Index

Adjoining a root of a polynomial, of a primitive, of an exponential of a primitive, to a differential field, 121 Adjoint (equation) of a differential linear equation, 186 Appell polynomials, 272 Approximate solution to within $\varepsilon$ of a differential equation, 166 Asymptotic expansion – more precise than another, 222 – of a function relative to a comparison scale, 222 – of a function to precision $g_\alpha$, 222 – precision of, 222 Asymptotic expansions, 220 – with variable coefficients, 226

Bernouilli numbers, 276 Bernouilli polynomials, 275 Binomial formula, 108 Binomial series, 109

Carleman’s inequality, 117 Carlson inequality, 116 Cauchy’s criterion for integrals, 65 Change of variables formula, 60 Character – local, 211 Coefficients – binomial, 21 Comparison scales, 220 Complements’ relation, 316 Composition operator, 270 – indicatrix, 278 – order of, 272 – regular, 279 Cosecant, 95 Cosine – hyperbolic, 102 Cosine of a complex number, 102 Cotangent, 95 Criteria – convergence – – of the second kind, 244 – integrals – – logarithmic, 229 Criterion – Cauchy’s criterion or integrals, 65 – Cauchy-Maclaurin, 237 – Ermakoff’s, 86

d’Alembert – criterion for convergence, 245 Definition sequence for an (H) function, 252 Derivative – first, 3 – infinite, 10 – logarithmic, 94 – $n^{th}$, 20 – second, 20 – symmetric, 39 Derivative of a function, 3 – left, 4 – right, 4 Determinant of $n$ integrals of a system of $n$ linear differential equations, 184 Differential equation – adjoint, 187 – integral, 163 – linear, 177 – linear homogeneous, 178 – linear of order $n$, 192 – Lipschitz, 171 – locally Lipschitz, 171 – of first order, 164 – of order $n$, 164 – real variable, 163 – scalar, 164 – solution, 163

– strict solution, 164 Differential equations – system of, 164 du Bois-Reymond theorems, 265

Euler integrals, 310 Euler’s constant, 242 Euler’s formulae, 99 Euler-Maclaurin summation formula, 282 Eulerian expansion of cot $z$, 286 Eulerian expansion of sin $z$, 287 Expansion – of log $\Gamma(z)$ – Stirling’s, 322 Exponential – complex, 97 Exponentials – iterated, 253 Extension – elementary of a differential field, 121 – of a Hardy field, 248

Field – differential, 121 Formula – change of variables, 60 – Euler-Maclaurin summation, 282 – Gauss’, 307 – integration by parts, 60 – integration by parts of order $n$, 60 – Legendre-Gauss multiplication, 318 – Leibniz’, 20 – Stirling’s, 243 – Taylor’s, 21 – Wallis’, 123 – Weierstrass’, 308 Formulae – Euler’s, 99 Function – $n$ times differentiable – at a point, 20 – on an interval, 20 – (H), 252 – concave, 26 – convex, 26 – derived, 4 – differentiable – at a point, 3 – indefinitely, 20 – on an interval, 4 – domination of one function by another, 212 – elementary, 122 – increasing to the right, 37 – left differentiable – at a point, 4 – on an interval, 4 – Lipschitz, 168, 171 – locally Lipschitz, 171 – logarithmically bounded, 214 – logarithmically convex, 310 – mean value of, 58 – negligible with respect to another, 214 – of bounded variation, 79 – of constant sign, 217 – preponderant over another, 214 – regularly convex, 260 – regulated, 54 – piecewise, 63 – right differentiable – at a point, 4 – on an interval, 4 – step, 53 – strictly concave, 26 – strictly convex, 25 – superadditive, 48 Functions – comparable – strongly, 217 – to order $k$, 232 – weakly, 214 – equivalent, 216 – hyperbolic, 102 – similar, 214

Gauss’ formula, 307 Gauss’ integral, 314 Generalized Taylor expansion of a function, 278 Generalized Taylor expansion of a polynomial, 274 Generating series of the Appell polynomials attached to a composition operator, 275

Hadamard’s inequality, 118 Hardy field, 247 Hardy inequality, 117 Hardy-Littlewood inequality, 89 Hardy-Littlewood tauberian theorem, 45 Hermite polynomials, 282 Hlawka’s inequality, 90 Hyperbolic functions, 102 Hölder’s inequality, 115

Identity – Redheffer, 88

Indefinitely differentiable function, 20 Indicatrix of a composition operator, 278 Inequality – Carleman’s, 117 – Carlson, 116 – Cauchy-Buniakowski-Schwarz, 115 – Cauchy-Schwarz, 88 – Hadamard’s, 118 – Hardy, 117 – Hardy-Littlewood, 89 – Hlawka’s, 90 – Hölder, 115 – Opial, 90 – Weyl, 90 Integral – absolutely convergent, 67 – convergent, 63 – Euler, 310 – Gauss’, 314 – normally convergent, 71, 73 – of a piecewise regulated function, 63 – of a regulated function over a noncompact interval, 63 – of differential equation, 163 – Raabe’s, 318 – uniformly convergent, 71 Integration by parts formula, 60 Integration by parts formula of order $n$, 60 Iterated exponentials, 253 Iterated logarithm, 229

Kummer’s congruences, 295

Leibniz’ formula, 20 Line – locally above a graph, 31 – locally below a graph, 31 – locally on a graph, 31 Liouville’s theorem, 122 Lipschitz condition, 168 Lipschitz differential equation, 171 Lipschitz function, 171 local character, 211 Logarithm – iterated, 229 – Naperian, 92 – natural, 92 – of complex number, 100 – principal value, 100 Logarithmic derivative, 94 Logarithmically bounded function, 214 Logarithmically convex function, 310

Maximum – relative, 11 – strict, 11 Mean – arithmetic, 93 – weighted, 93 – geometric, 93 – weighted, 93 Mean value of a function, 58 Mean value theorem, 14 Method of variation of parameters, 182 Minimum – relative, 11 – strict, 11 Naperian logarithm, 92 Natural logarithm, 92 Newton’s polygon, 259 Normally convergent integral, 71

Operator – composition, 270 – translation, 270 Opial inequality, 90 Order of one function with respect to another, 218 Peano’s theorem, 167 Piecewise regulated function, 63 Polygon – Newton, 259 Polynomials – Appell, 272 – Bernouilli, 275 – Hermite, 282 Preponderant – one function over another, 214 Prime number – irregular, 295 – regular, 295 Primitive – of a function on an interval of $\mathbf{R}$, 51 – of order $n$, 62 – second, 62 – strict, 51 Principal part – of a function relative to a comparison scale, 221 – of a function relative to a comparison scale and a domain of coefficients, 227 Principle of comparison of integrals, 66

Raabe’s integral, 318 Raabe’s test, 244 Redheffer identity, 88 Reduced to precision $g_\alpha$ – asymptotic expansion of a function, 222 Regular composition operator, 279 Regulated function, 54 Remainder – in asymptotic expansion, 222 – in the Euler-Maclaurin formula, 283 – in the Taylor formula, 22 Resolvent of a linear differential equation, 180 Rolle’s theorem, 12 Roots – characteristic, of a linear differential equation with constant coefficients, 190

Scales of comparison, 220 Secant, 95 Second mean value theorem, 82 Similar functions, 214 Sine – hyperbolic, 102 Sine of a complex number, 102 Solution to within $\varepsilon$ of a differential equation, 166 Stirling’s formula, 243 Strict primitive, 51 Strict solution of a differential equation, 164 Strictly – above a graph, 23 – below a graph, 23 Strictly concave function, 26 Strictly convex function, 25 Support line to graph of a convex function, 29 System of differential equations, 164 Systems – fundamental systems of integrals of a linear system of scalar differential equations, 183

Tangent – half – – left, 11 – – right, 11 – hyperbolic, 102 – to a graph, 11 Taylor’s expansion, 22 Taylor’s formula, 21 – of order $n$, 22 Test – d’Alembert’s, 245 – Raabe’s, 244 Theorem – Cauchy’s existence theorem, 171 – Clausen-von Staudt, 294 – Hardy-Littlewood tauberian, 45 – Liouville, 122 – Mean value, 14 – Peano, 167 – Rolle, 12 Theorems – du Bois-Reymond, 265 Translation operator, 270

Uniformly convergent integral, 71

Value – mean value of a function, 58 Variation of parameters – method of, 182

Wallis’ formula, 123 Weierstrass’ formula, 308 Weyl inequality, 90 Wronskian of $n$ integrals of a linear differential equation of order $n$, 193