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

This Analysis 1 course, taught by Prof Imamoglu, was very well-structured and effectively presented. I particularly appreciated his use of handwritten notes on paper. This approach felt more deliberate and natural for explaining complex proofs and concepts.

The primary difficulty for me was the language of instruction. As the course was conducted in German, it presented a significant challenge and made it harder to grasp the nuanced details. Despite the language barrier, the quality of the instruction was undeniable. Overall, it was a good course.

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