Lecturer	Alessio Figalli
Coordinator	Bernat Ramis Vich
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	Analysis 1

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

Week	Lecture	Date	Topics	Chapters
1	1	17.09.	Introduction	1.1
	2	18.09.	Set Theory, the axioms of the real numbers	1.2, 2.1
2	3	22.09.	Ordered fields, functions	2.1.1
	4	24.09.	Axiom of completeness, intervals	2.1.2, 2.1.3
	5	25.09.	Set operations, complex numbers	2.1.3, 2.2.1, 2.2.2
3	6	29.09.	Complex numbers, maximum and supremum	2.2.2, 2.3
	7	01.10.	Consequences of completeness	2.4.1, 2.4.2
	8	02.10.	Countability and uncountability, convergence of sequences of real numbers	2.4.2, 2.5.1
4	9	06.10.	Convergent subsequences and accummulation points, addition, multiplication and inequalities of sequences	2.5.2, 2.5.3
	10	08.10.	Boundeed and monotone sequences, superior and inferior limit	2.5.4, 2.5.5
	11	09.10.	Cauchy sequences, improper limits, sequences of complex numbers, boundedness and monotonicity of real-valued functions	2.5.6, 2.5.7, 2.6, 3.1.1
5	12	13.10.	Continuity, sequential continuity	3.1.2, 3.1.3
	13	15.10.	Intermediate Value Theorem, Inverse Function Theorem	3.2.1, 3.2.2
	14	16.10.	Continuous functions on compact intervals: Boundedness and extrema, uniform continuity. Existence of the exponential function	3.3, 3.4.1
6	15	20.10.	Properties of the exponential function, definition and properties of the logarithm, limits of functions (definition and limit of compositions)	3.4.2, 3.4.3, 3.5.1
	16	22.10.	Improper and one-sided limits, Landau notation	3.5.1, 3.5.2, 3.5.3
	17	23.10.	Sequences of functions (pointwise and uniform convergence), introduction to series of real numbers	3.6
7	18	27.10.	Series with nonnegative elements (comparison test and Cauchy condensation test), conditional convergence, Leibnitz and Cauchy criteria, absolute convergence implies convergence	4.1.1, 4.1.2, 4.1.3, 4.2.1
	19	29.10.	Cauchy root and D'Alembert's quotient criteria, reordering series, product theorem	4.2.1, 4.2.2, 4.2.3
	20	30.10.	Cauchy product, series of complex numbers, power series (radius of convergence and continuity), complex power series	4.2.3, 4.3, 4.4.1, 4.4.2
8	21	03.11.	The exponential and trigonometric functions as power series, the circle number, polar coordiantes in the complex plane, the complex logarithm and other trigonometric and hyperbolic functions	4.5.1, 4.5.2, 4.5.3, 4.5.4, 4.5.5, 4.5.6
		05.11.	Repetition	Notes
		06.11.	Repetition	Notes
9	22	10.11.	Definition and geometrical interpretation of the derivative, differentiation rules (sum, product, polynomials and chain rule)	5.1.1, 5.1.2
	23	12.11.	Quotient rule for derivatives, derivative of the inverse, local extrema, Mean Value Theorem	5.1.2, 5.2.1, 5.2.2
	24	13.11.	Lipschitz continuity vs bounded derivative, Cauchy Mean Value Theorem, l'Hôpital's rule, monotonicity and convexity via differential calculus	5.2.2, 5.2.3, 5.2.4
10	25	17.11.	Convexity and derivatives, differentiation of trigonometric functions	5.2.4, 5.3
	26	19.11.	Step functions and their integral, integrability of real-valued functions	6.1, 6.2.1
	27	20.11.	Linearity and monotonicity of the Riemann integral, integrablity of monotone functions, integrablity of continuous functions	6.2.2, 6.3.1, 6.3.2
11	28	24.11.	Integration and sequences of functions, the Fundamental Theorem of Calculus, integration by parts and by substitution	6.3.3, 7.1.1, 7.1.2
	29	26.11.	Improper integrals and integral test for series, integration and differentiation of power series	7.1.3, 7.2
	30	27.11.	Integration by parts and substitution in Leibniz notation, integration of rational functions	7.3.1, 7.3.2, 7.3.3, 7.3.4
12	31	01.12.	Definite integrals, the Gamma Function, Taylor approximation with integral remainder and with Lagrange remainder	7.3.5, 7.3.6, 7.4.1
	32	03.12.	Taylor approximation with little-o and big-O, analytic functions	7.4.1, 7.4.2
	33	04.12.	Introduction to Ordinary Differential Equations (classification and examples), linear first order ODEs (homogeneous and non-homogeneous)	8.1, 8.1.1
13	34	08.12.	Autonomous First Order ODEs (separation of variables), homogeneous linear second order ODEs with constant coefficients (characteristic polynomial)	8.1.2, 8.1.3
	35	10.12.	Existence and uniqueness (homogeneous 2nd order ODEs with constant coefficients), Wronskian and linear dependence, existence and uniqueness (non-homogeneous 2nd order ODEs with constant coefficients)	8.1.3, 8.1.4
		11.12.	Repetition	Notes (and extra examples)
14	36	15.12.	Extra material: existence and uniqueness for ODEs via Cauchy-Lipschitz and for higher order ODEs	8.2
		17.12.	Repetition
		18.12.	Repetition

Exercises

Schedule: Exercise Sheet n will be posted on Friday of the (n)th week of the semester and is due by Wednesday of the (n+2)th week at 12:00 (midday). For instance, Exercise Sheet 2 will be posted on Friday of the 2nd week and will be due on Wednesday of the 4th week at 12:00 (midday).

Solutions to bonus exercises

Exercise Classes

TA Notes

Analysis 1

What are the "Fokusgruppen"?
What is the "Study Center"?

Literature

• K. A. Ross Elementary Analysis
• O. Forster Analysis 1