Elements of Mathematics Functions of a Real Variable

I Derivatives.- 1. First Derivative.- 1. Derivative of a vector function.- 2. Linearity of differentiation.- 3. Derivative of a product.- 4. Derivative of the inverse of a function.- 5. Derivative of a composite function.- 6. Derivative of an inverse function.- 7. Derivatives of real-valued functions.- 2. The Mean Value Theorem.- 1. Rolle's Theorem.- 2. The mean value theorem for real-valued functions.- 3. The mean value theorem for vector functions.- 4. Continuity of derivatives.- 3. Derivatives of Higher Order.- 1. Derivatives of order n.- 2. Taylor's formula.- 4. Convex Functions of a Real Variable.- 1. Definition of a convex function.- 2. Families of convex functions.- 3. Continuity and differentiability of convex functions.- 4. Criteria for convexity.- Exercises on 1.- Exercises on 2.- Exercises on 3.- Exercises on 4.- II Primitives and Integrals.- 1. Primitives and Integrals.- 1. Definition of primitives.- 2. Existence of primitives.- 3. Regulated functions.- 4. Integrals.- 5. Properties of integrals.- 6. Integral formula for the remainder in Taylor's formula; primitives of higher order.- 2. Integrals Over Non-Compact Intervals.- 1. Definition of an integral over a non-compact interval.- 2. Integrals of positive functions over a non-compact interval.- 3. Absolutely convergent integrals.- 3. Derivatives and Integrals of Functions Depending on a Parameter.- 1. Integral of a limit of functions on a compact interval.- 2. Integral of a limit of functions on a non-compact interval.- 3. Normally convergent integrals.- 4. Derivative with respect to a parameter of an integral over a compact interval.- 5. Derivative with respect to a parameter of an integral over a non-compact interval.- 6. Change of order of integration.- Exercises on 1.- Exercises on 2.- Exercises on 3.- III Elementary Functions.- 1. Derivatives of the Exponential and Circular Functions.- 1. Derivatives of the exponential functions; the number e.- 2. Derivative of logax.- 3. Derivatives of the circular functions; the number ?.- 4. Inverse circular functions.- 5. The complex exponential.- 6. Properties of the function ez.- 7. The complex logarithm.- 8. Primitives of rational functions.- 9. Complex circular functions; hyperbolic functions.- 2. Expansions of the Exponential and Circular Functions, and of the Functions Associated with Them.- 1. Expansion of the real exponential.- 2. Expansions of the complex exponential, of cos x and sin x.- 3. The binomial expansion.- 4. Expansions of log(1 + x), of Arc tan x and of Arc sin x.- Exercises on 1.- Exercises on 2.- Historical Note (Chapters I-II-III).- IV Differential Equations.- 1. Existence Theorems.- 1. The concept of a differential equation.- 2. Differential equations admitting solutions that are primitives of regulated functions.- 3. Existence of approximate solutions.- 4. Comparison of approximate solutions.- 5. Existence and uniqueness of solutions of Lipschitz and locally Lipschitz equations.- 6. Continuity of integrals as functions of a parameter.- 7. Dependence on initial conditions.- 2. Linear Differential Equations.- 1. Existence of integrals of a linear differential equation.- 2. Linearity of the integrals of a linear differential equation.- 3. Integrating the inhomogeneous linear equation.- 4. Fundamental systems of integrals of a linear system of scalar differential equations.- 5. Adjoint equation.- 6. Linear differential equations with constant coefficients.- 7. Linear equations of order n.- 8 Linear equations of order n with constant coefficients.- 9 Systems of linear equations with constant coefficients.- Exercises on 1.- Exercises on 2.- Historical Note.- V Local Study of Functions.- 1. Comparison of Functions on a Filtered Set.- 1. Comparison relations: I. Weak relations.- 2. Comparison relations: II. Strong relations.- 3. Change of variable.- 4. Comparison relations between strictly positive functions.- 5. Notation.- 2. Asymptotic Expansions.- 1. Scales of comparison.- 2. Principal parts and asymptotic expansions.- 3. Sums and products of asymptotic expansions.- 4. Composition of asymptotic expansions.- 5. Asymptotic expansions with variable coefficients.- 3. Asymptotic Expansions of Functions of a Real Variable.- 1. Integration of comparison relations: I. Weak relations.- 2. Application: logarithmic criteria for convergence of integrals.- 3. Integration of comparison relations: II. Strong relations.- 4. Differentiation of comparison relations.- 5. Principal part of a primitive.- 6. Asymptotic expansion of a primitive.- 4. Application to Series with Positive Terms.- 1. Convergence criteria for series with positive terms.- 2. Asymptotic expansion of the partial sums of a series.- 3. Asymptotic expansion of the partial products of an infinite product.- 4. Application: convergence criteria of the second kind for series with positive terms.- 1. Hardy fields.- 2. Extension of a Hardy field.- 3. Comparison of functions in a Hardy field.- 4. (H)Functions.- 5. Exponentials and iterated logarithms.- 6. Inverse function of an (H) function.- Exercises on 1.- Exercises on 3.- Exercises on 4.- Exercises on Appendix.- VI Generalized Taylor Expansions. Euler-Maclaurin Summation Formula.- 1. Generalized Taylor Expansions.- 1. Composition operators on an algebra of polynomials.- 2. Appell polynomials attached to a composition operator.- 3. Generating series for the Appell polynomials.- 4. Bernoulli polynomials.- 5. Composition operators on functions of a real variable.- 6. Indicatrix of a composition operator.- 7. The Euler-Maclaurin summation formula.- 2. Eulerian Expansions of the Trigonometric Functions and Bernoulli Numbers.- 1. Eulerian expansion of cot z.- 2. Eulerian expansion of sin z.- 3. Application to the Bernoulli numbers.- 3. Bounds for the Remainder in the Euler-Maclaurin Summation Formula.- 1. Bounds for the remainder in the Euler-Maclaurin summation formula.- 2. Application to asymptotic expansions.- Exercises on 1.- Exercises on 2.- Exercises on 3.- Historical Note (Chapters V and VI).- VII The Gamma Function.- 1. The Gamma Function in the Real Domain.- 1. Definition of the Gamma function.- 2. Properties of the Gamma function.- 3. The Euler integrals.- 2. The Gamma Function in the Complex Domain.- 1. Extending the Gamma function to C.- 2. The complements' relation and the Legendre-Gauss multiplication formula.- 3. Stirling's expansion.- Exercises on 1.- Exercises on 2.- Historical Note.- Index of Notation.