Theorems in complex analysis | Articles containing proofs
In complex analysis, the open mapping theorem states that if U is a domain of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map (i.e. it sends open subsets of U to open subsets of C, and we have invariance of domain.). The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function f(x) = x2 is not an open map, as the image of the open interval (−1, 1) is the half-open interval [0, 1). The theorem for example implies that a non-constant holomorphic function cannot map an open disk onto a portion of any line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1. (Wikipedia).
Math 135 Complex Analysis Lecture 04 012915: Basic Topological Concepts part 2
Closed sets; closed sets and set operations; topological continuity (inverse image of open set is open); sequences; Cauchy sequence; closedness in terms of convergent sequences; continuity in terms of sequences; connected and path-connected sets; compact sets; Heine-Borel theorem (statemen
From playlist Course 8: Complex Analysis
Intro to Open Sets (with Examples) | Real Analysis
We introduce open sets in the context of the real numbers, along with examples and nonexamples of open sets. This is an important topic in the topology of the reals. We say a subset U of the reals is open if, for any x in U, there exists a delta-neighborhood of x that is contained in U. We
From playlist Real Analysis
How to Prove a Function is Not an Open Function
How to Prove a Function is Not an Open Function If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Thank you:)
From playlist Topology
Math 135 Complex Analysis Lecture 07 021015: Analytic Functions
Definition of conformal mappings; analytic implies conformal; Cauchy-Riemann equations are satisfied by analytic functions; partial converses (some proven, some only stated); definition of harmonic functions; harmonic conjugates
From playlist Course 8: Complex Analysis
Algebraic Topology - 5.4 - Mapping Spaces and the Compact Open Topology
We prove the adjunction between Top(X,-) and X\times- at least on the level of sets.
From playlist Algebraic Topology
All About Closed Sets and Closures of Sets (and Clopen Sets) | Real Analysis
We introduced closed sets and clopen sets. We'll visit two definitions of closed sets. First, a set is closed if it is the complement of some open set, and second, a set is closed if it contains all of its limit points. We see examples of sets both closed and open (called "clopen sets") an
From playlist Real Analysis
Jacob Tsimerman - o-minimality and complex analysis
This is the second talk in the Minerva Mini-course, Applications of o-minimality in Diophantine Geometry, by Jacob Tsimerman, University of Toronto and Princeton's Fall 2021 Minerva Distinguished Visitor
From playlist Minerva Mini Course - Jacob Tsimerman
Proof for Unions and Intersections of Open Sets | Real Analysis
We prove the union of a collection of open sets is open, and the intersection of a finite collection of open sets is open. To do this, we use basic set operation properties and the definition of open sets. #RealAnalysis Intro to Open Sets: https://youtu.be/pnWgj8jjs3w Real Analysis playl
From playlist Real Analysis
Hilbert Space Techniques in Complex Analysis and Geometry (Lecture 1) by Dror Varolin
PROGRAM CAUCHY-RIEMANN EQUATIONS IN HIGHER DIMENSIONS ORGANIZERS: Sivaguru, Diganta Borah and Debraj Chakrabarti DATE: 15 July 2019 to 02 August 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Complex analysis is one of the central areas of modern mathematics, and deals with holomo
From playlist Cauchy-Riemann Equations in Higher Dimensions 2019
8ECM Plenary Lecture: Franc Forstnerič
From playlist 8ECM Plenary Lectures
I created this video with the YouTube Video Editor (http://www.youtube.com/editor)
From playlist Complex Analysis Made Simple
Introduction to quadrature domains (Lecture 5) by Kaushal Verma
PROGRAM CAUCHY-RIEMANN EQUATIONS IN HIGHER DIMENSIONS ORGANIZERS: Sivaguru, Diganta Borah and Debraj Chakrabarti DATE: 15 July 2019 to 02 August 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Complex analysis is one of the central areas of modern mathematics, and deals with holomo
From playlist Cauchy-Riemann Equations in Higher Dimensions 2019
Strongly minimal groups in o-minimal structures - K. Peterzil - Workshop 3 - CEB T1 2018
Kobi Peterzil (Haifa) / 27.03.2018 Strongly minimal groups in o-minimal structures Let G be a definable two-dimensional group in an o-minimal structure M and let D be a strongly minimal expansion of G, whose atomic relations are definable in M. We prove that if D is not locally modular t
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Complex Analysis: Casorati Weierstrass Theorem (Intro)
Today, we introduce essential singularities and outline the Casorati-Weierstrass theorem. The proof of the theorem will be in the next video.
From playlist Complex Analysis
CTNT 2022 - p-adic Fourier theory and applications (by Jeremy Teitelbaum)
This video is one of the special guess talks or conference talks that took place during CTNT 2022, the Connecticut Summer School and Conference in Number Theory. Note: not every special guest lecture or conference lecture was recorded. More about CTNT: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2022 - Conference lectures and special guest lectures
Peter Bubenik - Lecture 2 - TDA: Theory
38th Annual Geometric Topology Workshop (Online), June 15-17, 2021 Peter Bubenik, University of Florida Title: TDA: Theory Abstract: In the second talk, I will discuss some of the theory of TDA. An important feature of TDA is that many of its constructions have been proven to be stable -
From playlist 38th Annual Geometric Topology Workshop (Online), June 15-17, 2021
Finiteness theorems for the space of holomorphic mappings by Jaikrishnan Janardhanan
PROGRAM CAUCHY-RIEMANN EQUATIONS IN HIGHER DIMENSIONS ORGANIZERS: Sivaguru, Diganta Borah and Debraj Chakrabarti DATE: 15 July 2019 to 02 August 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Complex analysis is one of the central areas of modern mathematics, and deals with holomo
From playlist Cauchy-Riemann Equations in Higher Dimensions 2019
MAXIMUM PRINCIPLE -- Part 1 -- Core Theorems of Complex Analysis
Part 2: https://www.youtube.com/watch?v=jmP4VlgZvb0 Part 3: https://www.youtube.com/watch?v=fLnRDhhzWKQ In this video, we give a proof of the Maximum Principle, which is a monumental result in the subject of complex analysis. The maximum principle is also referred to as the maximum modul
From playlist Complex Analysis
What are complex numbers? | Essence of complex analysis #2
A complete guide to the basics of complex numbers. Feel free to pause and catch a breath if you feel like it - it's meant to be a crash course! Complex numbers are useful in basically all sorts of applications, because even in the real world, making things complex sometimes, oxymoronicall
From playlist Essence of complex analysis