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From: Marko Amnell on 4 Jul 2010 18:42

"Timothy Murphy" <gayleard(a)eircom.net> wrote in message
kC7Yn.233$K4.126(a)news.indigo.ie...
> Marko Amnell wrote:
>
>>>> Well, if the OP wants to look at another introduction
>>>> by a Finn (and Ahlfors's teacher) Nevanlinna's book
>>>> is available for free:
>>>>
>>>> http://rapiddigger.com/download/introduction-to-complex-analysis-r-
>>> nevanlinna-v-paatero-djvu-4104964/
>>>>
>>>> but he will need an STDU viewer...
>>>
>>> Thanks for pointing out Nevanlinna's book, which looks quite nice.
>>>
>>> I was just glancing at it quickly,
>>> and was puzzled by his definition of a domain (page 9):
>>> "An open set of points [in the complex plane] forms a domain ...
>>> if it is possible to join any two points by a polygonal path ...
>>> (This condition makes the open set connected.)"
>>>
>>> Why doesn't he just say "a domain is a connected open set
>>> in the complex plane"?
>>
>> I believe this is because while all step-connected sets
>> are connected, not all connected sets are step-connected
>> (e.g. the deleted comb space http://en.wikipedia.org/wiki/Comb_space )
>> and the concept of a step-connected set is useful
>> in complex analysis, e.g. in defining integration.
>
> But isn't every connected open subset of the complex plane
> polygonally connected in this sense?

Yes, while not true for sets in general, it is true
for *open* subsets of the complex plane.

> If it is, I would have thought it would be simpler
> to define a domain as a connected open subset of the complex plane,
> and state as a lemma that a domain is polygonally connected.

I see what you mean. But Nevanlinna is not alone
is defining the domain this way. It's a question
of logical presentation. I also think the notion
of the polygonal path is right there in the
definition of the domain because it is easy
to define and grasp. To be honest, I'm not
really sure. Maybe there are historical reasons
involved.




From: Frederick Williams on 5 Jul 2010 06:07
Marko Amnell wrote:

> Brown and Churchill's _Complex Variables and Applications_
> is geared more towards engineers.
> http://www.amazon.com/Complex-Variables-Applications-James-Brown/dp/0072872527

.... but is unreliable.

Has anyone mentioned Remmert's _Theory of Complex Functions_? It's a
GTM but it starts at the beginning.

--
I can't go on, I'll go on.
From: David C. Ullrich on 5 Jul 2010 11:15
On Sun, 04 Jul 2010 23:07:12 +0100, Timothy Murphy
<gayleard(a)eircom.net> wrote:

>Marko Amnell wrote:
>
>>>> Well, if the OP wants to look at another introduction
>>>> by a Finn (and Ahlfors's teacher) Nevanlinna's book
>>>> is available for free:
>>>>
>>>> http://rapiddigger.com/download/introduction-to-complex-analysis-r-
>>> nevanlinna-v-paatero-djvu-4104964/
>>>>
>>>> but he will need an STDU viewer...
>>>
>>> Thanks for pointing out Nevanlinna's book, which looks quite nice.
>>>
>>> I was just glancing at it quickly,
>>> and was puzzled by his definition of a domain (page 9):
>>> "An open set of points [in the complex plane] forms a domain ...
>>> if it is possible to join any two points by a polygonal path ...
>>> (This condition makes the open set connected.)"
>>>
>>> Why doesn't he just say "a domain is a connected open set
>>> in the complex plane"?
>>
>> I believe this is because while all step-connected sets
>> are connected, not all connected sets are step-connected
>> (e.g. the deleted comb space http://en.wikipedia.org/wiki/Comb_space )
>> and the concept of a step-connected set is useful
>> in complex analysis, e.g. in defining integration.
>
>But isn't every connected open subset of the complex plane
>polygonally connected in this sense?

Yes - that follows easily from the fact that the plane is
locally polygonally connected.

My conjecture is simply that Narasimhan was going to
be using the notion of polygonal connectedness
and simply didn't need to worry the reader about
the fact that it's equivalent to connectedness.

>If it is, I would have thought it would be simpler
>to define a domain as a connected open subset of the complex plane,
>and state as a lemma that a domain is polygonally connected.

From: Edson on 7 Jul 2010 06:41
> 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
> t 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)

Thank you all for your help.

And I bought Complex Made Simple! Hope I can follow it!
Edson
From: Stephen Montgomery-Smith on 7 Jul 2010 11:11
Edson wrote:

> 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 have heard high praise about the book "Applied Complex Analysis with
Partial Differential Equations" by Nakhle H. Asmar. It is at the
undergraduate rather than graduate level. And I should add that Nakhle
is a good friend of mine. But I really do hear a lot of unbiased praise
for this book from students.

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