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 π -- 3. Transcendence of the values of the exponential function at algebraic points -- 4. Approximation of ez by rational functions -- 5. Linear approximating forms for eρ1z,..., eρmz -- 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 Ε-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(ξ),..., fm(ξ) -- 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 Ε-functions which are connected by algebraic equations over the field of rational functions -- 1. Rank of the m-tuple f1(ξ),..., fm(ξ) -- 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 Ε-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) -- 7. Ε-functions connected by special types of equations -- 8. Ε-functions connected by algebraic equations with constant coefficients -- 9. Ε-functions which are connected by a single algebraic equation over C(z) -- 10. Minimal equations -- 11. Dimension of the vector spaces spanned by monomials in the elements of a field extension -- 12. Algebraic independence of the values of IE-functions -- 13. Algebraic independence of the values of KE-functions -- Remarks -- Chapter 5. Transcendence and algebraic independence of the values of E-functions which satisfy first order linear differential equations -- 1. Hypergeometric E-functions -- 2. The simplest hypergeometric E-functions -- 3. Sets of solutions of first order linear differential equations -- 4. Some lemmas -- 5. Proof of the theorems -- Remarks -- Chapter 6. Algebraic independence of the values of E-functions which satisfy second order linear differential equations.

1. A general theorem on algebraic independence of the values of an Ε-function and its derivative -- 2. The functions Κλ(z) associated to Bessel functions -- 3. The functions Κλ(z) and -- 4. Kummer functions -- 5. Solutions of non-homogeneous linear differential equations -- Remarks -- Chapter 7. Solutions of certain linear differential equations of arbitrary order -- 1. Solutions of non-homogeneous differential equations -- 2. Solutions of homogeneous differential equations -- 3. Corollaries of Theorems 1 and 2 -- Remarks -- Chapter 8. Arithmetic methods applied to solutions of linear differential equations of arbitrary order -- 1. Statement of the theorems -- 2. Auxiliary lemmas -- 3. Proof of Theorems 1-5 -- 4. Proof of Theorems 6 and 7 -- 5. Further results -- Chapter 9. Siegel's Theorem -- 1. Statement of the theorem and some basic auxiliary results -- 2. Some lemmas -- 3. Some properties of solutions of second order homogeneous linear differential equations -- 4. Algebraic independence of solutions of a set of second order homogeneous linear differential equations -- 5. Proof of Siegel's Theorem -- 6. Solutions of non-homogeneous linear differential equations -- 7. Generalizations of Siegel's Theorem -- Chapter 10. Solutions of linear differential equations of prime order p -- 1. Statement of the basic results -- 2. The homogeneous ideal I -- 3. Algebraic functions of several variables -- 4. The differential operator G -- 5. The differential operators S and δ -- 6. A lemma on linear approximation -- 7. End of the proof of Theorem 7 -- 8. Linear reducibility -- 9. Proof of Theorems 6 and 5 -- Remarks -- Chapter 11. The algebraic independence measure of values of IE-functions -- 1. Definition of the measures -- 2. The linear independence measure of values of IE-functions.

3. The algebraic independence measure of values of IE-functions which are not connected by algebraic equations over C(z) -- 4. Auxiliary results -- 5. The algebraic independence measure of values of IE-functions which are connected by algebraic equations over C(z) -- 6. Some applications of the general theorems -- Remarks -- Chapter 12. The algebraic independence measure of values of KE-functions -- 1. The fundamental lemma -- 2. Bounds for the measures of the values of E-functions which are not connected by algebraic equations over C(z) -- 3. Bounds for the measures of the values of E-functions which are connected by a single algebraic equation over C(z) -- 4. Bounds for the measures of the values of E-functions which are connected by arbitrary algebraic equations over C(z) -- 5. Algebraic independence of the values of E-functions in conjugate fields -- 6. An auxiliary theorem -- 7. Consequences of the auxiliary theorem -- 8. Some applications of the general theorems -- Remarks -- Chapter 13. Effective bounds for measures -- 1. Definitions and notation -- 2. Refinement of the fundamental lemmas -- 3. Bounds for linear independence measures -- 4. Bounds for algebraic independence measures -- 5. Some applications of the general theorems -- Remarks -- Concluding remarks -- Supplementary remarks on recent work for the English edition -- Bibliography.