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From: Marko Amnell on 4 Jul 2010 08:03

"David C. Ullrich" <ullrich(a)math.okstate.edu> wrote in message
67p0361ir2cavqsei5aob4k4cgbs0rrifv(a)4ax.com...
> 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...)

Yes, _Visual Complex Analysis_ is more of a companion
to a basic textbook. Given the OP's interests, one obvious
choice for the basic texbook would be Bak and Newman.
http://www.amazon.com/Complex-Analysis-Joseph-Bak/dp/0387947566

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



From: Marko Amnell on 4 Jul 2010 08:06

"Gc" <gcut667(a)hotmail.com> wrote in message
b38cac6b-e9fb-44ea-8f26-c7e3bb8e3c04(a)j8g2000yqd.googlegroups.com...
> 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...

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...



From: Timothy Murphy on 4 Jul 2010 10:16
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"?
Doesn't any such set have this property?

[Nb His complex plane includes the point at infinity.]






--
Timothy Murphy
e-mail: gayleard /at/ eircom.net
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
From: Marko Amnell on 4 Jul 2010 14:39

"Timothy Murphy" <gayleard(a)eircom.net> wrote in message
rJ0Yn.229$K4.172(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.

> Doesn't any such set have this property?
>
> [Nb His complex plane includes the point at infinity.]
>
>
>
>
>
>
> --
> Timothy Murphy
> e-mail: gayleard /at/ eircom.net
> tel: +353-86-2336090, +353-1-2842366
> s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland


From: Timothy Murphy on 4 Jul 2010 18:07
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?

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.



--
Timothy Murphy
e-mail: gayleard /at/ eircom.net
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
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