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To the Reader | |

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Contents | |

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Derivatives | |

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First Derivative | |

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Derivative of a vector function | |

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Linearity of differentiation | |

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Derivative of a product | |

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Derivative of the inverse of a function | |

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Derivative of a composite function | |

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Derivative of an inverse function | |

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Derivatives of real-valued functions | |

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The Mean Value Theorem | |

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Rolle's Theorem | |

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The mean value theorem for real-valued functions | |

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The mean value theorem for vector functions | |

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Continuity of derivatives | |

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Derivatives of Higher Order | |

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Derivatives of order n | |

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Taylor's formula | |

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Convex Functions of a Real Variable | |

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Definition of a convex function | |

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Families of convex functions | |

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Continuity and differentiability of convex functions | |

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Criteria for convexity | |

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Exercises on ï¿½1 | |

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Exercises on ï¿½2 | |

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Exercises on ï¿½3 | |

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Exercises on ï¿½4 | |

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Primitives and Integrals | |

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Primitives and Integrals | |

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Definition of primitives | |

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Existence of primitives | |

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Regulated functions | |

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Integrals | |

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Properties of integrals | |

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Integral formula for the remainder in Taylor's formula; primitives of higher order | |

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Integrals over Non-Compact Intervals | |

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Definition of an integral over a non-compact interval | |

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Integrals of positive functions over a non-compact interval | |

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Absolutely convergent integrals | |

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Derivatives and Integrals of Functions Depending on a Parameter | |

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Integral of a limit of functions on a compact interval | |

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Integral of a limit of functions on a non-compact interval | |

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Normally convergent integrals | |

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Derivative with respect to a parameter of an integral over a compact interval | |

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Derivative with respect to a parameter of an integral over a non-compact interval | |

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Change of order of integration | |

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Exercises on ï¿½1 | |

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Exercises on ï¿½2 | |

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Exercises on ï¿½3 | |

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Elementary Functions | |

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Derivatives of the Exponential and Circular Functions | |

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Derivatives of the exponential functions; the number e | |

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Derivative of log<sub>a</sub> x | |

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Derivatives of the circular functions; the number ï¿½ | |

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Inverse circular functions | |

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The complex exponential | |

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Properties of the function e<sup>z</sup> | |

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The complex logarithm | |

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Primitives of rational functions | |

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Complex circular functions; hyperbolic functions | |

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Expansions of the Exponential and Circular Functions, and of the Functions Associated with them | |

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Expansion of the real exponential | |

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Expansions of the complex exponential, of cos x and sin x | |

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The binomial expansion | |

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Expansions of log(1 + x), of Arc tan x and of Arc sin x | |

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Exercises on ï¿½1 | |

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Exercises on ï¿½2 | |

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Historical Note (Chapters I-II-III) | |

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Bibliography | |

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Differential Equations | |

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Existence Theorems | |

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The concept of a differential equation | |

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Differential equations admitting solutions that are primitives of regulated functions | |

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Existence of approximate solutions | |

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Comparison of approximate solutions | |

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Existence and uniqueness of solutions of Lipschitz and locally Lipschitz equations | |

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Continuity of integrals as functions of a parameter | |

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Dependence on initial conditions | |

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Linear Differential Equations | |

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Existence of integrals of a linear differential equation | |

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Linearity of the integrals of a linear differential equation | |

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Integrating the inhomogeneous linear equation | |

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Fundamental systems of integrals of a linear system of scalar differential equations | |

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Adjoint equation | |

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Linear differential equations with constant coefficients | |

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Linear equations of order n | |

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Linear equations of order n with constant coefficients | |

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Systems of linear equations with constant coefficients | |

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Exercises on ï¿½1 | |

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Exercises on ï¿½2 | |

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Historical Note | |

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Bibliography | |

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Local Study of Functions | |

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Comparison of Functions on a Filtered Set | |

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Comparison relations: I. Weak relations | |

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Comparison relations: II. Strong relations | |

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Change of variable | |

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Comparison relations between strictly positive functions | |

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Notation | |

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Asymptotic Expansions | |

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Scales of comparison | |

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Principal parts and asymptotic expansions | |

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Sums and products of asymptotic expansions | |

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Composition of asymptotic expansions | |

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Asymptotic expansions with variable coefficients | |

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Asymptotic Expansions of Functions of a Real Variable | |

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Integration of comparison relations: I. Weak relations | |

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Application: logarithmic criteria for convergence of integrals | |

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Integration of comparison relations: II. Strong relations | |

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Differentiation of comparison relations | |

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Principal part of a primitive | |

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Asymptotic expansion of a primitive | |

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Application to Series with Positive Terms | |

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Convergence criteria for series with positive terms | |

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Asymptotic expansion of the partial sums of a series | |

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Asymptotic expansion of the partial products of an infinite product | |

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Application: convergence criteria of the second kind for series with positive terms | |

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Appendix | |

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Hardyfields | |

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Extension of a Hardy field | |

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Comparison of functions in a Hardy field | |

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(H) Functions | |

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Exponentials and iterated logarithms | |

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Inverse function of an (H) function | |

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Exercises on ï¿½1 | |

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Exercises on ï¿½3 | |

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Exercises on ï¿½4 | |

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Exercises on Appendix | |

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Generalized Taylor Expansions. Euler-Maclaurin Summation Formula | |

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Generalized Taylor Expansions | |

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Composition operators on an algebra of polynomials | |

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Appell polynomials attached to a composition operator | |

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Generating series for the Appell polynomials | |

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Bernoulli polynomials | |

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Composition operators on functions of a real variable | |

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Indicatrix of a composition operator | |

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The Euler-Maclaurin summation formula | |

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Eulerian Expansions of the Trigonometric Functions and Bernoulli Numbers | |

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Eulerian expansion of cot z | |

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Eulerian expansion of sin z | |

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Application to the Bernoulli numbers | |

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Bounds for the Remainder in the Euler-Maclaurin Summation Formula | |

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Bounds for the remainderin the Euler-Maclaurin Summation Formula | |

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Application to asymptotic expansions | |

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Exercises on ï¿½1 | |

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Exercises on ï¿½2 | |

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Exercises on ï¿½3 | |

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Historical Note (Chapters V and VI) | |

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Bibliography | |

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The Gamma Function | |

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The Gamma Function in the Real Domain | |

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Definition of the Gamma function | |

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Properties of the Gamma function | |

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The Euler integrals | |

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The Gamma Function in the Complex Domain | |

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Extending the Gamma function to C | |

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The complements' relation and the Legendre-Gauss multiplication formula | |

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Stirling's expansion | |

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Exercises on ï¿½1 | |

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Exercises on ï¿½2 | |

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Historical Note | |

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Bibliography | |

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Index of Notation | |

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Index | |