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<!DOCTYPE html>
<html lang="en">
<head>
<title> Complex Analysis
</title>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<!-- Main info: Dublin core -->
<meta name="DC.Title" content="Complex Analysis">
<meta name="DC.Creator" content="Juan Carlos Ponce Campuzano">
<meta name="DC.Date" content="26-01-2019">
<meta name="DC.Identifier" content="ISBN-13: 978-0-6485736-0-9">
<meta name="DC.Type" content="Interactive Resource">
<meta name="DC.Format" content="HTML">
<meta name="DC.Language" content="en-US">
<meta name="DC.Rights" content="https://creativecommons.org/licenses/by-nc-sa/4.0/">
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<meta property="og:title" content="Complex Analysis">
<meta property="og:description" content="An online interactive introduction to the study of complex analysis.">
<meta property="og:url" content="https://complex-analysis.com/">
<meta property="og:image" content="https://complex-analysis.com/images/complex_analysis_card.jpg">
<meta property="og:site_name" content="Complex Analysis">
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<meta name="twitter:card" content="An online interactive introduction to the study of complex analysis.">
<meta name="twitter:title" content="Complex Analysis">
<meta name="twitter:description" content="An online interactive introduction to the study of complex analysis.">
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<body data-type="book" id="top">
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<div class="container">
<section data-type="chapter" id="index">
<!-- Starts main content -->
<header>
<div>
<div class="hide-small hide-medium">
<br>
<iframe frameborder="0" width="800px" height="400px" scrolling="no" src="applets/p5js/joukowsky/intro.html" style="border: 1px solid black; overflow: hidden;">
</iframe>
</div>
<div class="hide-large ">
<br><br>
<img width="100%" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgP3eKwgEWJyXPaQoaywHwOTkfeEqu2NXxpYJW1RSjOikjsWAJhq4cnTUFJhOCi71XplIlf7llqI1qcyP5rkRp9Qd6L94vrboAinrBLOCfzsK02Camre4MjicGzkY45aUNJJ5v4IuOMuqNKTnSHxBCtwB-_KGcRkZ9ZK5gVp8azqhRHqYI/s1600/complex_prev.gif">
</div>
<div class="">
<h1><b>Complex Analysis</b>
</h1>
<h1 style="margin-top: -40px;">
<span style="font-size: 0.5em;">A Visual and Interactive Introduction</span>
</h1>
<p class="hide-small author" style="margin-top: -30px;">Juan Carlos Ponce Campuzano <br>
2019 - <script>document.write(new Date().getFullYear());</script>
<br>
<span class="dark-mode-button" id="dark-mode-toggle">
<i class="fa-regular fa-sun"></i>
</span>
</p>
<p class="hide-large hide-medium author" style="font-size:16px">Juan Carlos <br> Ponce Campuzano
<br>
2019 - <script>document.write(new Date().getFullYear());</script>
<br>
<span class="dark-mode-button" id="dark-mode-toggle">
<i class="fa-regular fa-sun"></i>
</span>
</p>
<br>
</div>
</div>
</header>
<hr>
<main>
<article>
<div class="language-link">
<a href="es.html">- Español -</a>
</div>
<p>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.</p>
<p>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).</p>
<p class="callout" style="text-align:left"><strong>UPDATES!</strong>
There is a new design in celebration to the 5th anniversary of this site.
I have also updated many of the JavaScript applets for better performance.
Don't forget to join me in
<a href="https://www.patreon.com/posts/celebrating-5-of-99495115?utm_medium=clipboard_copy&utm_source=copyLink&utm_campaign=postshare_creator&utm_content=join_link" target="_blank">Patreon</a>
to get your complex functions gallery.
<br>
A few of sections were added and updated:
<a href="content/limits.html">Limits</a>,
<a href="content/complex_integration.html">Integration of complex functions</a>
and <a href="content/laurent_series.html">Laurent series</a>,
<a href="content/topology_complex_plane.html">Topology of the complex plane</a> and
<a href="content/complex_differentiation.html">Complex differentiation</a>.
<br>
<br>
Recently, I started working in other projects to create more online
interactive books. I already have a preview for Real Analysis.
If you want to know more about it, check my
<a href="https://www.patreon.com/posts/real-analysis-98129332?utm_medium=clipboard_copy&utm_source=copyLink&utm_campaign=postshare_creator&utm_content=join_link" target="_blank">Patreon</a>.
</p>
<p>Ready to get started? Skip ahead to the <a href="content/table_of_contents.html">Table of Contents.</a>
</p>
<hr>
<h2 style="text-align: center;">WHAT IT IS ABOUT</h2>
<p>This book is an interactive introduction to the theory and applications of complex functions
from a visual point of view. However, it <em>does not</em> 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.</p>
<p>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.</p>
<p>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:</p>
<div style=" text-wrap: wrap; display: flex; justify-content: center;">
<div style="padding: 0 5px;">
<a target="_blank" href="https://www.geogebra.org">
<strong>GeoGebra</strong></a>
-
<a target="_blank" href="https://p5js.org/">
<strong>p5.js</strong></a>
-
<a target="_blank" href="https://cindyjs.org/">
<strong>Cindy.js</strong>
</a>
-
<a target="_blank" href="https://mathcell.org/">
<strong>MathCell.js</strong>
</a>
</div>
</div>
<p>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.</p>
<p>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.</p>
<p class="callout"><strong>NOTE!</strong> If you're on a mobile device, some of the applets might not work
super well for you.
They are designed for mouse input, so if you're getting frustrated or your finger is in the way, try the
site on a computer.</p>
<a href="https://www.acds.edu.au/resource/complex-analysis-a-visual-and-interactive-introduction/" target="_blank">
<img src="images/acds-logo.png" style="width: 40%;float:right; padding-top: 20px; padding-left: 20px;padding-bottom: 20px;">
</a>
<p>
This book was included in the
<a href="https://www.acds.edu.au/resource/complex-analysis-a-visual-and-interactive-introduction/" target="_blank">Australian Council
of Deans of Science's repository</a>
as a teaching and assessment resource
to support online learning.
</p>
<hr>
<h2 style="text-align: center;">ISSUES?</h2>
<p>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
<a href="https://github.com/complex-analysis/complex-analysis.github.io/issues">report them at this
project's GitHub repository</a>.
Thanks for your help!
</p>
<hr>
<h2 style="text-align: center;">DESIGN CREDITS</h2>
<p>
The initial design of this site was inpired by the fantastic book
<a href="https://www.jeffreythompson.org/collision-detection/index.php" target="_blank">Collision
Detection</a>
written by <a href="https://www.jeffreythompson.org/" target="_blank">Jeffrey Thompson</a>.
The current version, released in 2024, was built thanks to the following projects:
</p>
<ul>
<li>
<a href="https://github.com/vincentdoerig/latex-css" target="_blank">LaTeX.CSS</a> by Vincent Dörig
</li>
<li>
<a href="https://github.com/davidrzs/latexcss" target="_blank">LatexCSS</a> by David Zollikofer
</li>
<li>
<a href="https://github.com/magicbookproject" target="_blank">The Magic Book project</a>.
</li>
</ul>
<hr>
<h2 style="text-align: center;" id="support">SUPPORT THIS PROJECT</h2>
<p>If you like my work, you can support it using the links below:</p>
<div style="text-wrap: wrap; display: flex; justify-content: center;">
<div style="padding: 0 15px;">
<i class="fa-brands fa-patreon"></i> <a href="https://www.patreon.com/jcponce">
<strong>Patreon</strong></a>
</div>
<div style="padding: 0 15px;">
<i class="fa-brands fa-paypal"></i> <a href="https://www.paypal.com/cgi-bin/webscr?cmd=_donations&business=WTMTR7D8RYZ26&item_name=Your+support+will+help+me+to+improve+and+keep+up+to+date+the+applets+of+this+book.+Sincerely+appreciate+it%21&currency_code=AUD&source=url">
<strong>PayPal</strong></a>
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<i class="fa-solid fa-shop"></i> <a target="_blank" href="https://jcponcemath.secure-decoration.com/">
<strong>T-shirts</strong>
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<i class="fa-brands fa-square-x-twitter"></i> <a target="_blank"
href="https://twitter.com/share?ref_src=twsrc%5Etfw">
<strong>Post</strong>
</a><script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script>
</div>
</div>
<p style="text-align: center;">
Thank you for your support! ❤️
</p>
<hr>
<h2 style="text-align: center;">NAVIGATION</h2>
<p>OK, let's begin! Click the link at the bottom the page, or the arrows at the top, to
move to the next chapter. The <strong>Complex Analysis</strong> link at the top will
take you back to the <a href="content/table_of_contents.html">Table of Contents</a>.</p>
</article>
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<a href="content/table_of_contents.html" style="text-decoration: none;">
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<p>
This interactive book is <strong>free for everybody</strong>,
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:
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Your contributions help me to maintain this site
and, crucially, <strong>ensure it remains freely accessible</strong>
to anyone eager to explore this captivating field of mathematics
<strong>—without any distracting ads</strong>.
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<p>
You can't contribute this time, no worries! Maybe you just want to
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