Introduction to Complex Analysis

Go to Course: https://www.coursera.org/learn/complex-analysis

Introduction

## Course Review: Introduction to Complex Analysis on Coursera If you're fascinated by mathematics and want to delve into the profound and intriguing world of complex analysis, the Coursera course titled **Introduction to Complex Analysis** is an excellent choice. This course lays a strong foundation in the theory of complex functions and equips you with the tools needed to explore the complex plane, its algebra, and geometry. ### Course Overview This course offers a comprehensive introduction to complex analysis, starting with the basics of complex numbers and progressing through differentiability and integration, finally leading you to advanced topics like power series and the Laurent series. The course is structured over several modules, each comprising five engaging video lectures accompanied by quizzes that reinforce your understanding. ### Syllabus Breakdown 1. **Introduction to Complex Numbers**: - The course kicks off with a historical context, explaining the inception of complex numbers. You'll explore how complex numbers arise not only in quadratic equations but also in the realm of cubic equations. The module focuses on both algebraic and geometric computation in the complex plane, including polar representation and topology. 2. **Complex Functions and Iteration**: - Here, you handle functions with complex arguments and outputs. The module introduces concepts fundamental to complex dynamics and covers quadratic polynomials. You'll also learn to create stunning fractal images and gain insights into the famous Mandelbrot set. 3. **Analytic Functions**: - This module dives deep into the world of complex differentiation, where you'll encounter the Cauchy-Riemann equations and discover the beauty of analytic functions. The exploration of complex exponential and trigonometric functions will help bridge the gap between real and complex analysis. 4. **Conformal Mappings**: - Discover the properties of analytic functions, particularly their capability to preserve angles between curves. You will explore Möbius transformations and culminate in studying the Riemann mapping theorem, which is pivotal in understanding simply connected regions in the complex plane. 5. **Complex Integration**: - Building on your understanding of differentiation, this module tackles curve integrals in the complex plane. You'll delve into Cauchy's integral theorem and formula, learning how complex analysis leads to stunning results, including the Fundamental Theorem of Algebra. 6. **Power Series**: - This section focuses on representing analytic functions via power series. You'll gain insights into the convergence properties and learn how power series can be applied in approximating analytic functions, leading you into intriguing discussions around the Riemann zeta function and its conjectures. 7. **Laurent Series and the Residue Theorem**: - Laurent series extend your understanding of analytic functions near singularities. This module sheds light on isolated singularities, famous theorems, and the implications of the Residue Theorem on complex integration. 8. **Final Exam**: - After seven weeks of rigorous study, the final exam tests your cumulative knowledge across all modules. It's structured to challenge your understanding and application of the topics covered, making it a fitting conclusion to the course. ### Recommendation The **Introduction to Complex Analysis** course on Coursera is ideal for anyone who has a basic understanding of calculus and is eager to engage with higher-level mathematics. The carefully structured content, coupled with interactive quizzes, provides an opportunity for practical learning and self-assessment. The course material is well-presented, maintaining a balance between theory and application, making complex analysis accessible and engaging. Whether you're a student looking to deepen your knowledge, a professional in a math-related field seeking to fortify your skills, or simply a curious mind captivated by the elegance of mathematics, this course is a highly recommended resource. It's a journey through the beautiful landscapes of complex functions that will leave you with a profound appreciation for this essential topic in mathematics. Enroll today on Coursera, and take your first steps into the captivating world of complex analysis.

Syllabus

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.

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.

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!

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.

Now that we are familiar with complex differentiation and analytic functions we are ready to tackle integration. But we are in the complex plane, so what are the objects we’ll integrate over? Curves! We’ll begin this module by studying curves (“paths”) and next get acquainted with the complex path integral. Then we’ll learn about Cauchy’s beautiful and all encompassing integral theorem and formula. Next we’ll study some of the powerful consequences of these theorems, such as Liouville’s Theorem, the Maximum Principle and, believe it or not, we’ll be able to prove the Fundamental Theorem of Algebra using Complex Analysis. It's going to be a week filled with many amazing results!

In this module we’ll learn about power series representations of analytic functions. We’ll begin by studying infinite series of complex numbers and complex functions as well as their convergence properties. Power series are especially easy to understand, well behaved and easy to work with. We’ll learn that every analytic function can be locally represented as a power series, which makes it possible to approximate analytic functions locally via polynomials. As a special treat, we'll explore the Riemann zeta function, and we’ll make our way into territories at the edge of what is known today such as the Riemann hypothesis and its relation to prime numbers.

Laurent series are a powerful tool to understand analytic functions near their singularities. Whereas power series with non-negative exponents can be used to represent analytic functions in disks, Laurent series (which can have negative exponents) serve a similar purpose in annuli. We’ll begin this module by introducing Laurent series and their relation to analytic functions and then continue on to the study and classification of isolated singularities of analytic functions. We’ll encounter some powerful and famous theorems such as the Theorem of Casorati-Weierstraß and Picard’s Theorem, both of which serve to better understand the behavior of an analytic function near an essential singularity. Finally we’ll be ready to tackle the Residue Theorem, which has many important applications. We’ll learn how to find residues and evaluate some integrals (even some real integrals on the real line!) via this important theorem.

Congratulations for having completed the seven weeks of this course! This module contains the final exam for the course. The exam is cumulative and covers the topics discussed in Weeks 1-7. The exam has 20 questions and is designed to be a two-hour exam. You have one attempt only, but you do not have to complete the exam within two hours. The discussion forum will stay open during the exam. It is against the honor code to discuss answers to any exam question on the forum. The forum should only be used to discuss questions on other material or to alert staff of technical issues with the exam.