Start Date: 02/23/2020
Course Type: Common Course |
Course Link: https://www.coursera.org/learn/complex-analysis
Explore 1600+ online courses from top universities. Join Coursera today to learn data science, programming, business strategy, and more.This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today. Each module consists of five video lectures with embedded quizzes, followed by an electronically graded homework assignment. Additionally, modules 1, 3, and 5 also contain a peer assessment. The homework assignments will require time to think through and practice the concepts discussed in the lectures. In fact, a significant amount of your learning will happen while completing the homework assignments. These assignments are not meant to be completed quickly; rather you'll need paper and pen with you to work through the questions. In total, we expect that the course will take 6-12 hours of work per module, depending on your background.
We’ll begin this module by briefly learning about the history of complex numbers: When and why were they invented? In particular, we’ll look at the rather surprising fact that the original need for complex numbers did not arise from the study of quadratic equations (such as solving the equation z^2+1 = 0), but rather from the study of cubic equations! Next we’ll cover some algebra and geometry in the complex plane to learn how to compute with and visualize complex numbers. To that end we’ll also learn about the polar representation of complex numbers, which will lend itself nicely to finding roots of complex numbers. We’ll finish this module by looking at some topology in the complex plane.
Introduction to Complex Analysis This course provides an introductory overview of complex analysis, including an introduction to exponential and logistic regression models. The course is intended for people who are curious about complex analysis, who is looking for mathematical concepts and techniques to deal with complex problems, and for people who are familiar with linear algebra and programming. The course assumes some prior knowledge of programming, and it is strongly recommended that you have some programming skills already. In this course, you will learn how to use complex linear models to study complex problems. We will introduce the calculus of linear models, which allows you to solve problems using linear algebra. You will need to know the integral calculus, and it is strongly recommended that you have some programming skills already. It is strongly recommended that you have a basic knowledge of linear algebra, including calculus. The course is designed to follow the linear algebra and programming techniques introduced in Linear Algebra for Engineers, and it is strongly recommended that you have some programming skills already. This course is designed to follow the linear algebra and programming techniques introduced in Linear Algebra for Engineers, and it is strongly recommended that you have some programming skills already. The course consists of a series of lectures with online assignments, and it is highly recommended that you take the quizzes. The lectures contain a lot of mathematical information, and the online assignments will test your understanding of the material. The course is designed to follow the linear algebra and programming techniques introduced in Linear
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Cauchy–Hadamard theorem | The proof can be found in the book Introduction to Complex Analysis Part II functions in several Variables by B. V. Shabat |
Introduction to Economic Analysis | Introduction to Economic Analysis was the first published complete textbook being openly available online. McAfee was named SPARC innovator for year 2009 for making the book freely accessible. |
Complex analysis | As a differentiable function of a complex variable is equal to the sum of its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). |
Introduction to Economic Analysis | Version 3 is co-authored with Professor Tracy Lewis of the Fuqua School of Business and was published by Flat World Knowledge in 2009 under a Creative Commons license. "Introduction to Economic Analysis" is already in use on campuses from Harvard to New York University. |
Complex analysis | Complex analysis is one of the classical branches in mathematics, with roots in the 19th century and just prior. Important mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has become very popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory. |
Introduction to Economic Analysis | Introduction to Economic Analysis is a university microeconomics textbook by Caltech Professor Preston McAfee. It is available free of charge under Creative Commons license (an open source); under this "license that requires attribution, users can pick and choose chapters or integrate with their own material". |
Complex analysis | The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponential functions, logarithmic functions, and trigonometric functions) into the complex domain. |
Complex analysis | Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including hydrodynamics and thermodynamics and also in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. |
Complex analysis | All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more than one complex dimension in which the analytic properties such as power series expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality) do not carry over. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions. |
Princeton Lectures in Analysis | The Princeton Lectures in Analysis is a series of four mathematics textbooks, each covering a different area of mathematical analysis. They were written by Elias M. Stein and Rami Shakarchi and published by Princeton University Press between 2003 and 2011. They are, in order, "Fourier Analysis: An Introduction"; "Complex Analysis"; "Real Analysis: Measure Theory, Integration, and Hilbert Spaces"; and "Functional Analysis: Introduction to Further Topics in Analysis". |
Zero (complex analysis) | In complex analysis, a zero (sometimes called a root) of a holomorphic function "f" is a complex number "a" such that "f"("a") = 0. |
Complex analysis | Holomorphic functions are complex functions, defined on an open subset of the complex plane, that are differentiable. In the context of complex analysis, the derivative of formula_1 at formula_2 is defined to be formula_3.Although superficially similar in form to the derivative of a real function, the behavior of complex derivatives and differentiable functions is significantly different. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach formula_2 in the complex plane. Consequently, complex differentiability has much stronger consequences than usual (real) differentiability. For instance, holomorphic functions are infinitely differentiable, whereas most real differentiable functions are not. For this reason, holomorphic functions are also referred to as "analytic functions". Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, extended appropriately to complex arguments as functions formula_5, are holomorphic over the entire complex plane, making them "entire" "functions", while rational functions formula_6, where "p" and "q" are polynomials, are holomorphic on domains that exclude points where "q" is zero. Such functions that are holomorphic everywhere except a set of isolated points are known as "meromorphic functions". On the other hand, the functions formula_7, formula_8, and formula_9 are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy-Riemann conditions (see below). |
Antiderivative (complex analysis) | In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function "g" is a function whose complex derivative is "g". More precisely, given an open set formula_1 in the complex plane and a function formula_2 the antiderivative of formula_3 is a function formula_4 that satisfies formula_5. |
König's theorem (complex analysis) | In complex analysis and numerical analysis, König's theorem, named after the Hungarian mathematician Gyula Kőnig, gives a way to estimate simple poles or simple roots of a function. In particular, it has numerous applications in root finding algorithms like Newton's method and its generalization Householder's method. |
Introduction | Introduction, The Introduction, Intro, or The Intro may refer to: |
Real analysis | Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line. Real analysis is closely related to complex analysis, which studies broadly the same properties of complex numbers. In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressability as power series, and satisfying the Cauchy integral formula. |
Complex number | When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra. |
Complex geometry | In mathematics, complex geometry is the study of complex manifolds and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. |
Advanced Introduction to Finality | This is the second "Introduction to Finality" episode of the series, following season three's finale, "Introduction to Finality". |
Complex segregation analysis | Complex segregation analysis (CSA) is a technique within genetic epidemiology to determine whether there is evidence that a major gene underlies the distribution of a given phenotypic trait. CSA also provides evidence to whether the implicated trait is inherited in a Mendelian dominant, recessive, or codominant manner. |