Lagrange Inversion
Diplomarbeit
Verfat von
Markus Rosenkranz
Matr.-Nr. 9057164
Lindenstrae 12
4600 Wels
und eingerei ht bei
Univ.-Doz. Dr. Peter Paule
Institut f
ur Mathematik / RISC
Te hnis h-Naturwissens haftli he Fakultat
Johannes Kepler Universitat Linz
im
Februar 1997
zur Erlangung des akademis hen Grades
eines Diplom-Ingenieurs in der
Studienri htung Te hnis he Mathematik.
,Dedi ated to my dear family.
,Contents
Prefa e 2
1 Introdu tion 3
1.1 The Combinatorial Context . . . . . . . . . . . . . . . . . . 3
1.2 What Is a Solution? . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Elementary Con epts . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Combinatorial Con epts . . . . . . . . . . . . . . . . . . . . 25
2 Lagrange's Theorem 38
2.1 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Elementary Proof . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Analyti al Proof . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Algebrai Proof . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Combinatorial Proof . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Multivariate Proof . . . . . . . . . . . . . . . . . . . . . . . 61
3 Appli ations 65
3.1 Some Combinatorial Results . . . . . . . . . . . . . . . . . . 65
3.2 Solving Analyti al Problems . . . . . . . . . . . . . . . . . . 70
3.3 Inverse Relations . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4 Binomial Sequen es . . . . . . . . . . . . . . . . . . . . . . 79
Bibliography 92
A knowledgements 95
1
, For whi h of you, intending to build a tower, sitteth not down
rst, and ounteth the ost, whether he have suÆ ient to nish
it? Lest haply, after he hath laid the foundation, and is not able
to nish it, all that behold it begin to mo k him.
Luke 14:28{29
Prefa e
This diploma thesis analyzes the phenomenon of Lagrange inversion, also
known as Lagrange's theorem, rst published by the Fren h mathemati ian
Joseph Louis Lagrange in 1869. Its development in the following 125 years
has shown to be enormously fertile in numerous bran hes of mathemati s,
revealing deep onne tions in the proofs and appli ations thereof. Trying
to exhaust this depth would be an utterly futile attempt. Therefore our
goal is to simply lead the reader to some of the most beautiful spots so
he an have a glimpse of this ne mesh and taste the intelle tual joy of
understanding the underlying stru tures.
In Chapter 1, we set the stage by sket hing the ombinatorial framework
into whi h Lagrange's theorem is embedded. The theorem itself is pre-
sented and investigated from various sides in Chapter 2. It is put to work
in Chapter 3 for solving a vast array of di erent problems, ranging from
short stand-alone problems to an extensive theory of ertain transforma-
tions.
All formal units (de nition, notation, lemma, proposition, theorem, orol-
lary, proof, example) are numbered in a single sequen e. As labels, the
numbers appear in the margin so that they an be lo ated easily. For rea-
sons of style, however, they are pla ed behind when used as referen es, like
in \De nition 5". All formal units are terminated by a 2 symbol, whi h is
also put in the margin for better readability.
O asionally I introdu e some spe ial names that are not found in the
literature. Su h terms are marked by a ir le-supers ript when they rst
Æ
appear.
The hapter mottos are taken from the Bible (King James Version). The
S riptures often use metaphors from the natural world in order to illustrate
spiritual truths. We quote these verses here be ause they ontain universal
prin iples that are also valid in the world of mathemati s.
2
Diplomarbeit
Verfat von
Markus Rosenkranz
Matr.-Nr. 9057164
Lindenstrae 12
4600 Wels
und eingerei ht bei
Univ.-Doz. Dr. Peter Paule
Institut f
ur Mathematik / RISC
Te hnis h-Naturwissens haftli he Fakultat
Johannes Kepler Universitat Linz
im
Februar 1997
zur Erlangung des akademis hen Grades
eines Diplom-Ingenieurs in der
Studienri htung Te hnis he Mathematik.
,Dedi ated to my dear family.
,Contents
Prefa e 2
1 Introdu tion 3
1.1 The Combinatorial Context . . . . . . . . . . . . . . . . . . 3
1.2 What Is a Solution? . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Elementary Con epts . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Combinatorial Con epts . . . . . . . . . . . . . . . . . . . . 25
2 Lagrange's Theorem 38
2.1 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Elementary Proof . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Analyti al Proof . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Algebrai Proof . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Combinatorial Proof . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Multivariate Proof . . . . . . . . . . . . . . . . . . . . . . . 61
3 Appli ations 65
3.1 Some Combinatorial Results . . . . . . . . . . . . . . . . . . 65
3.2 Solving Analyti al Problems . . . . . . . . . . . . . . . . . . 70
3.3 Inverse Relations . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4 Binomial Sequen es . . . . . . . . . . . . . . . . . . . . . . 79
Bibliography 92
A knowledgements 95
1
, For whi h of you, intending to build a tower, sitteth not down
rst, and ounteth the ost, whether he have suÆ ient to nish
it? Lest haply, after he hath laid the foundation, and is not able
to nish it, all that behold it begin to mo k him.
Luke 14:28{29
Prefa e
This diploma thesis analyzes the phenomenon of Lagrange inversion, also
known as Lagrange's theorem, rst published by the Fren h mathemati ian
Joseph Louis Lagrange in 1869. Its development in the following 125 years
has shown to be enormously fertile in numerous bran hes of mathemati s,
revealing deep onne tions in the proofs and appli ations thereof. Trying
to exhaust this depth would be an utterly futile attempt. Therefore our
goal is to simply lead the reader to some of the most beautiful spots so
he an have a glimpse of this ne mesh and taste the intelle tual joy of
understanding the underlying stru tures.
In Chapter 1, we set the stage by sket hing the ombinatorial framework
into whi h Lagrange's theorem is embedded. The theorem itself is pre-
sented and investigated from various sides in Chapter 2. It is put to work
in Chapter 3 for solving a vast array of di erent problems, ranging from
short stand-alone problems to an extensive theory of ertain transforma-
tions.
All formal units (de nition, notation, lemma, proposition, theorem, orol-
lary, proof, example) are numbered in a single sequen e. As labels, the
numbers appear in the margin so that they an be lo ated easily. For rea-
sons of style, however, they are pla ed behind when used as referen es, like
in \De nition 5". All formal units are terminated by a 2 symbol, whi h is
also put in the margin for better readability.
O asionally I introdu e some spe ial names that are not found in the
literature. Su h terms are marked by a ir le-supers ript when they rst
Æ
appear.
The hapter mottos are taken from the Bible (King James Version). The
S riptures often use metaphors from the natural world in order to illustrate
spiritual truths. We quote these verses here be ause they ontain universal
prin iples that are also valid in the world of mathemati s.
2
