Transcendental numbers /

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Shidlovskiĭ, A. B.
Otros Autores: Koblitz, Neal, 1948-
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Berlin : W. de Gruyter, 1989.
Colección:De Gruyter studies in mathematics ; 12.
Temas:
Transcendental numbers.
Number theory.
Nombres transcendants.
Théorie des nombres.
MATHEMATICS > Algebra > Intermediate.
Number theory
Transcendental numbers
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Foreword
  • Preface to the English edition
  • Preface
  • Notation
  • Introduction
  • 1. Approximation of algebraic numbers
  • 2. The classical method of Hermite-Lindemann
  • 3. Methods arising from the solution of Hilbert's Seventh Problem, and their subsequent development
  • 4. Siegel's method and its further development
  • Chapter 1. Approximation of real and algebraic numbers
  • 1. Approximation of real numbers by algebraic numbers
  • 2. Simultaneous approximation
  • 3. Approximation of algebraic numbers by rational numbers.
  • 4. Approximation of algebraic numbers by algebraic numbers
  • 5. Further refinements and generalizations of Liouville's Theorem
  • Remarks
  • Chapter 2. Arithmetic properties of the values of the exponential function at algebraic points
  • 1. Transcendence of e
  • 2. Transcendence of s
  • 3. Transcendence of the values of the exponential function at algebraic points
  • 4. Approximation of ez by rational functions
  • 5. Linear approximating forms for eu1z ..., eumz
  • 6. A set of linear approximating forms
  • 7. Lindemann's Theorem.
  • 8. Linear approximating forms and the Newton interpolation series for the exponential function
  • Remarks
  • Chapter 3. Transcendence and algebraic independence of the values of F-functions which are not connected by algebraic equations over the field of rational functions
  • 1. E-functions
  • 2. The First Fundamental Theorem
  • 3. Some properties of linear and fractional-linear forms
  • 4. Properties of linear forms in functions which satisfy a system of homogeneous linear differential equations
  • 5. Order of zero of a linear form at z = 0.
  • 6. The determinant of a set of linear forms
  • 7. Passing to linearly independent numerical linear forms
  • 8. Auxiliary lemmas on solutions of systems of homogeneous linear equations
  • 9. Functional linear approximating forms
  • 10. Numerical linear approximating forms
  • 11. Rank of the m-tuple f1(q) ..., fm(q)
  • 12. Proof of the First Fundamental Theorem
  • 13. Consequences of the First Fundamental Theorem
  • Remarks.
  • Chapter 4. Transcendence and algebraic independence of the values of F-functions which are connected by algebraic equations over the field of rational functions
  • 1. Rank of the m-tuple f1(q) ..., fm(q)
  • 2. Some lemmas
  • 3. Estimate for the dimension of a vector space spanned by monomials in elements of a field extension
  • 4. The Third Fundamental Theorem
  • 5. Transcendence of the values of F-functions connected by arbitrary algebraic equations over C(z)
  • 6. Algebraic independence of the values of E-functions which are connected by arbitrary algebraic equations over C(z)

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