Basic notions. Axioms of the real numbers

1.0.4.(3)LetH?Rbe a subset. Formalize the following statements and

their negations. Is there a set with the given property?

1.Hhas at most 3 elements.

2.Hhas no least element.

3. Between any two elements ofHthere is a third one inH.

4. For any real number there is a greater one inH.

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