# Real Analysis – Serlo

## Introduction

• What is analysis? • Why study analysis? • Propositional logic • Mathematical induction • Real numbers ## Complex numbers

• Introduction and motivation • Definition of complex numbers • Absolute value and conjugation • Polar representation • Drawing complex-valued functions • Exercises ## Supremum and infimum

• Supremum and infimum • The infinite case • How to prove existence of a supremum or infimum • Properties of supremum and infimum ## Sequences

• Sequences • Explicit and recursive description • Examples and properties of sequences • Exercises ## Convergence and divergence

• Definition of limit • How to prove convergence and divergence • Examples for limits • Unbounded sequences diverge • Limit theorems • The squeeze theorem • Monotony criterion • How to prove convergence for recursive sequences • Exercises ## Subsequences, Accumulation points and Cauchy sequences

• Subsequence • Accumulation points of sequences • Accumulation points of sets • The Bolzano-Weierstrass theorem • Divergence to infinity • Divergence to infinity: rules • Lim sup and lim inf • Cauchy sequences • Exercises ## Series

• Series • Computation rules for series • Telescoping sums and series • Geometric series • Harmonic series • Exponential series • Absolute convergence of a series • Rearrangement theorem for series • Exercises ## Convergence criteria for series

• Overview: convergence criteria • Cauchy criterion • Term test • Bounded series and convergence • Direct comparison test • Root test • Ratio test • Alternating series test • Cauchy condensation test • Application of convergence criteria • Exercises ## Exponential and Logarithm functions

• Derivation and definition of the exponential series • Properties of the exponential series • Logarithmic function • Real exponents • Exp and log functions for complex numbers • Exercises ## Trigonometric and Hyperbolic functions

• Sine and cosine ## Continuity

• Continuity of functions • Epsilon-delta definition of continuity • Sequential definition of continuity • Limit of functions • Proving continuity • Proving discontinuity • Composition of continuous functions • Extreme value theorem • Intermediate value theorem • Continuity of the inverse function • Uniform continuity • Lipschitz continuity • Exercises ## Differential Calculus

• Derivatives • Computing derivatives • Computing derivatives - special • Derivative - inverse function • Examples for derivatives • Derivatives of higher order • Rolle's theorem • Mean value theorem • Constant functions • Monotone functions • Derivative and local extrema • L'Hôspital's rule • Overview: continuity and differentiability • Exercises 1 • Exercises 2 • Exercises 3 • Exercises 4 