Complex Analysis

Reader's Comments

Visitors from around the world

Many thanks to all the people around the world who have kindly shared their thoughts about this educational resource. Thanks!

This looks fantastically useful for anyone who is teaching or taking a course in complex analysis. Thank you, Juan Carlos! Beautiful work! And very generous of you to share it with the world 👏
— Steven Strogatz @stevenstrogatz

I like the way each page has careful and readable descriptions of the maths, while not shying away from the technical details. The particular strength here is that alongside the readable mathematics are regular interactive animations in Geogebra or p5.js (on every page!?). These really help the reader *literally* get to grips with the material. It's lovely stuff. I believe that making maths *move* in this way is the key for any of us trying to understand this subject better.
— Ben Sparks

The applets are beautiful! You found the right amount of interactivity: some sites have too much (overwhelming, or clunky), most have too little. It's also a great service to provide the source code on GitHub.
Marcus Berg

It is very innovative as there are customizable graphs and software that allow me to visualize the concepts.
Student (US)

I'm barely in and it already seems fantastic. It seems like the future of education, and already represents lucidly the model that I want instruction to look like.
Nate English (Standford)

This is the best mathematical resource I have ever come across! The visualisations are so helpful and the content is presented in a way that is concise and easy to understand. I wish all of my modules taught in this way!
Callum Barnett

t's very good. Being able to fully visualise abstract concepts is incredibly fulfilling. Hope you continue to add to this wonderful site.
— Niaz (London, UK)

A clean-looking website that describes complex analysis concepts in a pretty straigthforward manner, good addition to textbook
— Jennifer (Sweeden)

I am a fourth-year undergrad (BS Physics). We deal with complex analysis at several junctures. Although I have managed to have some understanding of it, I just wish that I had come across your resource earlier. I just spent 3 hours devouring your book, and I must say - I feel like I know complex analysis like the back of my hand.
Chandan Relekar

Was super helpful for checking what the Principal Argument was. Loved the visualisation of the different branches "spreading out" or "expanding" to represent the branch cuts. Helped with my intuitive understanding of a branch cut, and the issues that arise with multifunction's in the complex plane.
Blake Urban

I'm very grateful for the work that has been done on this website. It helps me to understand the potential complex theory easily. The way the creators of this elegant website worked is something to appreciate a lot. I hope that all courses teach us like this website does, especially with the applications that are facility the understanding of the equations and physical phenomena. Thank you very much.

Pretty good example of *right* use of *right* computing powers and open-source technologies for teaching people some interesting topics and overcoming some difficulties that aren't easily being "tackled" using "common ways" of studying! :)
Georgy (Russia)

Absolutely beautiful, elegant and removes the complexity of complex numbers.
Aedan Yates (South Africa)

Briefly, what came up immediately was exactly the help I needed.
Paul St. Jean (Michigan)

Absolutely beautiful. Being able to interact with visual representations is very helpful with understand what actually happens. Also the notes are well written and clear, so are very to understand. It does everything that a website about maths should do.
Student (England)

It was hard to find online resources for complex analysis when I was looking, and they usually would only state rules or theorems that I didn't understand the reasoning behind. I've only used this site for a few days, but it's been an easy way to start learning about complex series. The explanations flow nicely, and I like that the first example of something is put in a basic form to show what's happening. The web design & layout are great and are probably what first caught my attention and made me stay. The only thing I can think of in terms of improvement would be adding small expandable sections for something that might be related or notable, but not necessarily needed or at a different level of complexity that wouldn't match the topic.
Student (Oregon, USA)

I think it’s absolutely fantastic; it’s written so well, and it covers an impressive breadth of complex analysis. I find this resource to be more quality, more valuable than most textbooks I buy for my university courses. Thank you for providing such a great resource.

I think this is a fantastic resource for teaching Complex Analysis. I am an educator and currently working on an algorithm to turn musical excerpts into mathematical functions and vice versa and complex numbers are at the center of the treatise. Each pitch is represented as a cosine wave with a frequency phi representing the interval the particular note is from the key and a phase shift delta representing the key of the excerpt. This pitch is a Fourier series of the normal modes comprising the pitch keeping the instruments timbre in mind. Each rhythm is represented as a rational exponential with time signature in the numerator represented as a rational function tau = n/d and the subdivision of the beat in the denominator represented as a rational logarithmic function beta = ln(n)/ln(d). Therefore, each note N is equal to (p,r) = int(p(phi,delta)*r(tau,beta).
Dominic Ellis

Brilliant work! It is a true honor to be included within it.
— GeoGebra team

This is great. Would love to see more sections/chapters on other topics.
— Sultan Sial, Teacher

Handy to see all kind of topics , especialy now i am struggling with the notion of a branch. Using also Maple for doing math there I am looking now on the website and it can be used for training some exercises, and study. Lets say for the geomatrical meaning of the complex differentation is too much for explaining Good example : The logarithmic function to get a idea what a multi-valued function stands for Its only that i like to see the logarithmic function at the Riemann Surfaces examples too. Thanks for your website, very helpful to go into complex analysis
— Jan Douma, B Ed math

Love it!
— Thomas Blackwell, Engineer

It is a really good job, that helps me understand better the whole concept of complex analysis.
— Student

It was the best in complex analysis intuition so far(for me).
— Student

I have been trying to figure out how Complex Numbers worked and how they represented all three values of a 3D grid while only being given (x,y) or (r,t) for the longest time now. I believe I have now completed my search and onto going further with it! (I'd say about a 2-year self-journey at least. Though I've learned so much Math along the way.) Thank you so much!
— Matthew, Student

It looks lovely! I really like the interactive style and the approachable text style.
— Student

It's great, speacially the applets.
— Luana, Student

I developed an intuition about the complex surfaces.
— Oded Kafri, Physicist

Probably the best alternative for classical textbooks on complex analysis. Concise and playfull. Love the examples and interactive plots. This has something what "mainstream books" lacks, it is suitable for modern students.
— Teacher

Clear and aesthetically beautiful.
— Teacher

Absolutely wonderful.
— Anonymous

This looks like a great book. Thank you for making it available!
— George Sipos

Fantastic! Keep adding more content, specially graduate level topics.
— Teacher & Student (Malaysia)

Great way to learn about a difficult topic.
— Secondary School Student

Really well made website. wish there were more.
— Saurabh Dhingra

— Teacher

Very good, high quality, highly recommend
— Maya

It is incredible. Great visuals, examples, and gives intuition on how complex analysis works.
— Sam Sepiol

Fantastic concise introduction to the concepts, great to brush up on before lecture
— Student

Excellent - and my students agree!!
David Wohl

Very nice! It has embedded graphics/visualization, which are something conventional books could not afford.
— Student

This learning resource is very well made! It explains concepts simply and effectively, while still going through the working out and rigour of the subject.
— Student

I’m amazed at the availability and quality of such material. We live in such a wonderful time for learning.
— Matthew Mansfield @drunkengrass

Wow thank you @jcponcemath! I wish I had this when I took Complex Analysis ~20 years ago. Instead I treated a grad student friend to a meal so I could ask them 100 questions about how to visualize what we had to do for class.
— Federico Chialvo @FedericoChialvo

Complex analysis is imho perhaps the most “magical” corner of math, where the right amount of assumptions gives a counterintuitive amount of results... but for the same reason it is not easy to learn. This type of resources are really helpful!
— Fernando Rosas @_fernando_rosas

Te quería comentar nomas que estuve ojeando tu libro y me gustó mucho. Justamente este cuatrimestre estoy cursando una materia de análisis complejo y es un lindo complemento visual para muchas de las temáticas que vemos. En particular me fascinó el graficador interactivo de series de Taylor con domain coloring, resultó muy didáctico para entender el concepto de radio de convergencia!!!

Te quería agradecer por hacerlo material de libre acceso, lo compartiré con mis compañeros y docentes :)
— Agustín Brusco @fisplot

He leído tu libro de análisis complejo y me ha gustado mucho. Trata los contenidos de esta materia de una manera muy asequible que complementa visualmente con applets interactivos. Me parece idóneo para docentes, investigadores y estudiantes de ingenierías y matemáticas. Muchas gracias por esta aportación.
— Débora Pereiro @debora_pereiro

As Web-based open-access textbooks gain wider visibility, we need pioneers to demonstrate the possibilities of this emerging medium. In Complex Analysis: A Visual and Interactive Introduction, Juan Carlos Ponce Campuzano has already done something that was previously impossible: he’s seamlessly embedded beautiful, dynamic, interactive visualizations into the body of a math text.

Not only are these visualizations created with open-source software, but also the book itself is open to continual improvements. There is simply no need to wait for a new edition, and the contents are available to everyone with an internet connection. Authors, take note! This is how it’s done.
— Greg Stanton @HigherMathNotes

En mi opinión, este proyecto consigue el equilibrio perfecto entre el rigor matemático y la claridad. Está muy bien estructurado, pero lo que realmente le distingue de los libros al uso son sus applets interactivos, que permiten a los lectores ir jugando con distintos parámetros y ver el resultado que estos cambios provocan. De esta forma, es mucho más fácil para el lector entender la teoría que subyace y, sin duda, mucho más atractivo. Los gráficos interactivos se van haciendo cada vez más espectaculares a medida que avanza el libro y hacen justicia a lo fascinante que es esta rama de las matemáticas.

Mención especial merece también el hecho de que todas las animaciones hayan sido creadas con software libre, como GeoGebra, entre otros.

Lo recomiendo tanto para aquellos que vayan a impartir un curso en análisis complejo como para quienes quieran estudiarlo.
— Javier Arrospide Laborda

El tema de los números complejos es realmente apasionante y Juan Carlos Ponce nos organiza un recorrido por el mismo de manera clara, concisa y muy visual, incorporando aplicaciones con el programa GeoGebra para explicar los diferentes conceptos. El formato y la presentación hacen que la lectura sea muy amena, lejos de textos sobrecargados, densos y con demasiada información, muchas veces innecesaria. El lector puede ir descubriendo el mundo fascinante del análisis complejo interactuando con las aplicaciones del libro. Al final te acabas sintiendo como un viajero descubriendo un mundo nuevo... ¡y son matemáticas!
— Bernat Ancochea Millet

The applets and flow are a splendid way to get exposure to complex numbers and analysis. I have enjoyed reviewing a topic I haven’t played with in a long time. There are enthusiastic and eye-catching visuals throughout. Even if math is not your thing, I recommend that you take a look and play with the applets. This is a great book for anyone interested the complex plane, imaginary numbers, and jumping into some of the wonders of math.
— Sophia Wood @fractalkitty

A book (site) that explains mathematical theory that is difficult to understand intuitively with tools that can create interactive content such as GeoGebra and p5.js. You can actually move the parameters and see how they change.
— piqcy

Ideal para mi curso de Análisis Complejo. Gracias por compartirlo.
— Ana María Lucca

Por si alguno está estudiando análisis complejo, esta introducción está fetén.
— ThePurpleSensation

A neat interactive approach to learning about complex analysis.
— Sean Walker

Las matemáticas viven en el razonamiento del ser humano y, dado su nivel de abstracción, una simple lectura no es suficiente para comprender y asimilar los conceptos. Todo resulta más fácil cuando dichos conceptos vienen acompañados por ilustraciones y explicaciones breves y sencillas. Pues bien, Juan Carlos consigue en este libro precisamente eso. Nadie como él para sacar partido a todos los recursos visuales posibles en forma de aplicaciones e ilustraciones interactivas para ilustrar la magia de los números imaginarios. Alégrense la vista y la mente con este fantástico libro.
— Julio Mulero

I was just wanting to learn more about these holomorphic functions of the complex plane and conformal mapping and how this stuff relates to fluid flow myself, THANK YOU for making this!
— C010011012

Awesome work, thanks for sharing this! The maths is clearly written and - no surprise - the @geogebra applets are just perfect.
— Vincent Pantaloni

Complex Analysis: A visual and Interactive Introduction is like a virtual or imaginary candy-store for those interested in mathematical visualizations in general or complex numbers in particular. I have never seen so many mesmerizing applets in one place. The tools given to the user to explore the mathematics beyond the text and the attention to detail make a very captivating adventure.

Anyone interested in the Mandelbrot Set (which is probably just about everyone) should check out the chapter with the interactive Mandelbrot illustrations. I particularly like the applet that shows the changing character of the iterate orbitals as one moves among the various buds of the Mandelbrot set and another applet that connects the Mandelbrot set to the Julian sets.

After whetting one’s appetite on the Mandelbrot chapter, if you are like me, you will want to explore further. The colorful analytic landscapes and domain coloring apps are awesome. And don’t miss the cool conformal mapping applications in the last chapter!
— Ken Thele

Extraordinary....this will help a lot of guys.
— Azazaya

This is indeed one of the best educational content. Let's study. Thank you.
— ayush thakur

NEXT: A Brief History