\TIll
I>reface
reader, and to make clear the necessity of rigour by evidencing the doubtful
arguments, and sometimes false results, due to mathematicians like Newton,
the Bernoullis, Euler, Fourier, or Cauchy. Adopting this point of view lengthens the text palpably, but one of the ground principles of N. Bourbaki - no
economies of paper - is, I think, mandatory when one addresses students
embarking on a subject.
The other principle of this same author - to substitute ideas for computations - appears even more commendable to me whenever it can be
applied. All the same, one will, inevitably, find calculations in this book; but
I have essentially confined myself to those which, inherited from the great
mathematicians of the past, form an integral part of the theory and can be
considered as ideas.
Except occasionally, to round off the text, one will find no exercises here.
Working at exercises is indispensable when one learns mathematics, and one
will find them in profusion in many other books and specialised collections.
The majority of French students, obsessed by the string of examinations
imposed on them, have a very exaggerated tendency to consider the "lectures"
of little use and that only "practical work" and "formulae" count or pay. The
result is that the majority of them are able, up to errors in calculation, to
integrate a rational function but incapable of answering questions of a general
nature, e.g. why is a rational function integrable? To understand a theorem
is to be able to reconstruct its proof. To understand a block of mathematics
does not reduce to knowing how to apply its results; to understand a theory
is to be able to reconstruct its logical structure. Every mathematician knows
this.
One does not learn analysis or anything else from one single book; there
is neither Bible, nor Gospel nor Koran in Mathematics. The fact that the
spirit of my book is radically different from that of Serge Lang, Undergraduate Analysis (Springer, 2nd. ed., 1997) for example, should not dissuade from
reading it, quite the contrary; even less the books of E. Hairer and G. Wanner, Analysis by Its History, Wolfgang Walter, Analysis I (Springer, 1992,
in German) or Reinhold Remmert, Theory of Complex Functions (SpringerNew York, 1991, translation of Funktionentheorie 1, 4. Aufiage, 1995), which
I have often used, and cite when I do so. These excellent books present numerous exercises, as does Jean Dieudonne's Calcul Infinitesimal (Hermann,
1968) though his style enthuses me less.
I have not acceded to the new fashion which likes to decorate elementary
analysis textbooks with numerical calculations to fifteen decimal places under the pretext they will be useful to future computer scientists or applied
mathematicians. Everyone knows that the mathematicians of the xvn th and
X\TIIIth centuries loved numerical computations - done by hand, not by
tapping the keys of an electronic gadget - that enabled them to verify
their theoretical results or to demonstrate the power of their methods. This
childhood sickness of analysis disappeared when in the XIX th century one
I>reface
reader, and to make clear the necessity of rigour by evidencing the doubtful
arguments, and sometimes false results, due to mathematicians like Newton,
the Bernoullis, Euler, Fourier, or Cauchy. Adopting this point of view lengthens the text palpably, but one of the ground principles of N. Bourbaki - no
economies of paper - is, I think, mandatory when one addresses students
embarking on a subject.
The other principle of this same author - to substitute ideas for computations - appears even more commendable to me whenever it can be
applied. All the same, one will, inevitably, find calculations in this book; but
I have essentially confined myself to those which, inherited from the great
mathematicians of the past, form an integral part of the theory and can be
considered as ideas.
Except occasionally, to round off the text, one will find no exercises here.
Working at exercises is indispensable when one learns mathematics, and one
will find them in profusion in many other books and specialised collections.
The majority of French students, obsessed by the string of examinations
imposed on them, have a very exaggerated tendency to consider the "lectures"
of little use and that only "practical work" and "formulae" count or pay. The
result is that the majority of them are able, up to errors in calculation, to
integrate a rational function but incapable of answering questions of a general
nature, e.g. why is a rational function integrable? To understand a theorem
is to be able to reconstruct its proof. To understand a block of mathematics
does not reduce to knowing how to apply its results; to understand a theory
is to be able to reconstruct its logical structure. Every mathematician knows
this.
One does not learn analysis or anything else from one single book; there
is neither Bible, nor Gospel nor Koran in Mathematics. The fact that the
spirit of my book is radically different from that of Serge Lang, Undergraduate Analysis (Springer, 2nd. ed., 1997) for example, should not dissuade from
reading it, quite the contrary; even less the books of E. Hairer and G. Wanner, Analysis by Its History, Wolfgang Walter, Analysis I (Springer, 1992,
in German) or Reinhold Remmert, Theory of Complex Functions (SpringerNew York, 1991, translation of Funktionentheorie 1, 4. Aufiage, 1995), which
I have often used, and cite when I do so. These excellent books present numerous exercises, as does Jean Dieudonne's Calcul Infinitesimal (Hermann,
1968) though his style enthuses me less.
I have not acceded to the new fashion which likes to decorate elementary
analysis textbooks with numerical calculations to fifteen decimal places under the pretext they will be useful to future computer scientists or applied
mathematicians. Everyone knows that the mathematicians of the xvn th and
X\TIIIth centuries loved numerical computations - done by hand, not by
tapping the keys of an electronic gadget - that enabled them to verify
their theoretical results or to demonstrate the power of their methods. This
childhood sickness of analysis disappeared when in the XIX th century one