Transcendental Numbers
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Transcendental Numbers

  1. 485 pages
  2. English
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eBook - PDF

Transcendental Numbers

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The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.

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Information

Publisher
De Gruyter
Year
2011
ISBN
9783110889055
Edition
1
Topic
Mathematics
Subtopic
Mathematics General
Index
Mathematics

Table of contents

  1. Foreword
  2. Preface to the English edition
  3. Preface
  4. Notation
  5. Introduction
  6. § 1. Approximation of algebraic numbers
  7. § 2. The classical method of Hermite-Lindemann
  8. § 3. Methods arising from the solution of Hilbert’s Seventh Problem, and their subsequent development
  9. § 4. Siegel’s method and its further development
  10. Chapter 1. Approximation of real and algebraic numbers
  11. § 1. Approximation of real numbers by algebraic numbers
  12. § 2. Simultaneous approximation
  13. § 3. Approximation of algebraic numbers by rational numbers
  14. § 4. Approximation of algebraic numbers by algebraic numbers
  15. § 5. Further refinements and generalizations of Liouville’s Theorem
  16. Remarks
  17. Chapter 2. Arithmetic properties of the values of the exponential function at algebraic points
  18. § 1. Transcendence of e
  19. § 2. Transcendence of π
  20. § 3. Transcendence of the values of the exponential function at algebraic points
  21. § 4. Approximation of ez by rational functions
  22. § 5. Linear approximating forms for eρ1z,..., eρmz
  23. § 6. A set of linear approximating forms
  24. § 7. Lindemann’s Theorem
  25. § 8. Linear approximating forms and the Newton interpolation series for the exponential function
  26. Remarks
  27. Chapter 3. Transcendence and algebraic independence of the values of Ε-functions which are not connected by algebraic equations over the field of rational functions
  28. § 1. E-functions
  29. § 2. The First Fundamental Theorem
  30. § 3. Some properties of linear and fractional-linear forms
  31. § 4. Properties of linear forms in functions which satisfy a system of homogeneous linear differential equations
  32. § 5. Order of zero of a linear form at z = 0
  33. § 6. The determinant of a set of linear forms
  34. § 7. Passing to linearly independent numerical linear forms
  35. § 8. Auxiliary lemmas on solutions of systems of homogeneous linear equations
  36. § 9. Functional linear approximating forms
  37. § 10. Numerical linear approximating forms
  38. § 11. Rank of the m-tuple f1(ξ),..., fm(ξ)
  39. § 12. Proof of the First Fundamental Theorem
  40. § 13. Consequences of the First Fundamental Theorem
  41. Remarks
  42. Chapter 4. Transcendence and algebraic independence of the values of Ε-functions which are connected by algebraic equations over the field of rational functions
  43. § 1. Rank of the m-tuple f1(ξ),..., fm(ξ)
  44. § 2. Some lemmas
  45. § 3. Estimate for the dimension of a vector space spanned by monomials in elements of a field extension
  46. § 4. The Third Fundamental Theorem
  47. § 5. Transcendence of the values of Ε-functions connected by arbitrary algebraic equations over C(z)
  48. § 6. Algebraic independence of the values of E-functions which are connected by arbitrary algebraic equations over C(z)
  49. § 7. Ε-functions connected by special types of equations
  50. § 8. Ε-functions connected by algebraic equations with constant coefficients
  51. § 9. Ε-functions which are connected by a single algebraic equation over C(z)
  52. § 10. Minimal equations
  53. § 11. Dimension of the vector spaces spanned by monomials in the elements of a field extension
  54. § 12. Algebraic independence of the values of IE-functions
  55. § 13. Algebraic independence of the values of KE-functions
  56. Remarks
  57. Chapter 5. Transcendence and algebraic independence of the values of E-functions which satisfy first order linear differential equations
  58. § 1. Hypergeometric E-functions
  59. § 2. The simplest hypergeometric E-functions
  60. § 3. Sets of solutions of first order linear differential equations
  61. § 4. Some lemmas
  62. § 5. Proof of the theorems
  63. Remarks
  64. Chapter 6. Algebraic independence of the values of E-functions which satisfy second order linear differential equations
  65. § 1. A general theorem on algebraic independence of the values of an Ε-function and its derivative
  66. § 2. The functions Κλ(z) associated to Bessel functions
  67. § 3. The functions Κλ(z) and
  68. § 4. Kummer functions
  69. § 5. Solutions of non-homogeneous linear differential equations
  70. Remarks
  71. Chapter 7. Solutions of certain linear differential equations of arbitrary order
  72. § 1. Solutions of non-homogeneous differential equations
  73. § 2. Solutions of homogeneous differential equations
  74. § 3. Corollaries of Theorems 1 and 2
  75. Remarks
  76. Chapter 8. Arithmetic methods applied to solutions of linear differential equations of arbitrary order
  77. § 1. Statement of the theorems
  78. § 2. Auxiliary lemmas
  79. § 3. Proof of Theorems 1–5
  80. § 4. Proof of Theorems 6 and 7
  81. § 5. Further results
  82. Chapter 9. Siegel’s Theorem
  83. § 1. Statement of the theorem and some basic auxiliary results
  84. § 2. Some lemmas
  85. § 3. Some properties of solutions of second order homogeneous linear differential equations
  86. § 4. Algebraic independence of solutions of a set of second order homogeneous linear differential equations
  87. § 5. Proof of Siegel’s Theorem
  88. § 6. Solutions of non-homogeneous linear differential equations
  89. § 7. Generalizations of Siegel’s Theorem
  90. Chapter 10. Solutions of linear differential equations of prime order p
  91. § 1. Statement of the basic results
  92. § 2. The homogeneous ideal I
  93. § 3. Algebraic functions of several variables
  94. §4. The differential operator G
  95. § 5. The differential operators S and δ
  96. § 6. A lemma on linear approximation
  97. § 7. End of the proof of Theorem 7
  98. § 8. Linear reducibility
  99. § 9. Proof of Theorems 6 and 5
  100. Remarks
  101. Chapter 11. The algebraic independence measure of values of IE-functions
  102. § 1. Definition of the measures
  103. § 2. The linear independence measure of values of IE-functions
  104. § 3. The algebraic independence measure of values of IE-functions which are not connected by algebraic equations over C(z)
  105. § 4. Auxiliary results
  106. § 5. The algebraic independence measure of values of IE-functions which are connected by algebraic equations over C(z)
  107. § 6. Some applications of the general theorems
  108. Remarks
  109. Chapter 12. The algebraic independence measure of values of KE-functions
  110. § 1. The fundamental lemma
  111. § 2. Bounds for the measures of the values of E-functions which are not connected by algebraic equations over C(z)
  112. § 3. Bounds for the measures of the values of E-functions which are connected by a single algebraic equation over C(z)
  113. § 4. Bounds for the measures of the values of E-functions which are connected by arbitrary algebraic equations over C(z)
  114. § 5. Algebraic independence of the values of E-functions in conjugate fields
  115. § 6. An auxiliary theorem
  116. § 7. Consequences of the auxiliary theorem
  117. § 8. Some applications of the general theorems
  118. Remarks
  119. Chapter 13. Effective bounds for measures
  120. § 1. Definitions and notation
  121. § 2. Refinement of the fundamental lemmas
  122. § 3. Bounds for linear independence measures
  123. § 4. Bounds for algebraic independence measures
  124. § 5. Some applications of the general theorems
  125. Remarks
  126. Concluding remarks
  127. Supplementary remarks on recent work for the English edition
  128. Bibliography