complex-analysis / complex-analysis.github.io

Complex Analysis: A Visual and Interactive Introduction

MAJOR UPDATE COMPLETED!!!

Enjoy the new design with improved performance for the applets.

Complex Analysis

A visual and interactive introduction to Complex Analysis.

ISBN 978-0-6485736-0-9

---

INTRO

The study of complex analysis is important for students in engineering and the physical sciences and is a central subject in mathematics. In addition to being mathematically elegant, complex analysis provides powerful tools for solving problems that are either very difficult or virtually impossible to solve in any other way.

In this book you will find examples, problems and applets that allow you to explore complex analysis by using the visual interactive power of the computer (or tablet).

UPDATES! This site has been getting lots of visits which is fantastic. I have fixed some typos and modified some applets for better performance. I am also planning to add some new sections covering integration of complex functions. Finally, I will start the Spanish version so more people can enjoy learning and playing with complex analysis.

Ready to get started? Skip ahead to the Table of Contents

---

WHAT IT IS ABOUT

This book is an interactive introduction to the theory and applications of complex functions from a visual point of view. However, it does not cover all the topics of a standard course. In fact, it is a collection of selected topics and interactive applets that can be used as a supplementary learning resource by anyone interested in learning this fascinating branch of mathematics.

Some of the topics covered here are basic arithmetic of complex numbers, complex functions, Riemann surfaces, limits, derivatives, domain coloring, analytic landscapes and some applications of conformal mappings.

What distinguishes this online book from other traditional texts in the first instance is the use of interactive applets that allow you to explore properties of complex numbers geometrically and analyze complex functions by using different techniques to visualize them. For the design of applets I used the following open-source softwares: GeoGebra, p5.js, Cindy.js and MathCell.

Although I advocate for the use of computers as an aid to geometric reasoning, I highly encourage you to practice your problem solving skills by solving the suggested exercises (or filling the missing details) that you will encounter throughout the sections.

Think of the computer as a physicist would his laboratory. It may be used to check existing ideas about our world, or as a tool to discover new phenomena which then poses new ideas or challenges for their explanation. Throughout the sections I have provided detailed instructions (in some cases) to explore concepts and relationships about complex numbers using specific softwares, nevertheless you must still keep in mind that computer hardware and software are ephemeral things in comparison with mathematical ideas, which are timeless.

---

ISSUES?

Of course, no project such as this can be free from errors and incompleteness. If you find applets that don't run correctly, or the examples that aren't explained quite right, or a typo, please report them at this project's GitHub repository. Thanks for your help!

---

DESIGN CREDITS

The initial design of this book was based upon the fantastic book Collision Detection written by Jeffrey Thompson. If you want to learn about the algorithms behind collisions using basic geometrical shapes, you should check it! The current version, released in 2024, was built thanks to the following projects:

• LaTeX.CSS by Vincent Dörig
• LatexCSS by David Zollikofer
• The Magic Book project.

---

❤️ Support this project to keep it alive!

This interactive book is free for everybody, as I believe that everyone should have access to high-quality learning resources to explore and learn mathematics. That's why many users choose to contribute to my efforts. If you believe in what I do and want to help me on my journey, you can do so now with the links below:

• Patreon

• PayPal

• T-shirts

Your contributions help me to maintain this site and, crucially, ensure it remains freely accessible to anyone eager to explore this captivating field of mathematics —without any distracting ads.

---

LICENSE

This book is released under the license: Attribution-NonCommercial-ShareAlike 4.0 International.

complex-analysis.github.io's Issues

The "exp" function in Roots of Complex Numbers may be confusing to new readers

The paragraph in https://complex-analysis.com/content/roots_complex_numbers.html where equation (1) is introduced for the exercise introduces a bit of notation that I didn't notice anywhere earlier in the book:

Thus, the $n$th roots of a nonzero complex number $z\neq 0$ can also be expressed as
\begin{eqnarray}\label{expform}
z=\sqrt[n]{r};\mbox{exp}\left[i\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right)\right]
\end{eqnarray}
where $k=0, 1, 2, \ldots , n-1$.

You've been using e^x as the standard notation for the complex exponential up to this point, so the sudden introduction of the "exp" function will probably confuse people. Perhaps a short introduction of that notation is in order in case you use it in the future.

Typo in math next to Figure 6 on the "A Brief History Page"

The the first line starting with "x = " is not equal to the line below it. Perhaps the line below it should have 11*sqrt(-1) instead of the sqrt(-11) under each cubed root.

Typos/recommendation for geometric_interpretation_add_mult

• s/substraction/subtraction
• s/observe the behaviour the angles/observe the behaviour of the angles
• s/analize/analyze
• s/behaivour/behaviour
• s/As you already have noticed the geometric/As you already have noticed, the geometric

Possible typo in "The Logarithmic Function"

At least one of the following appears to be the case:

1. There is an typo in "The Logarithmic Function" section just after  "Some basic properties of the function logz are the following:" OR
2. I have not had nearly enough caffeine today.

The second concern is probably beyond the intended scope of your project, , but a spot-check on the first could prove helpful. Thanks for sharing!

Typo in a brief history

Leonard Euler (1707-1783) standirezed

Should be standardized

Typos/recommendations

riemann_surfaces

• s/diferent/different

power_function

• s/This is reason why/This is the reason why

curves_in_the_complex_plane

• s/countour/contour
• s/counter-clockwise/counterclockwise

complex_integration

• s/analize/analyze (a couple of times)
• s/countour/contour

integrals_of_functions_with_branch_cuts

• s/appropiate/appropriate

taylor_series

• s/convergece/convergence

Typo/recommendations for riemann_sphere

• s/can lead to confusion an abuse/can lead to confusion and abuse

Missing word in Chapter 1, A Brief History

Currently reads "How do interpret this value?" Could be corrected either as "How do we interpret this value?" or "How to interpret this value?"

Typo issue in "Complex functions"

Third paragraph

A multiple-valued function can be considered as a collection of single-valued functions, each member of which is called a branch of the function. In general, we consider on particular member as a principal branch of the multiple-valued function and the value of the function corresponding to this branch as the principal branch.

I think the last word should be "value". The sentence goes as "the value of the function corresponding to this branch as the principal value".

Analytic Landscapes: applet issue

Hi, on the "Analytic Landscapes" page, there are some minor issues with the "Dynamic Exploration" applet:

By default (i.e. right when you open that page), the slider a changes "something" on the plot, even though the label says that it's (z-1)/(z^2+z+1) (without any a) that should be represented. This is a bit puzzling (at least in the first moment) since the text only says that there's some slider "a" without explaining what it does and there's some unexpected action when you change its value.

Only a few moments later did I realize that "a" is just some parameter for some of the other predefined functions and if I switch to one of the other examples and then back again to the first one, a is disabled as it should be.

Typos / recommendations

in brief_history

• s/how we do interpret this value?/how do we interpret this value?
• s/italian/Italian
• s/standarized/standardized

---

in domain_coloring

• s/funcion/function

---

in integrals_of_functions_with_branch_cuts

• s/analize/analyze.

---

in taylor_series

• s/centred/centered

---

in laurent_series

• s/centred/centered

---

in classification_of_singularities

• s/centred/centered

---

in mandelbrot_set

• s/to nearest point/to the nearest point
• s/Every pixel that does not cotain/Every pixel that does not contain
• s/computer-genereted/computer-generated

---

in complex_potential_basic_examples

• s/funcion/function
• s/It is easy to generalise/It is easy to generalize

---

in flow_around_circle

• s/singularites/singularities

Why did you copy this page on complex integrals with branch cuts from `https://math.libretexts.org/` and still claim copywrite in your site?

I found the same entries in math.libretext site.

Errores tipográficos en breve_historia

• o en la escuela elementa / o en la escuela elemental
• la parábola y=x^2 y la y=−1 do no se intersecan / la parábola y=x^2 y la línea y=1 no se intersecan
• no har solución / no hay solución
• En este caso p=−6 and q=20 / En este caso p=-6 y q=20
• expresión extraña (\ ref {cubic004}) que / expresión extraña (8) que
• contribuciones a números imaginarios / contribuciones a los números imaginarios
• Por ejemplo René Descartes / Por ejemplo, René Descartes
• “número complejo” refiriendose / “número complejo” refiriéndose
• complejos se dio fue dada por William / complejos fue dada por William
• en el curso de desarrollo de análisis complejo / en el curso del desarrollo del análisis complejo
• surgimiento y desarrollo de números complejos / surgimiento y desarrollo de los números complejos

Complex Potential - Stagnation point flow applet not working

Nothing happens when one clicks on the screen to start this applet.

Errores de Tipeo/recomendaciones en raices_numeros_complejos

• s/entereo/entero
• s/conrrespondientecorrespondiente
• s/eponencial/exponencial
• s/ráices/raíces

Felicitaciones por el gran trabajo!

Rodrigo Chappa

Mandelbrot set applet not working for z=i

Check the applet to fix the orbits!

Possible Typo in Topology... > Classification...

The last two examples of the classification section state that $$S_2=\overline{B}_{\frac{7}{8}}\left(-1-\sqrt{2}i\right)$$ and

$$S_3=\overline{B}_{\frac{1}{2}}\left(2+\sqrt{3}i\right)\setminus \lbrace2+\sqrt{3}i\rbrace $$

where $$\text{Int } S_2=B_{\frac{7}{8}}\left(-1-\sqrt{2}\right)$$ and $$\text{Int } S_3=B_{\frac{1}{2}}\left(2+\sqrt{3}\right)$$

Am I misunderstanding the math or are we missing an i in the Int S statements?

Loving the course so far as someone that learned just enough z-plane to be dangerous in signal processing in college but had forgotten a lot of it since ;)

Typos/recommendations for complex_differentiation

• s/to the the derivative/to the derivative
• s/correspoding/corresponding
• s/can exists/can exist
• s/hyperboic/hyperbolic
• s/holomorfic/holomorphic
• s/synonimous/synonymous

Incorrect Power Series Definitions

On the Taylor Series page, I believe the series for sin, cos, cosh and sinh are incorrectly defined. In further pages the cos series is defined as (-1)^n * (z^2n)/(2n!) but on the Taylor series page the factorial value is only n!

Some suggestions

Request to use

Is it alright if I use your one of the applet in my website?

Errores de tipeo/recomendaciones en breve_historia

• s/ desarrollada por el antiguos griegos/ desarrollada por los antiguos griegos
• s/ ¿cuándo tomarón/ ¿cuándo tomaron
• s/intersecará/intersectará
• s/solición/solución
• s/usandi/usando
• s/architecto/arquitecto
• s/Esta la intuición no fue fácil/Esta intuición no fue fácil
• s/artiméticas/aritméticas
• s/Then he showed that the squared roots of negative numbers cancel each other// (remover texto en Ingles ya traducido)
• s/aparezcen/aparecen
• s/equación/ecuación
• s/podrían ser generado/podrían ser generados
• s/es la matemática misma que nos habla/es la matemática misma la que nos habla
• s/geométircamente/geométricamente
• s/la raíz cuadrada de un negativo número/la raíz cuadrada de un número negativo
• s/la teoría de las parejas es mucho más satisfactorio/la teoría de las parejas es mucho más satisfactoria
• s/la consistencia del teoría/la consistencia de la teoría

Missing exponent in equation for circle/line

Missing exponents in section Transformation 1/z for the equation of circle/line

Typo in "Topology of the Complex Plane"?

On the "Topology of the Complex Plane" page, at the end of the "Classification of points" section, it says $$\partial S_3=\left\{z:|z-\left(2+\sqrt{3}i\right)|=\frac{1}{2}\right\}\cup\{0\}$$ but shouldn't it be $$\partial S_3=\left\{z:|z-\left(2+\sqrt{3}i\right)|=\frac{1}{2}\right\}\cup\{\left(2+\sqrt{3}i\right)\}$$ instead?

Typos / recommendations for brief_history.html

• s/rewritting/rewriting
• s/invetion/invention
• s/Excercise/Exercise (a couple of times)
• s/nowdays/nowadays
• s/substituing/substituting
• s/refering/referring
• s/stuning/stunning
• s/to solution of the cubic equation/to the solution of the cubic equation
• s/Termonology/Terminology

Typo issue in

Problem with limits example 3 in chapter 2

I think there's a problem with example 3 in chapter 2, with both the
'D' and the D_2' domains. In short, I don't think that either has a
limit.

1. Let z=a+bi. Then f(z) = e^(a+bi) = (e^a)(e^(bi)).
2. But then |f(z)| = e^a, since e^(bi) represents the argument/phase of f(z).
3. Using our limits definition, let's choose M=e^2.
4. Now let's choose an arbitrarily-large R, which will go unspecified for now.
5. Let z = 1+Ri. Clearly |z| > R
6. But |f(z)| = e^1 = e, which is not going to surpass M, no matter how large
   a value we choose for R.
   We'll have a similar problem with D_2, since -1+Ri will circle zero without
   approaching it.

Errores de tipeo/recomendaciones en topologia_plano_complejo

• s/extrerior/exterior
• s/extrerior/exterior
• s/políginal/poligonal
• s/talque/tal que
• s/conjuto/conjunto

Nuevamente felicitaciones por el material!

Rodrigo Chappa.