By Tristan Needham
Now on hand in paperback, this radical first direction on advanced research brings a stunning and robust topic to existence by way of always utilizing geometry (not calculation) because the technique of clarification. even if geared toward the total newbie, expert mathematicians and physicists also will benefit from the clean insights afforded by way of this strange procedure.
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Discontinuous Groups of Isometries in the Hyperbolic Plane
This ebook via Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and complex historical past. In 1938-39, Nielsen gave a chain of lectures on
discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of
World battle II - to put in writing the 1st chapters of the e-book (in German). whilst Fenchel,
who needed to break out from Denmark to Sweden as a result German profession,
returned in 1945, Nielsen initiated a collaboration with him on what turned identified
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) used to be entire in 1948 and it was once deliberate to be released within the Princeton
Mathematical sequence. although, as a result of quick improvement of the topic, they felt
that great alterations needed to be made prior to booklet.
When Nielsen moved to Copenhagen collage in 1951 (where he stayed till
1955), he used to be a lot concerned with the foreign association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The records of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity comprise unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (
I -15), and an entire manuscript (manuscript I) in
English containing Chapters I-V (
1-27). The records additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes distinctive reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen collage in 1956 (and stayed there
until 1974), used to be greatly concerned with an intensive revision of the curriculum in al-
gebra and geometry, and centred his study within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly two decades
he additionally positioned a lot attempt into his activity as editor of the newly began magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this job left him little time to complete the
Fenchel-Nielsen undertaking the best way he desired to.
After his retirement from the college, Fenchel - assisted by means of Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - chanced on time to
finish the e-book user-friendly Geometry in Hyperbolic house, which used to be released by means of
Walter de Gruyter in 1989 presently after his demise. at the same time, and with an identical
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal some of the vague issues that have been within the unique
manuscript. Fenchel instructed me that he meditated elimination elements of the introductory
Chapter I within the manuscript, on the grounds that this may be coated via the ebook pointed out above;
but to make the Fenchel-Nielsen publication self-contained he finally selected to not do
so. He did choose to omit
27, entitled Thefundamental workforce.
As editor, i began in 1990, with the consent of the felony heirs of Fenchel and
Nielsen, to supply a TEX-version from the newly typewritten model (manuscript 2).
I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed a superb activity of typing this manuscript in AMS- TEX. i've got additionally had
much support from my colleague J0rn B0rling Olsson (himself a pupil of Kate Fenchel
at Aarhus collage) with the evidence interpreting of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a common dialogue of the difference to the fashion
of TEX. In such a lot respects we determined to stick with Fenchel's intentions. in spite of the fact that, turning
the typewritten version of the manuscript into TEX helped us to make sure that the notation,
and the spelling of yes key-words, will be uniform in the course of the ebook. additionally,
we have indicated the start and finish of an evidence within the traditional kind of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice aid and delight they agreed to post the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this optimistic and speedy response. One
particular challenge with the ebook became out to be the replica of the various
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to convey those in ultimate digital shape, yet through 1997 it turned transparent that he
would no longer manage to locate the time to take action. in spite of the fact that, the writer provided an answer
whereby I should still convey specified drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they'd manage the creation of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his fantastic
collaboration in regards to the real creation of the figures.
My colleague Bent Fuglede, who has personaHy recognized either authors, has kindly
written a brief biography of the 2 of them and their mathematical achievements,
and which additionally locations the Fenchel-Nielsen manuscript in its right viewpoint. In
this connection i need to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this publication reproductions of photos of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a few certain symbols, an inventory of notation with brief
explanations and connection with the particular definition within the booklet has been incorporated. additionally,
a complete index has been additional. In either instances, all references are to sections,
not pages.
We thought of including a whole record of references, yet made up our minds opposed to it because of
the overwhelming variety of study papers during this region. as an alternative, a miles shorter
list of monographs and different accomplished bills correct to the topic has been
collected.
My ultimate and so much honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this booklet into lifestyles.
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Additional info for Visual Complex Analysis
Example text
Qar •,q b1 , ... 49). 6, and the corresponding elliptic (and theta) hypergeometric series and their summation and transformation formulas are considered in Chapter 11. 3 The q-binomial theorem One of the most important summation formulas for hypergeometric series is given by the binomial theorem: 2FI (a, C; C; z) where Izl = IFo(a;-; z) = f (a),n zn n=O n. 1) < 1. We shall show that this formula has the following q-analogue rI-. ( . _ . 3 The q-binomial theorem 9 which was derived by Cauchy [1843]' Heine [1847] and by other mathematicians.
32) they observed that Eq(x; -i, -it/2) is a q-analogue of ext. It is now standard to use the notation in Suslov [2003] for the slightly modified q-exponential function Eq(X; a) = (a 2. 17]). 1) 00 where Izl < 1 and Ibl < 1. 2) that (cqn; q)oo (bqn; q)oo Hence, for f = m= 0 (clb; q)m (bqn)m. (q; q)m Izl < 1 and Ibl < 1, '" ( b.. 1). 2) has a q-analogue of the form 2 2) . S. Chihara [1978], Henrici [1974], Luke [1969], Miller [1968], Nikiforov and Uvarov [1988], Vilenkin [1968], and Watson [1952]. Some techniques for using symbolic computer algebraic systems such as Mathematica, Maple, and Macsyma to derive formulas containing hypergeometric and basic hypergeometric series are discussed in Gasper [1990]. Also see Andrews [1984d, 1986, 1987b], Andrews, Crippa and Simon [1997], Andrews and Knopfmacher [2001], Andrews, Knopfmacher, Paule and Zimmermann [2001]' Andrews, Paule and Riese [2001a,b], Askey [1989f, 1990], Askey, Koepf and Koornwinder [1999], Baing and Koepf [1999], Garoufalidis [2003], Garoufalidis, Le and Zeilberger [2003], Garvan [1999], Garvan and Gonnet [1992]' Gosper [2001], Gosper and Suslov [2000], Koepf [1998], Koornwinder [1991b, 1993a, 1998], Krattenthaler [1995b], Paule and Riese [1997], Petkovsek, Wilf and Zeilberger [1996], Riese [2003], Sills [2003c], Wilf and Zeilberger [1990], and Zeilberger [1990b].