Lectures related to Holomorphic function

Holomorphic Functions: Taylor Series Expansion

Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.

Harmonic Forms and Riemann Surfaces

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

Complex Analysis: Holomorphic Functions

Explores holomorphic functions, Cauchy-Riemann conditions, and principal argument values in complex analysis.

Open Mapping Theorem

Explains the Open Mapping Theorem for holomorphic maps between Riemann surfaces.

Complex Derivatives: Cauchy-Riemann Equations

Explores complex derivatives, Cauchy-Riemann equations, rules of derivation, and properties of holomorphic functions.

Analytic Continuation and Uniqueness of Holomorphic Functions

Covers the concept of analytic continuation and the uniqueness of holomorphic functions, including the extension of holomorphic functions and the properties of entire and meromorphic functions.

Cauchy Theorem and Laurent Series

Covers the Cauchy theorem, the conditions to apply it, and the Laurent series.

Laurent Series: Definition and Properties

Covers the definition and properties of Laurent series, including convergence and function expansion.

Functional Equation of Zeta

Covers the functional equation of zeta function and Jensen's formula in complex analysis.

Complex Functions: Norm Equivalence

Explores norm equivalence in complex functions, covering homogeneity and triangular inequality.

Laurent Series and Residue Theorem: Complex Analysis Concepts

Discusses Laurent series and the residue theorem in complex analysis, providing examples and applications for evaluating complex integrals.

Analyse IV: Laurent Series and Singularities

Covers Laurent series, singularities, and meromorphic functions, addressing convergence, holomorphicity, and residue theorem applications.

Convex Functions: Analytical Definition and Geometric Interpretation

Explores convex functions, including properties, definitions, and analytical interpretations, demonstrating how to determine convexity and evaluate limits for different types of functions.

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Complex Analysis: Cauchy Theorem

Explores the Cauchy Theorem and its applications in complex analysis.

Complex Integration and Cauchy's Theorem

Discusses complex integration and Cauchy's theorem, focusing on integrals along curves in the complex plane.

Laurent Series and Convergence: Complex Analysis Fundamentals

Introduces Laurent series in complex analysis, focusing on convergence and analytic functions.

Residues Theorem

Explores the Residues Theorem and the classification of holomorphic functions.

Proper Actions and Quotients

Covers proper actions of groups on Riemann surfaces and introduces algebraic curves via square roots.

Residue Theorem: Calculating Integrals on Closed Curves

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