# Analysis I: One VariableAutumn 2023

Lecturer
Alessio Figalli
Coordinator
Reto Kaufmann
Lectures

Mo 08:15 - 10:00 in ETA F 5

We 08:15 - 10:00 in HG F 1 with Livestream in HG F 3

Th 08:15 - 10:00 in ETA F 5

Forum
Analysis I: eine Variable
Lecture Notes
Lecture Notes

## Content

Introduction to differential and integral calculus in one real variable: Basic concepts of mathematical thinking, numbers, sequences and series, continuous functions, differentiable functions, ordinary differential equations, Riemannian integration.

## Lecture Outline

Recordings: The lectures are recorded. The videos can be found on the Videoportal. The username and password was sent to all enrolled students on Friday, 15.12.2023.

Week # Date Topics Chapters
1 1 19.09. Introduction 1.1.
2 21.09. Ordered fields 2.1.1
2 3 25.09. Properties of ordered fields, Absolute value, Sign 2.1.1
4 27.09. Axiom of completeness, Real numbers, Square Root, Intervals 2.1.2 & 2.1.3
5 28.09. Neighbourhoods, Open and closed sets in $$\mathbb{R}$$, Field of complex numbers, Complex conjugation 2.1.3 & 2.2.1
3 6 02.10. Open and closed sets in $$\mathbb{C}$$, Bounded Sets in $$\mathbb{R}$$, Maxima and Minima, Suprema and Infima, Undefinite Values 2.2.2 & 2.3.1
7 04.10. Archimidean principle, Uncountability of $$\mathbb{R}$$, Sequences and Convergence 2.4.1, 2.4.2 & 2.5.1
8 05.10. Subsequences, Accumulation points 2.5.1, 2.5.2 & 2.5.3
4 9 09.10. Sequences and Inequalities, Bounded sequences, Monotone sequences, Superior and Inferior limits 2.5.3 & 2.5.4
10 11.10. Superior and Inferior limit vs. Accumulation points, Bounded sequences and convergent subsequences, Cauchy Sequences 2.5.4 & 2.5.5
11 12.10. Improper limits, Sequences of complex numbers, Bounded and Monotone Functions, Continuous Functions 2.5.6, 2.5.7, 3.1.1 & 3.1.2
5 12 16.10. Combinations and Compositions of Continuous Functions, Sequential Continuity 3.1.2 & 3.1.3
13 18.10. Intermediate Value Theorem, Inverse Function Theorem 3.2.1 & 3.2.2
14 19.10. Compact Intervals, Extreme Values, Uniform Continuity 3.3.1 & 3.3.2
6 15 23.10. Exponential and Logarithm, Limits of Functions 3.4.1, 3.4.2, 3.4.3 & 3.5.1
16 25.10. Improper limits, One-sided limits, Limits at Infinity, Jumps, Landau Notation 3.5.1, 3.5.2 & 3.5.3
17 26.10. Sequences of Functions, Pointwise convergence, Uniform convergence, Series of real numbers and convergence 3.6.1, 3.6.2 & 4.1.1
7 18 30.10. Conditional convergence, Rearrangement Theorem, Leibniz and Cauchy criterion, Absolute convergence, Root and quotient criteria 4.1.2, 4.1.3 & 4.2.1
19 01.11. Reordering series, Products of series, Series of complex numbers, Power series, Radius of convergence 4.2.2, 4.2.3, 4.3, 4.4.1 & 4.4.2
20 02.11. Exponential map as power series, Trigonometric functions, Circle number, Polar coordinates, Complex logarithm, Hyperbolic functions 4.5.1, 4.5.2, 4.5.3, 4.5.4, 4.5.5 & 4.5.6
8 06.11 & 08.11. Repetition
21 09.11. Derivative, Higher Derivatives, Class $$C^n$$, Smooth functions, Differentiation rules 5.1.1 & 5.1.2
9 22 13.11. Derivative of the inverse, Local Extrema and the first Derivative, Rolle's Theorem and (Cauchy) Mean Value Theorem, L'Hopital's rule 5.1.2, 5.2.1, 5.2.2 & 5.2.3
23 15.11. L'Hopital's rule, Monotonicity and Convexity via Calculus, Differentiation of trigonometric functions 5.2.3, 5.2.4 & 5.3.1
24 16.11. Differentiation of Trigonometric functions, Decompositions and Step functions, Integral of step functions, Riemann integrability 5.3.1, 5.3.2, 5.3.3, 6.1.1, 6.1.2 & 6.2.1
10 25 20.11. Linearity and Monotonicity of integral, Triangle Inequality for integral, Integrability of monotone/continuous functions 6.2.1, 6.2.2, 6.3.1 & 6.3.2
26 22.11. Integration and sequences of functions, Primitve functions, Fundamental theorem of calculus 6.3.3 & 7.1.1
27 23.11. Integration by parts and integration by substitution, Improper integrals, Integral test for series 7.1.2 & 7.1.3
11 28 27.11. Integration and differentiation of power series, Integration methods 7.2, 7.3.1, 7.3.2, & 7.3.3
29 29.11. Integration methods, Improper integration limits 7.3.3, 7.3.4, 7.3.5 & 7.3.6
30 30.11. Gamma function, Taylor approximation with big-O and little-o 7.3.6 & 7.4.1
12 31 04.12. Taylor approximation with Lagrange remainder, Analytic functions 7.4.1 & 7.4.2
32 06.12. Ordinary differential equations, Classification and examples, Linear first order ODEs 7.5.1 & 7.5.2
33 07.12. Autonomous first order ODEs, Linear second order ODEs with constant coefficients 7.5.3, 7.5.4 & 7.5.5
13 11.12 & 13.12. Repetition & Exam preparation
34 14.12. Existence and Uniqueness for first order ODEs 8.2.1
14 35 18.12. Proof of Cauchy-Lipschitz (Extra-material), Higher Order ODEs (Extra-material) 8.2.2 & 8.2.3
20.12. & 21.12. Repetition & Questions

## Bonus Sheet

Exercises
Bonus Sheet
Solutions
Solutions for Bonus Sheet
Wednesday 13.12. 12:00
Guidelines
You can choose three exercises from the ten exercises in the Bonus Sheet. Each correct solution earns you 1 point, while an incorrect one yields 0 points. Please note:
• You're welcome to write your solutions in either German or English.
• Minor computational errors are accepted, as long as they don't simplify the exercise.
• Submitting more than three exercises will result in disregarding ALL exercises (earning 0 points).
Points Bonus
1 0.1
2 0.2
3 0.25

## Exercises

Schedule: Exercise Sheet n will be posted on Monday of the (n+1)th week of the semester and is due by Wednesday of the (n+2)th week at 12:00. For instance, Exercise Sheet 2 will be posted on Monday of the 3rd week and will be due on Wednesay of the 4th week at 12:00.

Hand in: To hand in your solutions, please use the provided upload link (How to). Your uploads can be accessed exclusively by the tutor of your exercise group, so it is crucial that you choose a group on myStudies. In order to upload your solutions, you must be connected to an ETH-WiFi or use a VPN (How to).

Exercise Sheet 0 Übungsserie 0 Wednesday 27.09. 12:00 Submission Lösungsblatt 0
Exercise Sheet 1 Übungsserie 1 Wednesday 04.10. 12:00 Submission Lösungsblatt 1
Exercise Sheet 2 Übungsserie 2 Wednesday 11.10. 12:00 Submission Lösungsblatt 2
Exercise Sheet 3 Übungsserie 3 Wednesday 18.10. 12:00 Submission Lösungsblatt 3
Exercise Sheet 4 Übungsserie 4 Wednesday 25.10. 12:00 Submission Lösungsblatt 4
Exercise Sheet 5 Übungsserie 5 Wednesday 01.11. 12:00 Submission Lösungsblatt 5
Exercise Sheet 6 Übungsserie 6 Wednesday 08.11. 12:00 Submission Lösungsblatt 6
Exercise Sheet 7 Übungsserie 7 Wednesday 15.11. 12:00 Submission Lösungsblatt 7
Exercise Sheet 8 Übungsserie 8 Wednesday 22.11. 12:00 Submission Lösungsblatt 8
Exercise Sheet 9 Übungsserie 9 Wednesday 29.11. 12:00 Submission Lösungsblatt 9
Exercise Sheet 10 Übungsserie 10 Wednesday 06.12. 12:00 Submission Lösungsblatt 10
Exercise Sheet 11 Übungsserie 11 Wednesday 13.12. 12:00 Submission Lösungsblatt 11
Exercise Sheet 12 Übungsserie 12 Wednesday 20.12. 12:00 Submission Lösungsblatt 12
Exercise Sheet 13 Übungsserie 13 -- -- Lösungsblatt 13

## Exercise Classes

Tutor
Benjamin Unger
Mail
beunger@student.ethz.ch
Language
German
Classes

We 12:15 - 13:00 in HG E 33.3

Fr 08:15 - 10:00 in CAB G 52

Material
Polybox
Tutor
Alexander Jürgens
Language
German
Classes

We 13:15 - 14:00 in meetngroom 440 019 6465 on Zoom

Fr 08:15 - 10:00 in meetingroom 440 019 6465 on Zoom

Tutor
Alexander Gillmann
Language
German
Classes

We 12:15 - 13:00 in HG E 33.5

Fr 08:15 - 10:00 in CHN D 46

Tutor
Samuel Oettl
Language
German
Classes

We 13:15 - 14:00 in HG E 33.5

Fr 08:15 - 10:00 in ETZ H 91

Tutor
Berno Binkert
Language
German
Classes

We 12:15 - 13:00 in HG F 26.5

Fr 08:15 - 10:00 in CLA E 4

Tutor
Tanish Patil
Language
English
Classes

We 13:15 - 14:00 in HG F 26.5

Fr 08:15 - 10:00 in HG G 26.3

Material
Polybox
Tutor
Christian Amend
Language
German
Classes

We 12:15 - 13:00 in ML F 40

Fr 08:15 - 10:00 in LFW C 4

Tutor
Carl Wolter
Language
German
Classes

We 12:15 - 13:00 in ML H 41.1

Fr 08:15 - 10:00 in IFW A 34

Material
Polybox
Tutor
Pepijn Cobben
Language
German
Classes

We 13:15 - 14:00 in ML H 41.1

Fr 08:15 - 10:00 in CHN D 44

Tutor
Mikhail Zaytsev
Mail
mzaytsev@student.ethz.ch
Language
German, English, French, Russian
Classes

We 13:15 - 14:00 in ML F 40

Fr 08:15 - 10:00 in IFW C 33

Material
Polybox
Tutor
Nicolo Massari
Language
English
Classes

We 12:15 - 13:00 in ML F 34

Fr 08:15 - 10:00 in LEE C 104

Material
Polybox
Focus
Physics (Cosmology)
Tutor
Giulio Caflisch
Language
Italian
Classes

We 13:15 - 14:00 in ML F 34

Fr 08:15 - 10:00 in LEE C 114

Material
Polybox
analisi1sa23
Tutor
Andy Disheng An
Language
German
Classes

We 12:15 - 13:00 in ML F 38

Fr 08:15 - 10:00 in LEE D 101

Material
Polybox
Zoom
Meeting for 22.09.
Tutor
Ömer Doruk Süder
Languages
English, French and Turkish
Classes

We 13:15 - 14:00 in ML F 38

Fr 08:15 - 10:00 in LEE D 105

Focus
Mathematics
Tutor
Micha Schmid
Language
German
Classes

We 12:15 - 13:00 in HG G 26.1

Fr 08:15 - 10:00 in LFW B 2

Tutor
Lukas Dundulis
Language
English
Classes

We 13:15 - 14:00 in LEE D 101

Fr 08:15 - 10:00 in CHN D 48

Tutor
Cyrill von Flüe
Language
German
Classes

We 12:15 - 13:00 in LFW C 5

Fr 08:15 - 10:00 in NO C 44

Material
Polybox
Tutor
Maurice Schmit
Language
German
Classes

We 13:15 - 14:00 in HG G 26.1

Fr 08:15 - 10:00 in ML J 34.3

Material
Polybox
Tutor
Vinzenz Neuner
Language
German
Classes

We 13:15 - 14:00 in HG E 33.1

Fr 08:15 - 10:00 in HG G 26.1

Tutor
Siddharth Setlur
Language
German
Classes

We 12:15 - 13:00 in HG E 33.1

Fr 11:45 - 13:30 in HCI H 8.1

Tutor
Céline Wallart
Language
German
Classes

We 13:15 - 14:00 in CAB G 52

Fr 11:45 - 13:30 in HIT J 53

## Pictures

You can see the pictures taken in the last lectures on Dropbox.