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<!doctype html>
<html>
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="icon" type="image/png" href="../images/infinity32.png" sizes="32x32">
<link rel="icon" type="image/png" href="../images/infinity16.png" sizes="16x16">
<title>Complex Analysis</title>
<!-- 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.isbn" 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/">
<!-- social media cards -->
<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:type" content="website">
<!--<meta property="og:image" content="">
<meta property="og:url" content="">-->
<meta name="twitter:card" content="An online interactive introduction to the study of complex analysis.">
<meta property="og:site_name" content="Complex Analysis">
<!-- FONTS AND CSS -->
<link href="https://fonts.googleapis.com/css?family=Raleway:400,600,800" rel='stylesheet' type='text/css'>
<link href="https://fonts.googleapis.com/css?family=Lora:400,400italic,700,700italic" rel='stylesheet'
type='text/css'>
<link href="../css/stylesheet.css" rel="stylesheet" type="text/css">
<link rel="stylesheet" href="../css/w3.css">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.13.0/css/all.min.css">
<!-- CODE PRETTIFY -->
<!-- via: https://github.com/google/code-prettify -->
<script src="../js/run_prettify.js"></script>
<!-- JQUERY FANCY ADDITIONS -->
<!-- add code class, hover text, etc -->
<script src="https://code.jquery.com/jquery-1.11.3.min.js"></script>
<script>
$(document).ready(function () {
// prettyprint the code, pls
$('pre').addClass('prettyprint lang-java');
// add word-breaks to anything with a / or - in it
$('h1').each(function () {
var t = $(this).text();
t = t.replace('/', '/<wbr>');
t = t.replace('-', '-<wbr>');
$(this).html(t);
});
// set formatting
resizeCommands();
});
// when the window gets resized, do some stuff to keep everything
// looking pretty
$(window).resize(function () {
resizeCommands();
});
// things to do when the page is loaded or window resized
function resizeCommands() {
// large-screen stuff...
if ($(window).width() > 600) {
// long headline on intro page
// $('#introHeadline').html('TABLE OF CONTENTS');
// menu hover stuff on larger screens
// (does weird formatting on larger screens)
// hover over title to show "Table of Contents"
$('#title a').hover(
function () {
$('#title a').text('Table of Contents');
},
function () {
$('#title a').text('Complex Analysis'); // set back when moving out
}
);
// set prev/next titles
$('#prev a').hover(
function () {
$('#title a').text('Prev: Acknowledgements');
},
function () {
$('#title a').text('Complex Analysis');
}
);
$('#next a').hover(
function () {
$('#title a').text('Next: A Brief History');
},
function () {
$('#title a').text('Complex Analysis');
}
);
}
// small-screen stuff
else {
// short headline on intro page
// $('#introHeadline').html('INTRO');
}
}
</script>
<!-- Global site tag (gtag.js) - Google Analytics -->
<script async src="https://www.googletagmanager.com/gtag/js?id=G-HVPF4FLZCB"></script>
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</head>
<body>
<div id="wrapper">
<header>
<p><span id="prev"><a href="thanks.html">←</a></span><span id="title"><a
href="table_of_contents.html">Complex Analysis</a></span><span id="next"><a
href="brief_history.html">→</a></span>
<div class="clear"></div>
</p>
</header>
<h1>Reader's Comments</h1>
<hr>
<br>
<p style="text-align:center;font-weight:bold;margin-bottom:-30px">Visitors from around the world since 2020</p>
<iframe width="100%" height="460px" src="../applets/maps-info/colors"></iframe>
<hr/>
<p>
Many thanks to all the people around the world who have kindly shared their thoughts about
this educational resource.
<!--If you want to share your thoughts, please complete this
<a href="https://forms.gle/gL8h1cTpzAvAFbbX6">brief survey</a>.-->
Thanks!
</p>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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 👏
<br>
<span class="w3-right"><strong>— Steven Strogatz</strong> <a
href="https://twitter.com/stevenstrogatz/status/1432296625431617542"
target="_blank">@stevenstrogatz</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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.
<br>
<span class="w3-right"><a href="https://www.bensparks.co.uk/" target="_blank">— Ben Sparks</a>
<!--<br><span class="w3-right"style="font-size:15px">22 Oct, 2020</span>--></span>
</p>
</div>
<br>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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.
<br>
<span class="w3-right"><strong>— </strong> <a
href="https://www.kau.se/en/about-university/about-karlstad-university/ceremonies/marcus-berg-professor-physics"
target="_blank">Marcus Berg</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
There might be some select few who are able to learn more advanced
areas of math purely from interaction with symbols and equations,
but for the rest of us (like me), mathematical intuition develops
best from tangible examples and playful exploration.
The presented content is a great first sight-seeing
tour of the beautiful landscapes of complex analysis,
and the well-chosen illustrations and interactive widgets
are essential for bringing the equations alive.
<br>
<span class="w3-right"><strong>— </strong> <a
href=""
target="_blank">Anton Pirogov</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
Just found it, Completely blown away. Fantastic work done
by the team. Great resource indeed, I honestly believe
these kind of educational resources truly change students
perspective and attract them towards an otherwise boring subject.
<br>
<span class="w3-right"><strong>— </strong> <a>Geek (Australia)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
A clean-looking website that describes complex analysis
concepts in a pretty straigthforward manner, good addition to textbook
<br>
<span class="w3-right"><strong>— </strong> <a>Jennifer (Sweden)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
Very helpful. I am an engineer with previous
knowledge and is very appropriate to brush up on it.
<br>
<span class="w3-right"><strong>— </strong> <a>Interested person (Germany)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
It is very innovative as there are customizable graphs
and software that allow me to visualize the concepts.
<br>
<span class="w3-right"><strong>— </strong> <a>Student (US)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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.
<br>
<span class="w3-right"><strong>— </strong> <a>Nate English (Standford)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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!
<br>
<span class="w3-right"><strong>— </strong> <a>Callum Barnett</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
It's very good. Being able to fully visualise abstract
concepts is incredibly fulfilling. Hope you continue
to add to this wonderful site.
<br>
<span class="w3-right"><a >— Niaz (London, UK)</a>
<!--<br><span class="w3-right"style="font-size:15px">22 Oct, 2020</span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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.
<br>
<span class="w3-right"><strong>— </strong> <a href="https://visualize-it.github.io/" target="_blank">
Chandan Relekar</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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.
<br>
<span class="w3-right"><strong>— </strong> <a>Blake Urban</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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.
<br>
<span class="w3-right"><strong>— </strong> <a>ESSAHRAOUI MAROUANE</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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! :)
<br>
<span class="w3-right"><strong>— </strong> <a href="https://github.com/geothecode"
target="_blank">Georgy (Russia)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
Absolutely beautiful, elegant and removes the complexity of complex numbers.
<br>
<span class="w3-right"><strong>— </strong> <a>Aedan Yates (South Africa)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
Briefly, what came up immediately was exactly the help I needed.
<br>
<span class="w3-right"><strong>— </strong> <a>Paul St. Jean (Michigan)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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.
<br>
<span class="w3-right"><strong>— </strong> <a>Student (England)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<br>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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.
<br>
<span class="w3-right"><strong>— </strong> <a>Student (Oregon, USA)</a>
<!--<br><span class="w3-right"style="font-size:15px"></span>--></span>
</p>
</div>
<hr>
<br>
<div class="w3-panel w3-leftbar">
<p>
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.
<br>
<span class="w3-right"><strong>— </strong> <a>Michael</a>
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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).
<br>
<span class="w3-right"><strong>— </strong> <a>Dominic Ellis</a>
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Brilliant work! It is a true honor to be included within it.
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<span class="w3-right"><a href="https://twitter.com/geogebra/status/1216162738415849472?s=20"
target="_blank">— GeoGebra team</a>
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This is great. Would love to see more sections/chapters on other topics.
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<span class="w3-right"><strong>— Sultan Sial, Teacher</strong>
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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
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<span class="w3-right"><strong>— Jan Douma,
B Ed math</strong>
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Love it!
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<span class="w3-right"><strong>— Thomas Blackwell, Engineer</strong>
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It is a really good job, that helps me understand better the whole concept of complex analysis.
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<span class="w3-right"><strong>— Student</strong>
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It was the best in complex analysis intuition so far(for me).
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<span class="w3-right"><strong>— Student</strong>
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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!
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<span class="w3-right"><strong>— Matthew, Student</strong>
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It looks lovely! I really like the interactive style and the approachable text style.
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<span class="w3-right"><strong>— Student</strong>
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It's great, speacially the applets.
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<span class="w3-right"><strong>— Luana, Student</strong>
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I developed an intuition about the complex surfaces.
<br>
<span class="w3-right"><strong>— Oded Kafri, Physicist</strong>
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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.
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<span class="w3-right"><strong>— Teacher</strong>
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Clear and aesthetically beautiful.
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<span class="w3-right"><strong>— Teacher</strong>
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Absolutely wonderful.
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<span class="w3-right"><strong>— Anonymous</strong>
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This looks like a great book. Thank you for making it available!
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<span class="w3-right"><strong>— George Sipos</strong>
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Fantastic! Keep adding more content, specially graduate level topics.
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<span class="w3-right"><strong>— Teacher & Student (Malaysia)</strong>
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Great way to learn about a difficult topic.
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<span class="w3-right"><strong>— Secondary School Student</strong>
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Really well made website. wish there were more.
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<span class="w3-right"><strong>— Saurabh Dhingra</strong>
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IT'S GOOD
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<span class="w3-right"><strong>— Teacher</strong>
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Very good, high quality, highly recommend
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<span class="w3-right"><strong>— Maya</strong>
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It is incredible. Great visuals, examples, and gives
intuition on how complex analysis works.
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<span class="w3-right"><strong>— Sam Sepiol</strong>
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Fantastic concise introduction to the concepts, great to brush up on before lecture
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<span class="w3-right"><strong>— Student</strong>
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Excellent - and my students agree!!
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<span class="w3-right"><strong>— </strong> <a
href="https://las.touro.edu/faculty/faculty-members/wohl-david.php" target="_blank">David
Wohl</a>
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Very nice! It has embedded graphics/visualization, which are something
conventional books could not afford.
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<span class="w3-right"><strong>— Student</strong>
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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.
<br>
<span class="w3-right"><strong>— Student</strong>
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I’m amazed at the availability and quality of such material.
We live in such a wonderful time for learning.
<br>
<span class="w3-right"><strong>— Matthew Mansfield</strong> <a
href="https://twitter.com/drunkengrass/status/1432456027010580480" target="_blank">@drunkengrass
</a>
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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.
<br>
<span class="w3-right"><strong>— Federico Chialvo</strong> <a
href="https://twitter.com/FedericoChialvo/status/1432362694187778053"
target="_blank">@FedericoChialvo</a>
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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!
<br>
<span class="w3-right"><strong>— Fernando Rosas</strong> <a
href="https://twitter.com/_fernando_rosas/status/1432338563115405313"
target="_blank">@_fernando_rosas</a>
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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!!!
<br><br>
Te quería agradecer por hacerlo material de libre acceso,
lo compartiré con mis compañeros y docentes :)
<br>
<span class="w3-right"><strong>— Agustín Brusco</strong> <a href="https://twitter.com/fisplot"
target="_blank">@fisplot</a>
<!--<br>Estudiante de física,
<br>Facultad de Ciencias Exactas,
<br>Universidad de Buenos Aires-->
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