Informações sobre o curso
4.8
475 classificações
146 avaliações
100% online

100% online

Comece imediatamente e aprenda em seu próprio cronograma.
Prazos flexíveis

Prazos flexíveis

Redefinir os prazos de acordo com sua programação.
Nível intermediário

Nível intermediário

Horas para completar

Aprox. 35 horas para completar

Sugerido: 8 weeks of study, 6-12 hours/week...
Idiomas disponíveis

Inglês

Legendas: Inglês...

Habilidades que você terá

Power SeriesComplex AnalysisMappingOptimizing Compiler
100% online

100% online

Comece imediatamente e aprenda em seu próprio cronograma.
Prazos flexíveis

Prazos flexíveis

Redefinir os prazos de acordo com sua programação.
Nível intermediário

Nível intermediário

Horas para completar

Aprox. 35 horas para completar

Sugerido: 8 weeks of study, 6-12 hours/week...
Idiomas disponíveis

Inglês

Legendas: Inglês...

Programa - O que você aprenderá com este curso

Semana
1
Horas para completar
5 horas para concluir

Introduction to Complex Numbers

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....
Reading
5 vídeos (Total de 119 min), 5 leituras, 2 testes
Video5 videos
Algebra and Geometry in the Complex Plane30min
Polar Representation of Complex Numbers32min
Roots of Complex Numbers14min
Topology in the Plane21min
Reading5 leituras
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Quiz1 exercício prático
Module 1 Homework10min
Semana
2
Horas para completar
3 horas para concluir

Complex Functions and Iteration

Complex analysis is the study of functions that live in the complex plane, that is, functions that have complex arguments and complex outputs. The main goal of this module is to familiarize ourselves with such functions. Ultimately we’ll want to study their smoothness properties (that is, we’ll want to differentiate complex functions of complex variables), and we therefore need to understand sequences of complex numbers as well as limits in the complex plane. We’ll use quadratic polynomials as an example in the study of complex functions and take an excursion into the beautiful field of complex dynamics by looking at the iterates of certain quadratic polynomials. This allows us to learn about the basics of the construction of Julia sets of quadratic polynomials. You'll learn everything you need to know to create your own beautiful fractal images, if you so desire. We’ll finish this module by defining and looking at the Mandelbrot set and one of the biggest outstanding conjectures in the field of complex dynamics....
Reading
5 vídeos (Total de 123 min), 5 leituras, 1 teste
Video5 videos
Sequences and Limits of Complex Numbers30min
Iteration of Quadratic Polynomials, Julia Sets25min
How to Find Julia Sets20min
The Mandelbrot Set18min
Reading5 leituras
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Quiz1 exercício prático
Module 2 Homework10min
Semana
3
Horas para completar
5 horas para concluir

Analytic Functions

When studying functions we are often interested in their local behavior, more specifically, in how functions change as their argument changes. This leads us to studying complex differentiation – a more powerful concept than that which we learned in calculus. We’ll begin this module by reviewing some facts from calculus and then learn about complex differentiation and the Cauchy-Riemann equations in order to meet the main players: analytic functions. These are functions that possess complex derivatives in lots of places; a fact, which endows them with some of the most beautiful properties mathematics has to offer. We’ll finish this module with the study of some functions that are complex differentiable, such as the complex exponential function and complex trigonometric functions. These functions agree with their well-known real-valued counterparts on the real axis!...
Reading
5 vídeos (Total de 135 min), 5 leituras, 2 testes
Video5 videos
The Cauchy-Riemann Equations29min
The Complex Exponential Function24min
Complex Trigonometric Functions21min
First Properties of Analytic Functions25min
Reading5 leituras
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Quiz1 exercício prático
Module 3 Homework10min
Semana
4
Horas para completar
3 horas para concluir

Conformal Mappings

We’ll begin this module by studying inverse functions of analytic functions such as the complex logarithm (inverse of the exponential) and complex roots (inverses of power) functions. In order to possess a (local) inverse, an analytic function needs to have a non-zero derivative, and we’ll discover the powerful fact that at any such place an analytic function preserves angles between curves and is therefore a conformal mapping! We'll spend two lectures talking about very special conformal mappings, namely Möbius transformations; these are some of the most fundamental mappings in geometric analysis. We'll finish this module with the famous and stunning Riemann mapping theorem. This theorem allows us to study arbitrary simply connected sub-regions of the complex plane by transporting geometry and complex analysis from the unit disk to those domains via conformal mappings, the existence of which is guaranteed via the Riemann Mapping Theorem....
Reading
5 vídeos (Total de 113 min), 5 leituras, 1 teste
Video5 videos
Conformal Mappings26min
Möbius transformations, Part 127min
Möbius Transformations, Part 217min
The Riemann Mapping Theorem15min
Reading5 leituras
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Lecture Slides10min
Quiz1 exercício prático
Module 4 Homework10min
4.8
146 avaliaçõesChevron Right
Benefício de carreira

83%

consegui um benefício significativo de carreira com este curso

Melhores avaliações

por RKApr 6th 2018

The lectures were very easy to follow and the exercises fitted these lectures well. This course was not always very rigorous, but a great introduction to complex analysis nevertheless. Thank you!

por NSJun 25th 2018

The prof makes it easy to understand yet fascinating. I enjoyed video checkpoints, quizzes and peer reviewed assignments. This course encourages you to think and discover new things.

Instrutores

Avatar

Dr. Petra Bonfert-Taylor

Former Professor of Mathematics at Wesleyan University / Professor of Engineering at Thayer School of Engineering at Dartmouth

Sobre Wesleyan University

At Wesleyan, distinguished scholar-teachers work closely with students, taking advantage of fluidity among disciplines to explore the world with a variety of tools. The university seeks to build a diverse, energetic community of students, faculty, and staff who think critically and creatively and who value independence of mind and generosity of spirit. ...

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