In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis. Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point z0" means not just differentiable at z0, but differentiable everywhere within some neighbourhood of z0 in the complex plane. (Wikipedia).
Complex analysis: Holomorphic functions
This lecture is part of an online undergraduate course on complex analysis. We define holomorphic (complex differentiable) functions, and discuss their basic properties, in particular the Cauchy-Riemann equations. For the other lectures in the course see https://www.youtube.com/playlist
From playlist Complex analysis
What are domains of holomorphy?
We define domains of holomorphy in C^n. We introduce holomorphically convex domains. We state the Cartan-Thullen theorem, and list consequences. One if them provides the existence of a smallest domain of holomorphy containing a fixed domain. For more details see Hormander's "An introducti
From playlist Several Complex Variables
Tangential Lipschitz Gain for Holomorphic Functions - Sivaguru Ravisankar
Sivaguru Ravisankar The Ohio State University; Member, School of Mathematics October 1, 2012 For more videos, visit http://video.ias.edu
From playlist Mathematics
Transcendental Functions 3 Examples using Properties of Logarithms.mov
Examples using the properties of logarithms.
From playlist Transcendental Functions
Transcendental Functions 19 The Function a to the power x.mp4
The function a to the power x.
From playlist Transcendental Functions
From playlist MATH2621 Higher Complex Analysis
This video explains what a mathematical function is and how it defines a relationship between two sets, the domain and the range. It also introduces three important categories of function: injective, surjective and bijective.
From playlist Foundational Math
(New Version Available) Inverse Functions
New Version: https://youtu.be/q6y0ToEhT1E Define an inverse function. Determine if a function as an inverse function. Determine inverse functions. http://mathispower4u.wordpress.com/
From playlist Exponential and Logarithmic Expressions and Equations
Complex Analysis (Advanced) -- The Schwarz Lemma
A talk I gave concerning my recent results on the Schwarz Lemma in Kähler and non-Kähler geometry. The talk details the classical Schwarz Lemma and discusses André Bloch. This is part 1 of a multi-part series. Part 1 -- https://youtu.be/AWqeIPMNhoA Part 2 -- https://youtu.be/hd7-iio77kc P
From playlist Complex Analysis
MATH 331: Compactifying CC - part 3 - Functions to PP^1
We describe three compactifications of the complex numbers: The one point compactification, the Riemann Sphere and the complex projective line. In a subsequent video we explain the following facts: *Why all holomorphic functions on the compactification are constant. *Why endomorphism of PP
From playlist The Riemann Sphere
https://www.math.ias.edu/files/media/agenda.pdf More videos on http://video.ias.edu
From playlist Mathematics
Complex analysis: Analytic continuation
This lecture is part of an online undergraduate course on complex analysis. We discuss analytic continuation, which is the extraordinary property that the values of a holomorphic function near one point determine its values at point far away. We give two examples of this: the gamma functi
From playlist Complex analysis
Complex Analysis - Part 4 - Holomorphic and Entire Functions [dark version]
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From playlist Complex Analysis [dark version]
Miroslav Englis: Analytic continuation of Toeplitz operators
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Analysis and its Applications
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
MATH 331: Compactifying CC - part 4 - Endomorphisms of PP^1 are Rational
In this video we finally prove that holomorphic maps from PP^1 to itself are rational.
From playlist The Riemann Sphere
Complex analysis: Singularities
This lecture is part of an online undergraduate course on complex analysis. We discuss the different sorts of singularities of a holomorphic function (removable singularities, poles, essential singularities, branch-points, limits of singularities, natural boundaries) and give examples of
From playlist Complex analysis
Positivity and algebraic integrability of holomorphic foliations – Carolina Araujo – ICM2018
Algebraic and Complex Geometry Invited Lecture 4.7 Positivity and algebraic integrability of holomorphic foliations Carolina Araujo Abstract: The theory of holomorphic foliations has its origins in the study of differential equations on the complex plane, and has turned into a powerful t
From playlist Algebraic & Complex Geometry