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

Deadline

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).

The bonus added to your grade (before rounding) follows the formula:

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 Classes

Literature

• H. Amann, J. Escher Analysis I
• J. Appell Analysis in Beispielen und Gegenbeispielen
• R. Courant Vorlesungen über Differential- und Integralrechnung
• O. Forster Analysis 1
• H. Heuser Lehrbuch der Analysis
• K. Königsberger Analysis 1
• W. Walter Analysis 1
• V. Zorich Mathematical Analysis I (englisch)
• A. Beutelspacher "Das ist o.B.d.A. trivial"
• H. Schichl, R. Steinbauer Einführung in das mathematische Arbeiten

Pictures

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