HDTenEightyP/NetNews huggingface dataset:
From: campbell@Aero.org (Andrew Campbell)
Subject: Re: Need help!
Organization: The Aerospace Corporation
Date: Thu, 30 Jul 1992 18:03:17 GMT

----------------------- 
In article <chepan.712505966@gsusgi1.gsu.edu>, chepan@gsusgi1.gsu.edu (Parthasarathy Nambi) writes:
|> 
|> 
|>  Hi,
|> 
|>  I need help in solving the equation:
|> 
|>   
|> 
|> 
|>    e^(-x) - x = e^(-y) + y
|> 
|> 
|>  For what values of x and y this equation is true?
|> 
|>  P.Nambi

f(x) = e^(-x) -x has derivative -e^(-x) - 1 so it is
monotone decreasing. Since the limits as x -> +/- oo
of f(x) are -/+ oo, f is a one-to-one map of R onto all of R.
Let g be the inverse function. Then the set of solutions
can be described by x = g(e^(-y)+y). Unfortunately, g has
no common name or elementary expression that I am aware of.
Presumably, this is the sort of answer you were after (with
the set of (x,y) satisfying the equation described in terms of
a function of one the variables). From the form of the answer
given and the fact that the function e^(-y)+y has a single
global minimum at y = 0, where its value is 1, and assumes
every value greater than 1 twice, one can deduce that:

for every y, there is a unique x satisfying the equation

every x for which there is a solution satisfies f(x) => 1

if f(x) = 1 then x = 0 and (0,0) is the only solution for this x

if f(x) > 1, there are exactly two values of y for which
there is a solution (x,y)

Hope this is what you were after.

 zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz
|L. Andrew Campbell     internet: campbell@aerospace.aero.org   |
|M1-102 PO Box 92957    organization: The Aerospace Corp.       |
|Los Angeles CA 90009   telephone: (310) 336-8642               |
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: Cardinarity of a Borel field--antichoice version
Date: 21 Jul 92 14:46:44 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: ara@zurich.ai.mit.edu (Allan Adler)

In article <ARA.92Jul20200638@camelot.ai.mit.edu>, ara@zurich (Allan Adler) writes:
>Solovay remarks that it is known that the theory ZFC+I has a transitive
>\epsilon-model if ZF+DC+P does (J.Nycielski, "On the Axiom of Determinateness")
>This by itself seems to be quite striking.

I believe that name is Mycielski.

>He goes on to write: "However it does seem likely that the existence of a
>transitive  model of ZF+DC+LM is a consequence, in ZFC, of the existence
>of a transitive model of ZFC".

As it turned out, Shelah showed this to be false--the inaccessible is
required.

>Thus, Solovay seems not to be claiming a proof of the assertion which David
>is attributing to him. Instead, he is stating that he believes it is true
>but does not have a proof.

David's version was inaccurate, yes, but he didn't make any claims true or
false regarding the converse consistency result--the above does not refer.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)

Newsgroups: sci.math
From: pradeep@engin.umich.edu (Pradeep Jain)
Subject: Fibre Bundle Theory
Date: Sat, 25 Jul 92 19:59:19 EDT
Organization: University of Michigan Engineering, Ann Arbor

Hi,
 
I understand that Fibre Bundle Theory is about formulating geometric 
problems as Fibre Bundle Theoritic problems (and then as algebric 
problems?). Can someone please give me some introductory refrences to
Fibre Bundle Theory? I would greatly appreciate the help. Also, what 
are the pre-requisites for gaining an understanding of Fibre Bundle
Theory?

Thanks in advance.

Pradeep

-- 

Pradeep Jain                              157 Advanced Technology Lab
Home:   (313) 662-4382                    The University of Michigan
Office: (313) 764-2138                    Ann Arbor, MI 48109
Newsgroups: sci.math,sci.math.num-analysis
From: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Subject: Triangulation
Followup-To: sci.math.num-analysis
Originator: rcpshdb@dutrun2.tudelft.nl
Reply-To: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Organization: Delft University of Technology
Date: Mon, 27 Jul 1992 06:48:33 GMT

Throw an arbitrary number of points at random in the plane.

Does there exist an algorithm that _triangulates_ such an set of points;
i.e. generates triangles in a "sensible" way, such that every point is a 
vertex of a triangle, and the triangles cover the whole area of points ?

I've heard there exists something like "Kolgomorov triangulation".
Has it to do with the above?

"Sci.math" readers, please respond by e-mail or note the followup line,
since I've become very picky in reading netnews.
-- 
* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354   | would be NO idleness in  * Oil
* 2600 AJ  Delft; The Netherlands.     | Mathematics" (HdB).      * for
* E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood
From: ohsie@cs.columbia.edu (David Ohsie)
Newsgroups: sci.math
Subject: Re: Group Theory Question
Keywords: group, coset
Date: 30 Jul 92 01:10:55 GMT
Distribution: usa
Organization: Columbia University Department of Computer Science

In article <36049@sdcc12.ucsd.edu> dmassey@sdcc3.ucsd.edu (Daniel Massey) writes:
>G is an abelian group and H a subgroup  of G. Prove their exists
>a subgroup of G which is iso. to G/H.
>
>Thanks,
>Dan Massey

counterexample:

G=integers under addition, H=subgroup generated by 2.

G/H is simply the integers under addition mod 2.  both elements are
their own inverse, but no integer but 0 is its own inverse so there is
no isomorphic subgroup.



 
-- 
david alan ohsie
internet: ohsie@cs.columbia.edu
usenet: ...!rutgers!columbia!ohsie
bitnet: ohsie%cs.columbia.edu@cuvmb
Newsgroups: sci.math
From: mhosea@seas.smu.edu (Mike Hosea)
Subject: Re: Computational aids.  Was:  Re: An interesting limit problem.
Organization: SMU School Of Engineering and Applied Science
Date: Wed, 29 Jul 1992 21:01:35 GMT

In article <55402@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
>In article <1992Jul29.000223.27339@massey.ac.nz> news@massey.ac.nz (USENET News System) writes:
>>In article <1992Jul28.191037.28756@gdr.bath.ac.uk>, mapsj@gdr.bath.ac.uk (Simon Juden) writes:
>
>>> >mattm@lure.latrobe.edu.au writes:
>
>>> [stuff deleted]
>
>>> >Will someone please back me up, and say that no one learning
>>> >Calculus should ever be allowed anywhere _NEAR_ Mathematica?
>>> >(When I say 'learning Calculus' I mean the first two or three
>>> >semesters)
>
>>> to the professional mathematician. But I believe such gadgets can also do
>>> irreparable harm unless used appropriately. I also believe children should not
>>> be given calculators until they can do sums without them. But doubtless these
>>> views will get me flamed...
>
>A six-year-old can understand the Peano postulates with little difficulty;
>they do not mention anything except counting.  Then the understanding of
>what the arithmetic operations mean can be taught.  Those who have
>manipulative ability will learn the time-saving mechanics, and those
>who do not are far better served by using calculators.
>
>Only researchers and teachers need to know the mechanics.  It is useless
>to know how to differentiate if one does not know what a derivative means.
>No amount of memorizing formulas and tricks helps in understanding.

I think it is unreasonable to teach mathematics under the pretext that it 
is useful and NOT to also teach the proper use of computational tools that 
they are likely to need when actually putting mathematics to use.  As a
numerical analyst it would be hard for me to lose sight of the fact that
some problems cannot be solved analytically.  Even when analytical 
solutions are possible, evaluating them may be far less efficient and
reliable than a numerical solution.  That numerical solutions may be
"incorrect" hardly needs comment. 

This question pops up from time to time, and the strongest reactions
occur when someone suggests a total abandonment of hand calculation
or a total abandonment of modern computational tools (in the context
of education).  Either extreme is a poor choice IMO.  Why must we 
go to an extreme?


-- 
Regards,
Mike Hosea (mhosea@seas.smu.edu)
From: brnstnd@nyu.edu (Dan Bernstein)
Newsgroups: sci.math
Subject: Re: Mathematical reality (was: You know, the integers)
Date: 31 Jul 92 02:24:09 GMT
Organization: IR

In article <1992Jul23.171232.6159@galois.mit.edu> tycchow@riesz.mit.edu (Timothy Y. Chow) writes:
> I was tempted to use the word "formalism" in the discussion several times
> but refrained because I didn't think it was necessary.

``Formalism'' is a loaded term. People called Ramanujan a ``formalist''
when he blindly manipulated the zeta function without realizing that it
had complex zeros. The ``formalism'' of a century ago was ushered out by
rigor.

In any case, the purely syntactic viewpoint which I advocate doesn't
have any difficulties. You may refuse for years to give in and admit
that you're a symbol-pusher too, but I'll be able to take any paper you
publish and keep the entire mathematical content while removing every
appeal to semantics. I win. [chuckle]

---Dan
From: jpat@hpsad.sad.hp.com (Jeff Patterson)
Date: Thu, 23 Jul 1992 19:45:06 GMT
Subject: Re: Mathematica Help
Organization: HP Signal Analysis Division - Rohnert Park, CA
Newsgroups: sci.math

Mathematica will not let you make assignments to an element of a list 
directly. You can build up lists using Prepend, Append, Insert etc.
However here's a one liner to do what you want:

  v=Thread[DD[f[x1,x2,x3],{x1,x2,x3}]] /. DD->D

Thread[f[arg1,{arg2,arg3}] forms {f[arg1,arg2],f[arg1,arg3]}. So the
in the above line Thread[DD[f[x1,x2,x3],{x1,x2,x3}]] forms:

 {DD[f[x1,x2,x3],x1],DD[f[x1,x2,x3],x2],DD[f[x1,x2,x3],x3]}.

The /. DD->D replaces all the DD heads with D heads which results in the
partial derivatives being taken. The last bit of subterfuge is necessary
because the derivitive operator D allows a list in the second position
as in D[f[x1,x2],{x1,5}] which yields the 5th partial of f wrt x1, so we
first get the final list in the form we want (no list in second position)
using the dummy head DD and then change the head to the desired D.

Another way is to use the Outer function as in:

 v=Outer[D,{f[x1,x2,x3]},{x1,x2,x3}]

Outer applies the fuction in arg 1 to a list formed by the outer product
of lists in arg2 and arg3. Thus

In[211]:= Outer[Plus,{a,b},{c,d}]
Out[211]= {{a + c, a + d}, {b + c, b + d}}

Note that using the above you can build a matrix of partials directly

 Outer[D,{f1[x1,x2,x3],f2[x1,x2,x3]},{x1,x2,x3}] produces a list of two lists.
The first list contains all the partials of f1 and the second all the partials
of f2.

Hope this helps-Jeff
From: clong@remus.rutgers.edu (Chris Long)
Newsgroups: sci.math
Subject: Re: A Divisibility Problem
Date: 28 Jul 92 07:28:03 GMT
Organization: Rutgers Univ., New Brunswick, N.J.

In article <9207270341.AA10422@.euclid.uoregon.edu.>, M Johnson writes:

> Given  N > 1  positive integers, it is always possible to find a subset of
> of the set of N positive integers such that its sum is divisible by N.

Consider the succesive results mod N of adding the numbers together;
by the pigeonhole principle either two are equal mod N or one is 0.
If the latter, we are done, if the former, it is clear that sum of
elements that are in one and not the other equals 0 mod N.
-- 
Chris Long, 265 Old York Rd., Bridgewater, NJ  08807-2618

"In a study of schoolboys, an educator discovered a correlation between size
 of feet and quality of handwriting.  The boys with the larger feet were,
 on the average, older."        Wallis & Roberts, _The Nature of Statistics_
Newsgroups: sci.math
From: cxm7@po.CWRU.Edu (Colin Mclarty)
Subject: Re: American mathematical hegemony
Reply-To: cxm7@po.CWRU.Edu (Colin Mclarty)
Organization: Case Western Reserve University, Cleveland, OH (USA)
Date: Wed, 22 Jul 92 12:57:18 GMT


In a previous article, dlm@sjfc.UUCP (Don Muench) says:

>From Craig Cockburn:
>> 
>> And how much longer before a Cauchy sequence becomes a "COW-SHEE sequence" 
>> rather than a "CO-SHAY sequence"
>> 
>I believe that most American ("Merkun", to some of you dweebs) mathematicians
>pronounce "Cauchy" as "co-SHEE", which I believe is correct.
>

	I also believe it is right.  Not all French words are first 
conjugation verbs.
From: rose@fsu1.cc.fsu.edu (Kermit Rose)
Newsgroups: sci.math
Subject: Re: ? Quick algorithm for random covering sets ?
Date: 23 Jul 92 18:01:41 GMT
Reply-To: rose@fsu1.cc.fsu.edu
Organization: Florida State University
News-Software: VAX/VMS VNEWS 1.3-4

In article <1992Jul18.131603.26814@mailer.cc.fsu.edu>, rose@fsu1.cc.fsu.edu (Kermit Rose) writes...
>In article <SQUASH.92Jul17215435@bourbaki.msri.org>, squash@bourbaki.msri.org (Jonathan King) writes...
>>Do any of you have a suggestion for a "fast" algorithm which:
>>   Given positive integers D > K > 2, the algorithm hands me K "random"
>>non-empty subsets of {1,...,D} whose union is all of {1,...,D} and with no
>>element common to all K sets.
>> 
>>"Fast" doesn't have a technical definition here, -the algorithm needs to go
>>inside a deeply nested loop.  "Random" isn't precise either but at least the
>>probability that element d is in set S_k should be close to independent of d.
>> 
>>I'm looking for a easily-programmable, practical algorithm, even if it has
>>theoretical shortcomings.        -Jonathan,  squash@math.ufl.edu
> 
> Here is a FAST algorithm.  Use a random number generator which returns an 
integer in the range {1,2,....K} each time it is called.  Then for j = 
1,2,3,.....D, call this random number generator to give the number s_j.  
Put j in subset number s_j.

It is possible that we could make your program run MUCH faster by figuring 
out how to make it unnecessary to have this random generation withing the 
deeply nested loop.  Is the problem too complicated to post or describe in 
e-mail.  Perhaps I or someone can suggest alternative ways to see the 
problem that will take less calculation.
> 
>rose@fsu1.cc.fsu.edu          To be sure I see your response, use e-mail.
-----------------------------------------------------------------------
Be of good cheer, for it is much more fun than being depressed.
From: wilson@ctron.com (David Wilson)
Newsgroups: sci.math
Subject: My paper
Date: 22 Jul 92 14:20:16 GMT
To: shallit@graceland.waterloo.edu (Jeffrey Shallit)



I sent you my paper "A Discussion of b-regular Sets".  Did you receive it?

David W. Wilson (wilson@ctron.com)

Disclaimer: "Truth is just truth...You can't have opinions about truth."
- Peter Schikele, introduction to P.D.Q. Bach's oratorio "The Seasonings."
Newsgroups: sci.math
From: jk87377@cc.tut.fi (Juhana Kouhia)
Subject: Re: sci.math archives
Organization: Tampere University of Technology
Date: Wed, 22 Jul 92 15:39:55 GMT


In article <9207220957.AA07402@cs.utexas.edu> whorf@vnet.ibm.com writes:
>I am interested to get information where the sci.math and
>sci.math.num.analysis fora are archived for the last months or years?


Storing articles without filtering and summarizing the information 
is not efficient. You could find articles on FTL from n+1 different
place.

And filtering and summarizing takes time, so there's no change to keep
a good archive.

If you have some special article or articles series, I could
check them for you; e-mail me some keywords.


Juhana Kouhia
From: davidc@panix.com (David C. Chorlian)
Newsgroups: sci.math
Subject: Re: The Lorentz transformation
Date: 31 Jul 92 03:37:55 GMT
Organization: PANIX Public Access Unix, NYC

In <Bs7JFz.Ewq@ux1.cso.uiuc.edu> ceblair@ux1.cso.uiuc.edu (Charles Blair) writes:


>   Some recent posts have suggested Poincare had the Lorentz transform
>equations.  I don't know about this, but there's a book by O'Rahilly
>called Electromagnetics or Electromagnetic  Theory which refers to
>special relativity as ``Einstein's use of Voigt's tranformation.''

Pais, in "Subtle is the Lord..." notes that in 1887 Voigt had
what was in essence the Lorentz transformation (as it is called
today).  Lorentz refers to Voigt in a Columbia U lecture of 1906.
Poincare published two papers in 1905 which have the Lorentz
transformations worked out in full.

David B. Chorlian
Newsgroups: sci.math
From: gjstephe@rodan.acs.syr.edu (Greg J Stephens)
Subject: Non-linear stabilty
Organization: Syracuse University, Syracuse, NY
Date: Tue, 28 Jul 92 22:38:59 EDT

I have a set of autonomous (no explicit time dependence), first-order
non-linear diffEQ.  Does anyone know of a better (i.e easier to do in
practice) way of determining the stability of the solution than the 
methods of Liapunov.  I can easily solve these numerically on a computer 
and all initial conditions seem to lead to a steady-state but I would
like to prove this analytically and I am having a hard time constructing
the required Liapunov function.  Any help would be appreciated.
greg     
--
Greg Stephens
201 Physics Bldg.
Syracuse, NY 13244
Internet: stephens@suhep.phys.syr.edu
Newsgroups: sci.math
From: clarke@acme.ucf.edu (Thomas Clarke)
Subject: Re: The Lorentz transformation
Organization: IST, University of Central Florida, Orlando
Date: Thu, 30 Jul 1992 16:29:02 GMT

Wasn't it Lorentz who first had the Lorentz transformation
equations?  


-- 
Thomas Clarke
Institute for Simulation and Training, University of Central FL
12424 Research Parkway, Suite 300, Orlando, FL 32826
(407)658-5030, FAX: (407)658-5059, clarke@acme.ucf.edu
Newsgroups: sci.math
From: dak@kaa.informatik.rwth-aachen.de (David Kastrup)
Subject: Re: Square roots
Organization: Rechnerbetrieb Informatik  /  RWTH Aachen
Date: 29 Jul 92 10:55:01 GMT

hpassaue@silver.ucs.indiana.edu (Billy Passauer) writes:

>I ran across this algorithm for finding the square root of a number a
>long time ago.  I was wondering if someone could tell me where I could
>find more information about it.

>To find the INTEGER portion of a square root:
>  Take the number you want the square root of, and successively
>subtract the odd numbers (starting with 1.)  Each time the result is
>positive, add 1 to a counter.  Repeat with the new number.  When the
>subtraction yields an odd number, the counter contains the answer.

Well, for natural number freaks negative numbers may seem odd, but
usually an odd number is one not dividable by 2. So say, until
subtraction yields a negative number (as shown by the example).
Now (n+1)^2= n^2 + (2n + 1), meaning we get successive square numbers
by successively adding odd numbers. All clear?
>(integer only)  For instance the square root of 24:

>	              Counter
>	24 - 1 = 23  :   1
>	23 - 3 = 20  :   2
>	20 - 5 = 15  :   3
>	15 - 7 =  8  :   4
>	 8 - 9 = <0 

>Any information about this would be greatly appreciated.
>-- 
>  _
> /_)  *  /    /    /  /     hpassaue@silver.ucs.indiana.edu
>/__) /  (__  (__  (__/      hpassaue@ucs.indiana.edu
>     Passauer    ___/
Xref: sparky sci.math:9442 soc.culture.british:10551
From: nick@cs.hw.ac.uk (Nick Taylor)
Newsgroups: sci.math,soc.culture.british
Subject: Re: American mathematical hegemony
Date: 23 Jul 92 09:06:15 GMT
Organization: Dept of Computer Science, Heriot-Watt University, Scotland

In article <2176@sjfc.UUCP> dlm@sjfc.UUCP (Don Muench) writes:

>I believe that most American ("Merkun", to some of you dweebs) mathematicians
>pronounce "Cauchy" as "co-SHEE", which I believe is correct.

I can't imagine the Cauchy pronouncing it any other way.

>I have never heard "CO-SHAY", 

Nor have I - in England, Scotland or Wales.

>although when I arrive in England for the autumn term, I'll hear it...maybe a
>billion times.

In that case, please correct them [apologies to Susan Bradley].

Nick.
From: petry@frobenius.math.washington.edu (David Petry)
Newsgroups: sci.math
Subject: Re: Prime quadruplet records?
Date: 28 Jul 92 18:12:25 GMT
Organization: University of Washington, Mathematics, Seattle

In article <1992Jul28.151305.8747@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
>In article <AeRJFQW00WBNI1kVwn@andrew.cmu.edu> ow0a+@andrew.cmu.edu (Oswald Wyler) writes:
>:A quadruplet of primes occurs when four out of five consecutive odd numbers
>:are prime.  Except for the first two quadruplets, 3,5,7,11 and 5,7,11,13,
>:all quadruplets of primes are of the form 30n + 11, 30n + 13, 30n + 17,
>:30n + 19.  It is well known that there are only finitely many prime
>:quadruplets.
> 
>Where did you get your information?
>
>It is NOT well known that there are only finitely many because almost
>certainly there are infinitely many. This is based upon the prime k-tuples
>conjecture (which in turn is a special case of Schinzel's conj. which in turn
>is a special case of the Bateman-Horn conj.)
>
>Heuristically, up to N, one should expect about 1/phi(30) N/(log N)^4 such
>quadruplets.

Just for the record, there are 4768 prime quadruplets less than 10^8.  The
largest starts with 99982241.


David Petry
Newsgroups: sci.math
Subject: Re: An interesting limit problem.
From: elkies@ramanujan.harvard.edu (Noam Elkies)
Date: 25 Jul 92 20:18:04 EDT
Organization: Harvard Math Department

In article <1992Jul25.212844.1@lure.latrobe.edu.au>
mattm@lure.latrobe.edu.au writes:

>A challenge to all mathematicians. A 100 years ago, this would probably have 
>been solved fairly simply in a natural way, but can you? I think that this
>problem was first posed by the Russian mathematician Arnold. Hope you find this
>problem as interesting as I did when I first solved it.
>
>                sin(tan x)    -    tan(sin x)
>	lim  ----------------------------------  = ???
>	x->0 arcsin(arctan x) - arctan(arcsin x)

I don't know how they sould have done this 100 years ago.  One direct
approach is to expand both numerator and denominator in Taylor series
about x=0 and compare leading terms.  Since we're doing this today instead
of 100 years ago, we can obtain the Taylor expansions instantaneously using
a symbolic manipulation package.  We find that both numerator and denominator
are -x^7/30+O(x^9), so the limit equals 1.  This also suggests that you
*don't* want to find this limit using the L'Hospital rule, especially if
you have to do all the derivatives by hand!

--Noam D. Elkies (elkies@zariski.harvard.edu)
  Dept. of Mathematics, Harvard University
From: carabalo@phoenix.Princeton.EDU (David G. Caraballo)
Newsgroups: sci.math
Subject: Re: Complex Analysis
Keywords: i,e,pi
Date: 24 Jul 92 13:32:12 GMT
Distribution: na
Organization: Princeton University
Originator: news@ernie.Princeton.EDU

In article <1992Jul23.195728.6153@eliot.uucp> andyc@eliot.uucp (Andy Panda Collins) writes:
>
>I've got some questions about complex analysis and I don't have access to a 
>good book store that carries heavy-duty math texts. Can someone suggest some 
>good texts? I have some basic knowledge of the subject, but would like some 
>information on how to analyze complex functions of complex variables. Thanks 
>a bunch. e^(i*_pi_)=(-1)!

I would recommend Ahlfors's _Complex Analysis_, Rudin's _Real and Complex
Analysis_, and Markushevich's _Theory of Functions_.  John B. Conway's
(not John H. Conway at Princeton) _Functions of One Complex Variable_ is
fairly detailed, but a large number of the proofs can be easily shortened,
either via a different line of development, or via careful "tricks".  This is
not necessarily a bad thing, of course.  The proofs, being straight-forward,
are quite easy to follow.  

I believe Ahlfors's book is the best overall, but Rudin's book is fantastic
and definitely worth having.  Markushevich's book, now published as one
huge volume by Chelsea (it's fairly inexpensive, too, as are most things 
published by Chelsea), is very detailed and easy to follow.  It's a good
reference, but I would not use it as a main text.

There are several other good books on the subject, but these complement
each other very well.  Still, if you are going to buy only one book on the
subject, I would say "buy Ahlfors's book" without hesitation.

David Caraballo
Newsgroups: sci.math