Complex Analysis Book

Question

I want a really good book on Complex Analysis, for a good understanding of theory. There are many complex variable books that are only a list of identities and integrals and I hate it. For example, I found Munkres to be a very good book for learning topology, and "Curso de Análise vol I" by Elon Lages Lima is the best Real Analysis book (and the best math book) that I have read with many examples, good theory and challenging exercises.

An intuitive and introductory approach is not very important if the book has good explanations and has correct proofs.

Added: If it is possible, tell me your experience with your recommended books and if you got a really good understanding of complex analysis with a deep reading.

See the question, "What is a good complex analysis textbook?". — Dylan Moreland, Jun 19 '12 at 01:16
probably any book of this topic satisfies the kind of book of what I was looking, I don't want an intermediary book with an application approach. — Gaston Burrull, Jun 19 '12 at 01:20
@GastónBurrull The analysis book by Lima that you have mentioned, does it have an english version? — Bhaskar Vashishth, Nov 04 '14 at 20:33
@BhaskarVashishth No, only in portuguese. But you can read "courant and john introduction to calculus and analysis" it is excellent and it is very similar to lima. — Gaston Burrull, Nov 05 '14 at 01:58
@GastónBurrull I do not like analysis. I am past under graduation. But I want to give it a try if I can find some gem of a book, otherwise I guess its too late for me.. — Bhaskar Vashishth, Nov 05 '14 at 02:18

score 23 · Answer 1 · answered Jun 19 '12 at 01:23

23

My biggest recommendation is Tristan Needham's Visual Complex Analysis. Although not a strict textbook, all of the traditional theorems in elementary complex analysis are covered. Proofs aren't thorough, but are instead explained geometrically in general outlines. The big advantage with this book is the massive amount of pictures, nearly on every page in some sections.

Other great classics are Rudin's Real and Complex Analysis, Conway's Functions of One Complex Variable. For a thorough but relatively intuitive approach, I also heavily recommend Sarason's Complex Function Theory.

answered Jun 19 '12 at 01:23

Greg Zitelli

856

I think that Tristan's book is not the book that I was looking. But Conways looks good. What about Rudin, which topics cover it?. – Gaston Burrull Jun 19 '12 at 01:26
Rudin is interested in showing you how clever he is. Conway gets you to believe that you can do this yourself. Sarason is an insightful man of few words, and every one of them counts. – Disintegrating By Parts Dec 26 '23 at 07:37

score 20 · Accepted Answer · answered Jun 19 '12 at 01:15

20

Conway, "Functions of One Complex Variable I" http://books.google.ca/books?id=9LtfZr1snG0C

answered Jun 19 '12 at 01:15

Robert Israel

448,999

Looks good, and it is avaible in my university library. I'll check it. – Gaston Burrull Jun 19 '12 at 01:23
Im looking Conway preview and looks very good with a topologycal and metric spaces early approachs. Probably this is my book. – Gaston Burrull Jun 19 '12 at 01:41
I second that. Excellent choice. – ncmathsadist Jun 19 '12 at 01:43
Thanks, I'll see it with Alfors in my university and I'll decide for one, probably I'll decide for Conway. – Gaston Burrull Jun 19 '12 at 02:06
5

I've always felt Conway was overrated. It's ok,but pretty dry and it misses a lot of the beautiful geometry of the complex plane. It's probably the second volume that really impressed most people-the material in THAT book isn't readily available in a lot of sources. – Mathemagician1234 Aug 27 '13 at 06:46

score 16 · Answer 3 · answered Jun 19 '12 at 01:27

16

The followings are very, very good. Note that you should start with the first one if you are a beginner.

Reinhold Remmert. Theory of complex functions. Springer 1991.
Reinhold Remmert. Classical topics in complex function theory. Springer 2010.

answered Jun 19 '12 at 01:27

timur

16,599

Can I read the second one helping with the first one? – Gaston Burrull Jun 19 '12 at 01:30
I would recommend starting with the first one, and you will know when you need to get the second one. – timur Jun 19 '12 at 01:32
Thanks, I think that I'll start with the both and Conway and I'll decide which book I will definitely read. – Gaston Burrull Jun 19 '12 at 01:39
3

In case it helps, here are some lecture notes I produced when I taught a complex analysis course: http://www.math.mcgill.ca/gantumur/math566f10/?Lecture_notes – timur Jun 19 '12 at 01:43
thanks for your proper material =) – Gaston Burrull Jun 19 '12 at 01:54

score 13 · Answer 4 · answered Jun 19 '12 at 01:28

13

The books below are excellent:

Invitation to Complex Analysis by Ralph P. Boas (second edition revised by Harold P. Boas).
Complex Made Simple by David C. Ullrich.

answered Jun 19 '12 at 01:28

lhf

216,483

Both looks so good, for my level, thanks. – Gaston Burrull Jun 19 '12 at 01:32
2

I haven't seen either,but I'm told Ulrich is outstanding. – Mathemagician1234 Aug 27 '13 at 06:56

score 13 · Answer 5 · edited Jun 19 '13 at 20:37

13

A very classic book to learn complex analysis from is Ahlfors's book (which I used). There is also Stein and Shakarchi's book, and Bak and Newman's book.

edited Jun 19 '13 at 20:37

Martin

8,491

answered Jun 19 '12 at 01:43

Eugene

7,612
4
33
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I've heard about Ahlfor's, is really a good book? Do you full understand the basic topics with a deep reading? Has challenger problems? Tell me about your own experience. – Gaston Burrull Jun 19 '12 at 01:56
@GastónBurrull I thought it was a good book but definitely on the tough side. It's not like Churchill's book which is more geared towards applications, this is intent on rigor. The exercises I thought were challenging but that's a relative opinion. It's pricey but there's an international edition which is a lot cheaper (about $20). – Eugene Jun 19 '12 at 02:00
1

@GastónBurrull Another thing is Ahlfors isn't heavy on examples. If you like lots of examples then maybe you won't like Ahlfors. In my opinion, which may be controversial, is that this is the baby Rudin of complex analysis. – Eugene Jun 19 '12 at 02:03
Ahlfors looks very good but some difficult and not many examples I'll get it in my library is avaible I'll check it with conway (both are avaible in my university library) and I'll decide for one. – Gaston Burrull Jun 19 '12 at 02:05
1

I strongly suggest the book by Bak and Newman. I learned complex analysis there, and I still believe it is a wonderful textbook for undergrads. – Siminore Jun 19 '12 at 07:39
@Gaston NO,Gaston,it's NOT a good book. It's just been enshrined by people who had the misfortune of being graduate students when there weren't a lot of choices for textbooks at that level.That and Ahlfors was one of the preeminient function theorists of the 20th century-which creates a lot of misplaced reverence. The third and final edition was a VAST improvement over the first-which was an unreadable abomination-but still not a great book by any means. – Mathemagician1234 Aug 27 '13 at 06:51

score 10 · Answer 6 · answered Jun 19 '12 at 06:21

10

Here's one that I love: Donald Sarason's book, Complex Function Theory. It's beautifully and economically written, so that it really flows. It was published by Henry Helson in his garage for a long time, but has been taken over by the American Math. Soc.

It covers complex analysis up to and including some advanced topics such as the Riemann mapping theorem, starting from basic real analysis.

answered Jun 19 '12 at 06:21

anthonyquas

1,017
5
15

Thanks for your opinion and experience. – Gaston Burrull Jun 19 '12 at 13:11
It's a bit terse for my tastes, but I agree,this is quite a nice book if that's your cup of potion. – Mathemagician1234 Aug 27 '13 at 06:52

score 8 · Answer 7 · answered Jul 09 '13 at 14:38

Ahlfors, Complex Analysis. It is an absolute classic and, while spartan-seeming, is a fantastic introduction to the course. It was actually my second introduction to the subject (I had it at an earlier undergraduate level using Churchill & Brown, which isn't bad, but no classic.) That book, coupled with an amazing instructor, made a huge impact.

score 7 · Answer 8 · answered Jun 19 '12 at 01:35

7

I had really good luck with Fisher's Complex Variables and Gamelin's Complex Analysis.

answered Jun 19 '12 at 01:35

Nicholas Kirchner

726

1

What do you mean with really good luck?, what is the approach? – Gaston Burrull Jun 19 '12 at 01:55
Books looks good and basic with many examples thanks! – Gaston Burrull Jun 19 '12 at 05:15
Really good luck means that I was taking a graduate complex analysis course that used Ahlfors. I floundered horribly until I studied from the two books mentioned. Fisher was thorough, but it wasn't comprehensive enough. Gamelin helped me make up the difference between Fisher and where I needed to be. – Nicholas Kirchner Jun 19 '12 at 21:33
+1 for 2 great books. The former is a classic in a wonderful cheap Dover edition and the latter is on it's way to becoming a classic. My one quibble with Gamelin is that it tends to be a little TOO soft for a graduate course at times, especially early in the book. But that's what makes it ideal for self study. It really starts from jump,unlike most books-even honors high school students could use the early chapters! – Mathemagician1234 Aug 27 '13 at 06:55

score 7 · Answer 9 · answered Sep 14 '13 at 15:21

7

Look up this free complex analysis book by Shabbat http://math.stanford.edu/~ryzhik/shabat-all.pdf

answered Sep 14 '13 at 15:21

ILoveMath

10,694

score 6 · Answer 10 · answered Jul 09 '13 at 14:46

6

Elias Wegert's book: Visual Complex Functions: An Introduction Using Phase Portraits might not be so good for analytic techniques, but I've found it to be really good for honing one's intuition. (it's the first of a planned two volume set).

answered Jul 09 '13 at 14:46

graveolensa

5,678

score 5 · Answer 11 · answered Nov 04 '14 at 20:26

5

I think Serge Lang's book on Complex Analysis is a good a one to go with.

http://www.amazon.com/Complex-Analysis-Graduate-Texts-Mathematics/dp/0387985921

answered Nov 04 '14 at 20:26

dustin

8,241

score 5 · Answer 12 · answered Nov 04 '14 at 20:35

5

No one has mentioned here, but 'A First Course in Complex Analysis with Applications' by zill is my favourite book. It is so clear and comprehensive, and much simpler and intutive explainantions.

answered Nov 04 '14 at 20:35

Bhaskar Vashishth

11,279

1

From the authors taste in Munkres, I expect he was looking for a graduate book. However, I did use this book as an undergraduate. I do still have Zills book on DE too. – dustin Nov 04 '14 at 20:41

score 5 · Answer 13 · answered Dec 23 '15 at 17:33

5

Complex Analysis, by Beardon is unmatched in quality. I put it above Ahlfors, because in the preface of Ahlfors CA, Allen F Beardon is mentioned with gratitude.

answered Dec 23 '15 at 17:33

R K Sinha

821
9
6

I've never seen that book! How could Beardon have written a complex analysis text and I didn't know? – Mathemagician1234 Jan 14 '16 at 19:46

score 4 · Answer 14 · answered Jun 03 '15 at 14:59

4

Complex Analysis by Kunihiko Kodaira

answered Jun 03 '15 at 14:59

bharath

169

score 4 · Answer 15 · edited Jul 22 '17 at 07:24

4

as my opinion s.ponnusamy ''foundation of complex analysis'' is the best book. same concepts are taught in simple and different way.

edited Jul 22 '17 at 07:24

Soham Chowdhury

1,797

answered Jul 09 '13 at 14:26

amit soni

11

score 3 · Answer 16 · answered Jan 09 '16 at 08:24

3

An Introduction to Complex Analysis by Ravi P. Agarwal , Kanishka Perera , Sandra Pinelas is a fantastic book!

answered Jan 09 '16 at 08:24

mike

667

I found this book to be the most infuriating. Eventually got tired of filling in the gaps in the steps. Does not develop any intuition. Produces key concepts like the exponential form like a rabbit out of the hat instead of motivating them. The only redeeming quality is solutions to exercises. – Apr 13 '19 at 22:41

Complex Analysis Book

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