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From: Edson on 3 Jul 2010 19:53
Hi

I'm somewhat young (15) but I have studied some math. I started at number theory, but then I was introduced to analysis and found it very interesting. And I know it's very important in several practical fields like engineering, economics and optimization.

I've been studying it for almost 2 years, in Rudin's and Bartle books. I've also studied some analysis in R^n. I consider to become a mathematician, though many people keep telling me to graduate in something else.

Anyway, my present goal is to study complex analysis, something fascinating. I've been introduced to complex numbers, I know they form a field with respect to addition and multiplications. I've grown out of that phase when i = sqrt(-1) seems something mystical. So, I'd like some opinions on books on complex analysis for someone at my level.

I've had the opportunity to browse through 2 best sellers. One of them is the classic Alhfors' Complex Analyis, the other (a moderm book, launched about 2 years ago, still in its 1st edition) is Ulrich's Complex Made Simple (so he claims!). I could read the first chapters of each book. Well, exaggerating a bit, to me they seem books on different subjects, the approaches are quite different. If I had to decide right now, I'd get Ulrichs's book. At least to me, it looks, say, more reader friendly, easier to understand. I could understand why the integral of a holomorphic function along a closed smooth curve is zero, though I couldn't get sure if the set where the curve lies must be convex or just connected (or simply path connected).

Alhfors is very clear in the first 2 chapters, but somewhat complicated when he introduces derivatives and integrals. I couldn't understand why every differentiable function is infinitely diferentiable. I think in Ulrich's book this is clearer and easier to understand. But I got somewhat shocked with the two completely different approaches.

I'd like to mention 2 other points:

1. The proofs that power series can be differentiated term by term are completely different. Ulrich's proof, maybe original, is kinda cool, although he implicitly assumes the reader knows something about functions from C^2 to C (but, after all, the concepts involved are the same as those of functions from R^2 to R).

2. In Alhfors book, the proof of the Cauchy Riemann equations are very easy (actually, I did it myself before reading the proof. I'm not boasting, it's not big deal, anyone who has studied limits can do it. All you have to know is the limit is the same no matter how you approach the point). But Ulrich's approach, though not so intuitive, is great, the reader sees the link between complex differentiability and differentiability in R^2. Really cool!

I'd like some help on this. What book should I choose? Actually, I have a 10-year-old edition of Alhfors book, my father was a mathematician (although he was a number theorist, he studied much analysis, including complex analysis, and even taught Calculus. Once I heard him say to some students that the zeta function had much to do with prime numbers. Then, I had no idea what the zeta function was. Now, I'm dying to know what that series has to do with prime numbers). My father can't help me any more, but if someone here can, I'll be grateful.

Edson

(Maybe Dr. David Ulrich himself. I know he contributes to this forum)
From: Marko Amnell on 4 Jul 2010 06:01

There are many good introductory textbooks on complex analysis.
You certainly don't have to pick just one since many are available
for free on the Internet. But don't miss Tristan Needham's
_Visual Complex Analysis_, a unique book about complex analysis.
http://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469

_An Imaginary Tale: The Story of i_ by Paul Nahin is a well-written
non-technical book about complex analysis.
http://www.amazon.com/Imaginary-Tale-Story-square-minus/dp/0691027951

You mentioned the Riemann zeta function. One of the best introductory
books is _Riemann's Zeta Function_ by Harold Edwards.
http://www.amazon.com/Riemanns-Zeta-Function-Harold-Edwards/dp/0486417409

The "Bible" of the theory of the Riemann zeta function is
Titchmarsh's _The Theory of the Riemann Zeta-Function_
http://www.amazon.com/Theory-Riemann-Zeta-Function-Science-Publications/dp/0198533691

John Derbyshire's _Prime Obsession_ is a very good non-technical book
about the Riemann Hypothesis.
http://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mathematics/dp/0309085497



From: Gc on 4 Jul 2010 06:29
Hi, years ago I also found out that Alhfors book was difficult. Like
Ulrich`s book (which I have an impression is more gentler) it`s a
graduate level book. Another gentle graduate level book is a Serge
Langs book. But why not first study some undergradute level complex
analysis book? I mean you are still very young, there is no rush...
From: David C. Ullrich on 4 Jul 2010 06:39
On Sat, 03 Jul 2010 23:53:14 EDT, Edson <edsonm37(a)yahoo.com> wrote:

>Hi

It seems a little tacky for me to comment here, but since you
specifically mention me as someone you'd like comments from:
It's probably obvious which book I think you should get, heh-heh.
About the details you mention, in general you seem to like the
ways a few things are done - I'd hope that if you read further
you'd continue to enjoy this and that, the whole point to the
book was to try to explain the "real reason" various things
work, etc.

>I'm somewhat young (15) but I have studied some math. I started at number theory, but then I was introduced to analysis and found it very interesting. And I know it's very
>important in several practical fields like engineering, economics and optimization.
>
>I've been studying it for almost 2 years, in Rudin's and Bartle books. I've also studied some analysis in R^n. I consider to become a mathematician, though many people keep
> telling me to graduate in something else.
>
>Anyway, my present goal is to study complex analysis, something fascinating. I've been introduced to complex numbers, I know they form a field with respect to addition and
>multiplications. I've grown out of that phase when i = sqrt(-1) seems something mystical. So, I'd like some opinions on books on complex analysis for someone at my level.
>
>I've had the opportunity to browse through 2 best sellers. One of them is the classic Alhfors' Complex Analyis, the other (a moderm book, launched about 2 years ago,
>still in its 1st edition) is Ulrich's Complex Made Simple (so he claims!). I could read the first chapters of each book. Well, exaggerating a bit, to me they seem books on different subjects,
> the approaches are quite different.

One would hope the approaches would be different or there'd be little
point to another book in a field where there are already so many
excellent texts (none of which ever seemed exactly right for the
course here at OSU, for various reasons...)

> If I had to decide right now, I'd get Ulrichs's book. At least to me, it looks, say, more reader friendly, easier to understand. I could understand
>why the integral of a holomorphic function along a closed smooth curve is zero, though I couldn't get sure if the set where the curve lies must be convex or just connected (or simply path connected).

You're just at the beginning. The domain doesn't have to be convex;
convexity is just needed for that particular proof of a preliminary
version of the theorem. Read on (in my book or any other) - you'll
find that connectedness is not enough, what's needed is exactly
something called "simple connectedness" (roughly speaking,
no holes in the domain).

>Alhfors is very clear in the first 2 chapters, but somewhat complicated when he introduces derivatives and integrals. I couldn't understand why every differentiable function is infinitely
>diferentiable. I think in Ulrich's book this is clearer and easier to understand. But I got somewhat shocked with the two completely different approaches.
>
>I'd like to mention 2 other points:
>
>1. The proofs that power series can be differentiated term by term are completely different. Ulrich's proof, maybe original, is kinda cool,

I have no idea whether it's original or not; seems unlikely given that
it's a simple proof of such a basic result. There are many places in
the book where I found what I thought was a nice argument but
I have no idea whether it's new or not.

Heh. For a long time I thought the proof of the Big Picard Theorem
at the end of the main section was both cool and original. It turns
out to be just a cleaned up/modernized version of the original
proof. Still cool, though.

>although he implicitly assumes the reader
>knows something about functions from C^2 to C (but, after all, the concepts involved are the same as those of functions from R^2 to R).
>
>2. In Alhfors book, the proof of the Cauchy Riemann equations are very easy (actually, I did it myself before reading the proof. I'm not boasting, it's not big deal, anyone who has
>studied limits can do it. All you have to know is the limit is the same no matter how you approach the point). But Ulrich's approach, though not so intuitive, is great, the reader sees the
>link between complex differentiability and differentiability in R^2. Really cool!
>
>I'd like some help on this. What book should I choose? Actually, I have a 10-year-old edition of Alhfors book, my father was a mathematician (although he was a number theorist,
>he studied much analysis, including complex analysis, and even taught Calculus. Once I heard him say to some students that the zeta function had much to do with prime numbers.
>Then, I had no idea what the zeta function was. Now, I'm dying to know what that series has to do with prime numbers).

You can find a lot about that in any book on "analytic number theory";
probably you want to learn some complex analysis first. (I know
nothing about number theory; there's a tiny hint regarding the
connection between the zeta function and prime numbers at the
end of the chapter on the Gamma function.)

>My father can't help me any more, but if someone here can, I'll be grateful.
>
>Edson
>
>(Maybe Dr. David Ulrich himself. I know he contributes to this forum)

From: David C. Ullrich on 4 Jul 2010 06:45
On Sun, 4 Jul 2010 13:01:04 +0300, "Marko Amnell"
<marko.amnell(a)kolumbus.fi> wrote:

>
>There are many good introductory textbooks on complex analysis.
>You certainly don't have to pick just one since many are available
>for free on the Internet. But don't miss Tristan Needham's
>_Visual Complex Analysis_, a unique book about complex analysis.
>http://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469

Heh. It's a shame that the one book you mention that's specifically
a text on complex analysis at the relevant level isn't one that I
can say is just awful for this or that reason. Heh.

Visual Complex Analysis is indeed another excellent example of a
text on the subject that's not just the same as the thousands
of other books on the subject. (One might point out that, for
example, there's no proof of the Riemann Mapping Theorem,
which is certainly something that one would expect to find
in a beginning text on the subject. The author says that he
couldn't find a way to illustrate the point to the proof with a
picture, so he left it out. I might sound biased if I suggested
that although this is a book nobody should miss it might be
better as a companion to another more complete book...)

>
>_An Imaginary Tale: The Story of i_ by Paul Nahin is a well-written
>non-technical book about complex analysis.
>http://www.amazon.com/Imaginary-Tale-Story-square-minus/dp/0691027951
>
>You mentioned the Riemann zeta function. One of the best introductory
>books is _Riemann's Zeta Function_ by Harold Edwards.
>http://www.amazon.com/Riemanns-Zeta-Function-Harold-Edwards/dp/0486417409
>
>The "Bible" of the theory of the Riemann zeta function is
>Titchmarsh's _The Theory of the Riemann Zeta-Function_
>http://www.amazon.com/Theory-Riemann-Zeta-Function-Science-Publications/dp/0198533691
>
>John Derbyshire's _Prime Obsession_ is a very good non-technical book
>about the Riemann Hypothesis.
>http://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mathematics/dp/0309085497
>
>

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