Lectures related to Applications of Convergence Theorems

Improper Integrals: Convergence and Comparison

Explores improper integrals, convergence criteria, comparison theorems, and solid revolution.

Residue Theorem: Calculating Integrals on Closed Curves

Covers the application of the residue theorem in calculating integrals on closed curves in complex analysis.

Generalized Integrals: Convergence and Divergence

Explores the convergence and divergence of generalized integrals using comparison methods and variable transformations.

Generalized Integrals and Convergence Criteria

Covers generalized integrals, convergence criteria, series convergence, and harmonic series in analysis.

Convergence of Integrals: Criteria and Examples

Explores the convergence of integrals through criteria and examples, emphasizing the importance of understanding both sides' convergence.

End of Semester Contact Session

Covers the transition to a blended format, student performance, and mathematical analysis methods.

Differentiation under Integral Sign

Explores differentiation under the integral sign, comparing it with the Riemann integral and discussing key assumptions and theorems.

Real Analysis: Summary

Covers real numbers, sequences, series, functions, limits, derivatives, Taylor series, integrals, and more.

Analysis IV: Convergence Theorems and Integrable Functions

Covers convergence theorems and integrable functions, including the Lebesgue integral and Borel-Cantelli sets.

Analytic Continuation: Residue Theorem

Covers the concept of analytic continuation and the application of the Residue Theorem to solve for functions.

Comparison Series and Integrals

Explores the relationship between series and integrals, highlighting convergence criteria and function examples.

Advanced Analysis II: Integrals and Functions

Covers advanced topics in analysis, focusing on integrals, functions, and their properties.

Monotone Convergence: Fatou's Lemma

Explores monotone convergence, dominated convergence, and Fatou's lemma with practical examples.

Curve Length and Function Definition

Explores curve length, function definition, continuity, derivatives, integrals, and graphical representations of functions in two variables.

Power series and Taylor series

Covers the properties of improper integrals, power series, and Taylor series.

Development of Taylor Polynomials

Covers the development of Taylor polynomials of order p around a point x.

Uniform Integrability and Convergence

Explores uniform integrability, convergence theorems, and the importance of bounded sequences in understanding the convergence of random variables.

Complex Integration: Fourier Transform Techniques

Discusses complex integration techniques for calculating Fourier transforms and introduces the Laplace transform's applications.

Generalized Integrals: Definition and Applications

Covers the definition and applications of generalized integrals in advanced analysis, including real functions, differential equations, and multiple integrals.

Generalized Integrals: Definitions and Criteria

Covers the definition of generalized integrals and comparison theorems for convergence.