Analysis 1

• Profs: Ö. Imamoglu
• Website: https://metaphor.ethz.ch/x/2025/fs/401-0212-16L/
• SAM : https://sam-up.math.ethz.ch/
• Admin: Admin
• Material: Material
• Videos:
  • 2024: Videos ETHZ
  • 2025: Videos ETHZ

Lecture Notes

• 01 Introduction, Real Numbers, Completeness
• 02 Real Numbers, Min, Max, Bounds, Supremum, Infimum, Cardinality, Euclidian Space
• 03 Supremum, Infimum, Cross Product, Complex Numbers, Sequences
• 04 Divergence, Limits, Monotony, Weierstrass Theorem, Limit Superior and Inferior
• 05 Sandwich Lemma, Cauchy Criterion, and Bolzano-Weierstrass Theorem
• 06 Accumulation Points, Sequences in Reals and Complex, and Series
• 07 Series Cauchy Criterion, Absolute and Conditional Convergence, Riemann Rearrangement Theorem, Dirichlet’s Theorem
• 08 More Convergence Criteria, Dirichlet’s Theorem, Ratio, Root Tests and Power Series
• 09 Double Series, Cauchy Products, Exponential Functions and Real-Valued Functions
• 10 Continuity, Intermediate Value Theorem
• 11 Continuity, Extreme Value Theorem, Composition of Continuous Functions, Continuity of Inverse Functions
• 12 Continuity, Exponential Function and Function Sequences
• 13 Function Sequences, Continuity of Limit Function under Uniform Convergence, Cauchy Criterion for Uniform Convergence, Series of Functions, Uniform Convergence of Power Series, Trig Functions
• 14 Exponential Function, Arc Measure and Radians, Defining Pi (Existence and Properties), Tangent and Cotangent, Limit of Functions
• 15 Limit of Functions and Rules, One-Sided Limits
• 16 Differentiation, Derivative and Rules
• 17 Chain Rule, Derivatives of Inverse Functions, Central Theorems (Local Extrema, Rolle’s Theorem, Mean Value Theorem)
• 18 Derivatives of Inverse Trig Functions, L’Hopital’s Rule, Convex Functions
• 19 Higher Order Derivatives, Smooth Functions, Power Series and Taylor Approximation
• 20 Taylor Approximation, Higher Derivative Test for Local Extrema, Riemann Integral, Darboux Sums and Integrals
• 21 Dirichlet’s Function, Integrability
• 22 Inequalities and MVT for Integrals, Fundamental Theorem of Calculus, Integration Techniques (by Parts, Substitution)
• 23 Area of Circle, Volume of Sphere, Indefinite Integrals, Partial Fraction Decomposition
• 24 Integration of Convergent Series and Swapping Limit and Integrals, Improper Integrals
• 25 Integral Test for Series Convergence, Stirling’s Formula to Approximate Factorial, Summary

Script

• Chapter 1 - Real Numbers, Euclidean Spaces, and Complex Numbers
• Chapter 2 - Sequences and Series, Approaching Infinity, Divergence, Infinity, Algebra of Limits, Monotone Sequences, Weierstrass, Cauchy, Series, Summing Infinitely, Special Series and Operations
• Chapter 3 - Continuous Functions, Smoothness and Limits, Continuity, Functions Without Jumps, Combining Continuous Functions, Fundamental Theorems of Continuity, Exponential and Trigonometric Functions
• Chapter 4 - Differentiable Functions, The Slope of a Curve, Rules for Differentiation, Mean Value Theorem and Beyond
• Chapter 5 - The Riemann Integral, Measuring Areas Under Curves, Properties and Classes of Integrable Functions, Properties of Integrals, Linearity, Monotonicity, and the Mean Value Theorem for Integrals, Fundamental Theorem and Applications