Book FVR · English

Index of statements

The 363 statements of Functions of a Real Variable, by chapter and §. Pick a kind, filter by a word, or put the most cited first.

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I. Derivatives

§ 1. First derivative

Definition 1 Let f be a vector function defined on an interval I ⊂ R which does not reduce to a single point. We say that f is differentiable at a point x₀ ∈ I if lim_x → x₀, x ∈ I, x ≠ x₀ (f(x… 082P
Remark 1 In Kinematics, if the point f(t) is the position of a moving point in the space R³ at time t, then fracf(t) - f(t₀)t - t₀ is termed the average velocity between the instants t₀ and… 082Q
Remark 2 If a function f, defined on I, is differentiable at a point x₀ ∈ I, it is necessarily continuous relative to I at this point. 082R
Definition 2 Let f be a vector function defined on an interval I ⊂ R, and let x₀ be a point of I such that the interval I ∩ [x₀, +∞[ (resp. I ∩ ]-∞, x₀]) does not reduce to a single point. We s… 082S
Example 1 A constant function has zero derivative at every point. 082T
Example 2 An affine linear function x ↦ ax + b has derivative equal to a at every point. 082U
Example 3 The real function 1/x (defined for x ≠ 0) is differentiable at each point x₀ ≠ 0, for we have ( 1/x - 1/x₀ ) / (x - x₀) = -1/(x x₀), and, since 1/x is continuous at x₀, the limit o… 082V
Example 4 The scalar function |x|, defined on R, has right derivative +1 and left derivative -1 at x = 0; it is not differentiable at this point. 082W
Example 5 The real function equal to 0 for x = 0, and to x sin 1/x for x ≠ 0, is defined and continuous on R, but has neither right nor left derivative at the point x ≠ 0. One can give examp… 082X
Definition 3 We say that a vector function f defined on an interval I ⊂ R is differentiable (resp. right differentiable, left differentiable) on I if it is differentiable (resp. right different… 082Y
Remark A function may be differentiable on an interval without its derivative being continuous at every point of the interval (cf. I, p. 36, exerc. 5); this is shown by the example of the… 082Z
Proposition 1 The set of vector functions defined on an interval I ⊂ R, taking values in a given topological vector space E, and differentiable at the point x₀, is a vector space over R, and the… 0830
Corollary The set of vector functions defined on an interval I, taking values in a given topological vector space E, and differentiable on I, is a vector space over R, and the map f ↦ Df is… 0831
Remark If one endows the vector space of mappings from I into E and its subspace of differentiable mappings (cf. Gen. Top., X, p. 277) with the topology of simple convergence (or the topo… 0832
Proposition 2 Let E and F be two topological vector spaces over R, and u a continuous linear map from E into F. If f is a vector function defined on an interval I ⊂ R, taking values in E, and di… 0833
Corollary If φ is a continuous linear form on E, then the real function φ ∘ f has a derivative equal to φ(f'(x₀)) at the point x₀. 0834
Example 1 Let f = (fᵢ)_1 ≤ i ≤ n be a function with values in Rⁿ, defined on an interval I ⊂ R; each real function fᵢ is none other than the composite function prᵢ ∘ f, so is differentiable… 0835
Example 2 In Kinematics, if f(t) is the position of a moving point M at time t, if g(t) is the position at the same instant of the projection M' of M onto a plane P (resp. a line D) with ker… 0836
Example 3 Let f be a complex-valued function defined on an interval I ⊂ R, and let a be an arbitrary complex number; prop. 2 shows that if f is differentiable at a point x₀ then so is af, an… 0837
Proposition 3 For each index i (1 ≤ i ≤ p) let fᵢ be a function defined on an interval I ⊂ R, taking values in Eᵢ, and differentiable at the point x₀ ∈ I. Then the function 0838
Example 1 Let f be a real function, g a vector function, both differentiable at a point x₀; the function gf has a derivative equal to g'(x₀)f(x₀) + g(x₀)f'(x₀) at x₀. In particular, if a is… 0839
Example 2 The real function xⁿ arises from the multilinear function 083A
Example 3 The euclidean scalar product (x| y) (Gen. Top , VI, p. 40) is a bilinear map (necessarily continuous) of Rⁿ × Rⁿ into R. If f and g are two vector functions with values in Rⁿ, and… 083B
Example 4 If E is a topological algebra over R (cf. Introduction), the product xy of two elements of E is a continuous bilinear function of (x, y); if f and g have their values in E and are… 083C
Example 5 The determinant det(x₁, x₂, …, xₙ) of n vectors xᵢ = (xᵢⱼ)_1 ≤ i ≤ n from the space Rⁿ (Alg., III, p. 522) being a (continuous) multilinear function of the xᵢ, one sees that if the… 083D
Remark If U(x) is a square matrix which is differentiable and invertible at the point x₀, then the derivative of its determinant Δ(x) = det(U(x)) can be expressed through the derivative o… 083E
Proposition 4 Let E be a complete normed algebra with a unit element over R and let f be a function defined on an interval I ⊂ R, taking values in E, and differentiable at the point x₀ ∈ I. If y… 083F
Example 1 The most important particular case is that where E is one of the fields R or C: if f is a function with real or complex values, differentiable at the point x₀, and such that f(x₀)… 083G
Example 2 If U = (αᵢⱼ(x)) is a square matrix of order n, differentiable at x₀ and invertible at this point, then U⁻¹ has derivative equal to -U⁻¹ U' U⁻¹ at x₀. 083H
Proposition 5 Let f be a real function defined on an interval I ⊂ R, and g a vector function defined on an interval of R containing f(I). If f is differentiable at the point x₀ and g is differen… ↩ 1 083I
Proposition 6 Let f be a homeomorphism of an interval I ⊂ R onto an interval J = f(I) ⊂ R, and let g be the inverse homeomorphism³. If f is differentiable at the point x₀ ∈ I, and if f'(x₀) ≠ 0… 083J
Corollary If f is differentiable on I and if f'(x) ≠ 0 on I, then g is differentiable on J and its derivative at each point y ∈ J is 1/f'(g(y)). 083K
Remark 1 All the preceding propositions, stated for functions differentiable at a point x₀, immediately yield propositions for functions which are right (resp. left) differentiable at x₀, w… 083L
Remark 2 The preceding definitions and propositions (except for those concerning right and left derivatives) extend easily to the case where one replaces R by an arbitrary commutative non-d… 083M
Example At the point x = 0 the function x^1/3 (the inverse function of x³, a homeomorphism of R onto itself) has a derivative, equal to +∞; at x = 0 the function |x|^1/3 has right derivati… 083N
Definition 4 We say that a real function f, defined on a subset A of a topological space E, has a relative maximum (resp. strict relative maximum, relative minimum, strict relative minimum) at… 083O
Proposition 7 Let f be a finite real function, defined on an interval I ⊂ R. If f admits a relative maximum (resp. relative minimum) at a point x₀ interior to I, and has both right and left deri… 083P

§ 2. The mean value theorem

Proposition 1 ("Rolle's theorem") Let f be a real function which is finite and continuous on a closed interval I = [a, b] (where a < b), has a derivative (finite or not) at every point of ]a, b[, and is such that f… 083Z
Corollary Let f be a real function which is finite and continuous on [a, b] (where a < b), and has a derivative (finite or not) at every point. Then there exists a point c of ]a, b[ such tha… 0840
Proposition 2 Let f be a real function which is finite and continuous on the closed bounded interval I = [a, b] (where a < b) and has a right derivative (finite or not) at all the points of the… ↩ 2 0841
Corollary Let f be a finite continuous real function on [a, b] (where a < b) and have a right derivative at all points of the complement in [a, b[ of a countable subset A of this interval. F… 0842
Remark 1 Prop. 2 remains true when one replaces the interval [a, b[ by ]a, b and the words "right derivative" by "left derivative". 0843
Remark 2 The hypothesis of continuity on f on the closed interval I (and not just right continuity ⁴ at every point of [a, b[) is essential for the validity of prop. 2 (cf. I, p. 36, exerc… 0844
Remark 3 The conclusion of prop. 2 is not guaranteed if one merely supposes that the set A of "exceptional" points is nowhere dense in I, but not countable (cf. I, p. 37, exerc. 3). 0845
Remark Th. 1 fails if one allows f'_d and g' to be simultaneously infinite on an uncountable subset of I (cf. I, p. 37, exerc. 3). 0846
Corollary Let f be a finite continuous function on [a,\ b] (where a < b) and have a right derivative (finite or not) at all points of the relative complement B in [a,\ b[ of a countable subs… 0847
Remark The inequalities (5) prove that a continuous function cannot have right derivative equal to +∞ at all points of an interval (cf. I, p. 38, exerc. 6). 0848
Theorem 2 Let f be a vector function defined and continuous on a closed bounded interval I = [a,\ b] of R (where a < b) and taking values in a normed space E over R; let g be a continuous in… ↩ 1 0849
Remark 1 Here again, in the statement of th. 2 one can replace the interval [a, b[ by ]a, b] and "right derivative" by "left derivative". 084A
Remark 2 We shall show later how to identify the case of equality in (7), and also how to generalize th. 2 to the case where E is an arbitrary locally convex space, with the help of another… 084B
Corollary For a continuous vector function on an interval I ⊂ R, with values in a normed space E over R, to be constant on I it suffices that it have zero right derivative at all points of t… 084C
Remark The proofs of ths. 1 and 2 rely in an essential manner on the special topological properties of the field R; one can give examples of valued fields K for which there are nonconstan… 084D
Proposition 3 Let f be a vector function with values in a normed space E over R, defined and continuous on an interval I ⊂ R, and right differentiable on the complement B (relative to I) of a co… 084E
Proposition 4 Let f be a continuous differentiable function of a complex variable defined on a convex open subset A of the field C, with values in a normed space E over the field C. If one has |… 084F
Corollary For a vector function f of a complex variable, defined and continuous on an open set A ⊂ C, and with values in a normed space over C, to be constant, it suffices that it have zero… 084G
Proposition 5 Let f be a vector function of a complex variable, defined, continuous and differentiable on a convex open set A ⊂ C, taking values in a normed space over the field C; then, no matt… 084H
Proposition 6 Let I be an open interval in R, let x₀ be one of the endpoints of I, and f a vector function defined and continuous on I, with values in a complete normed space E over R; suppose t… 084I
Remark A similar argument, based on th. 1, shows that if f is a real function such that f'_d(x) tends to +∞ at the point x₀ then the ratio 084J

§ 3. Derivatives of higher order

Proposition 1 The set of vector functions defined on an interval I ⊂ R, taking values in a given topological vector space E, and having an nᵗʰ derivative on I, is a vector space over R, and f ↦… 084Y
Proposition 2 ("Leibniz' formula") Let E, F, G be three topological vector spaces over R, and (x, y) ↦ [x.y] a continuous bilinear mapping of E × F into G. If f (resp. g) is defined on an interval I ⊂ R, takes its v… 084Z
Theorem 1 If the function f has an nᵗʰ derivative at the point a then 0850
Proposition 3 If |f^(n+1)(x)| ≤ M on I, then we have |rₙ(x)| ≤ M frac|x - a|ⁿ⁺¹(n + 1)! on I. 0851
Corollary If f is a finite real function with a derivative of order n + 1 on I, and if m ≤ f^(n+1)(x) ≤ M on I, then for all x ≥ a in I one has m frac(x - a)ⁿ⁺¹(n + 1)! ≤ rₙ(x) ≤ M frac(x -… 0852
Remark 1 We have already noticed in the proof of th. 1 that if f has a derivative of order n on I, and if f(x) = a₀ + a₁(x - a) + a₂(x - a)² + ⋯ + aₙ(x - a)ⁿ + rₙ(x) is its Taylor expansion… 0853
Remark 2 With the same hypotheses, the coefficients aᵢ in (11) are determined recursively by the relations 0854
Remark 3 The definition of the derivative of order n and the preceding results generalize immediately to functions of a complex variable; we shall not pursue this topic further here; it wil… 0855

§ 4. Convex functions of a real variable

Lemma Let A = (a, a'), B = (b, b'), C = (c, c') be three points in R² such that a < b < c. The following statements are equivalent: 085O
Definition 1 We say that a finite numerical function f, defined on an interval I ⊂ R, is convex on I if, no matter what the points x, x' of I, (x < x'), every point M_z of the graph G of f such… 085P
Remark On sees in the same way that the set of points lying strictly above G is convex. Conversely, if this set is convex one has 085Q
Example 1 Every (real) affine linear function ax + b is convex on R. 085R
Example 2 The function x² is convex on R, since one has 085S
Example 3 The function |x| is convex on R, since 085T
Definition 2 We say that a finite real function f defined on an interval I ⊂ R is strictly convex on I if, for any points x, x' of I (x < x'), every point M_z of the graph G of f such that x <… 085U
Proposition 1 Let f be a finite real function, convex (resp. strictly convex) on an interval I ⊂ R. For every family (xᵢ)_1 ≤ i ≤ p of p ≥ 2 distinct points of I, and every family (λᵢ)_1 ≤ i ≤ p… 085V
Proposition 2 Let fᵢ (1 ≤ i ≤ p) be p convex functions on an interval I ⊂ R, and cᵢ (1 ≤ i ≤ p) be p arbitrary positive numbers; then the function f = ∑ᵢ₌₁ᵖ cᵢ fᵢ is convex on I. Further, if for… 085W
Proposition 3 Let (f_α) be a family of convex functions on an interval I ⊂ R; if the upper envelope g of this family is finite at every point of I then g is convex on I. 085X
Proposition 4 Let H be a set of convex functions on an interval I ⊂ R; if 𝔉 is a filter on H which converges pointwise on I to a finite real function f₀, then this function is convex on I. 085Y
Proposition 5 For a real finite function f to be convex (resp. strictly convex) on an interval I it is necessary and sufficient that for all a ∈ I the gradient 085Z
Proposition 6 Let f be a finite real function convex on an interval I ⊂ R. Then at every interior point a of I the function f is continuous, has finite right and left derivatives, and f'_g(a) ≤… 0860
Corollary 1 Let f be a convex (resp. strictly convex) function on I; if a and b are two interior points of I such that a < b one has (fig. 3) 0861
Corollary 2 If f is convex (resp. strictly convex) on I then f'_d and f'_g are increasing (resp. strictly increasing) on the interior of I; the set of points in I at which f is not differentia… 0862
Remark If f is a strictly convex function on an open interval I then f'_d is strictly increasing on I, so there are three possible cases, according to prop. 2 of I, p. 13: f is strictly d… 0863
Proposition 7 Let f be a finite real function defined on an interval I ⊂ R. For f to be convex on I it is necessary and sufficient that for every pair of numbers a, b of I such that a < b, and f… 0864
Proposition 8 For a finite real function f to be convex (resp. strictly convex) on an open interval I ⊂ R it is necessary and sufficient that it be continuous on I, have a derivative at every po… 0865
Corollary Let f be a finite real function, continuous and twice differentiable on an interval I ⊂ R; for f to be convex on I it is necessary and sufficient that f''(x) geqslant 0 for all x ∈… 0866
Example On the interval ]0, +∞ [ the function xʳ (r any real number) has a second derivative equal to r(r-1)x⁻²; thus it is strictly convex if r > 1 or r < 0, and strictly concave if 0 < r… 0867
Proposition 9 Let f be a real finite function which is upper semi-continuous on an open interval I ⊂ R. For f to be convex on I it is necessary and sufficient that for every point Mₓ of the grap… 0868
Corollary 1 For a real finite function f defined on an open interval I ⊂ R and upper semi-continuous on I to be convex on I it is necessary and sufficient that for all x ∈ I there should exist… 0869
Corollary 2 Let f be a finite real function defined on an open interval I ⊂ R. If for every point x ∈ I there is an open interval Jₓ ⊂ I containing x and such that the restriction of f to Jₓ i… 086A

II. Primitives and integrals

§ 1. Primitives and integrals

Definition 1 Given a vector function f defined on an interval I ⊂ R we say that a function g defined on I is a primitive of f if g is continuous on I and has a derivative equal to f(x) at every… 0870
Proposition 1 Let f be a vector function defined on I with values in E; if f admits a primitive g on I then the set of primitives of f on I is identical to the set of functions g + a, where a is… 0871
Theorem 1 Let A be a set filtered by a filter 𝔉, and (f_α)_α ∈ A a family of vector functions with values in a complete normed space E over R, defined on an interval I ⊂ R: for each α ∈ A le… ↩ 1 0872
Corollary 1 The set H of maps from I into E which admit a primitive on an interval I is a closed (and so complete) vector subspace of the complete vector space F_c(I; E) of maps from I into E… ↩ 1 0873
Corollary 2 Let x₀ be a point of I, and for each function f ∈ H let P(f) be the primitive of f which vanishes at the point x₀; the map f ↦ P(f) of H into F_c(I; E) is a continuous linear mappi… 0874
Definition 2 One says that a map f of an interval I ⊂ R into a set E is a step function if there is a partition of I into a finite number of intervals Jₖ such that f is constant on each of the… 0875
Proposition 2 The set of step functions defined on I, with values in a vector space E over R, is a vector subspace E of the vector space F(I; E) of all maps of I into E. 0876
Corollary The vector subspace E is generated by the characteristic functions of intervals. 0877
Definition 3 One says that a vector function, defined on an interval I, with values in a complete normed space E over R, is a regulated function, if it is the uniform limit of step functions on… 0878
Theorem 2 Every regulated function on an interval I admits a primitive on I. 0879
Theorem 3 For a vector function f defined on an interval I, with values in a complete normed space E over R, to be regulated, it is necessary and sufficient that it have a right limit and a… 087A
Corollary 1 Let f be a regulated function on I; at every point of I, apart from the right-hand endpoint (resp. the left-hand endpoint) of I, every primitive of f has a right derivative equal t… 087B
Corollary 2 Let fi (1 \leq i \leq n) be n regulated functions on an interval I, each fi having its values in a complete normed space Ei over \mathbf{R} (1 \leq i \leq n). If g is a continuous… 087C
Remark 1 If f is a real regulated function on I, and g is a regulated vector function on an interval containing f(I), then the composite function g ∘ f is not necessarily regulated (cf. II… 087D
Proposition 3 Every continuous vector function on an interval I ⊂ R taking its values in a complete normed space E over R is regulated, and admits a primitive on I, of which it is the derivative… 087E
Remark 2 To show that a continuous function admits a primitive, one can use the fact that every polynomial function of a real variable (with coefficients in E) admits a primitive; since fro… 087F
Remark 3 The principle of the preceding remark extends without significant modification to vector functions of a complex variable taking values in a complete normed space over C. If U is an… 087G
Proposition 4 Every monotone real function f on an interval I ⊂ R is regulated, and every primitive of f is convex on I. 087H
Remark 4 One must not think that the regulated functions on an interval I are the only functions having a primitive on I (cf. II, p. 80, exercises 7 and 8). 087I
Proposition 5 Let f be a regulated function on an interval I, let g be a primitive of f on I, and [x₀,\ x] a compact interval contained in I. For every ε > 0 there exists a number ρ > 0 such tha… 087J
Remark 1 When f is continuous prop. 5 can be proved more simply: since f is uniformly continuous on [x₀,\ x] there exists a ρ > 0 such that on every interval of length ≤ ρ contained in [x₀… 087K
Remark 2 For an arbitrary function h defined on I, with values in E, the element h(x) - h(x₀) is also written as h(t)|_x₀ˣ; with this notation one sees that if g is any primitive of a regul… 087L
Remark 3 The expressions ∫_x₀ˣ f(t) dt and g(t)|_x₀ˣ are abbreviating symbols representing assemblies in which the letters x, x₀, f, g, but not the letter t, appear (cf. Set Theory, I, p. 1… 087M
Remark 4 The approximation of an integral by Riemann sums is closely connected to one of the historical origins of the concept of an integral, the problem of the measure of areas. We shall… 087N
Proposition 6 (theorem of the mean) Let f be a real regulated function on a compact interval l = [a, b]; if J is the set of points of I where f is continuous, and m = inf_x ∈ J f(x), M = sup_x ∈ J f(x), then 087O
Corollary 1 If a real regulated function f on I is such that f(x) ≥ 0 at the points where f is continuous, then 1/(b-a) ∫ₐᵇ f(t) dt > 0 unless f(x) = 0 at the points where f is continuous. 087P
Corollary 2 Let f and g be two real regulated functions on I, such that g(x) ≥ 0 at the points where g is continuous; if m and M are the greatest lower bound and least upper bound of f over th… 087Q
Proposition 7 Let f be a regulated vector function on a compact interval I = [a, b], with values in a complete normed space E, and let g be a real regulated function on I, such that g(x) ≥ 0 at… 087R

§ 2. Integrals over non-compact intervals

Definition 1 One says that a vector function f, piecewise regulated on an interval [a,\ b[ of overlineR, admits an integral on this interval if f admits a primitive on [a,\ b[; if g is any one… 088F
Example i) The integral ∫₁^+∞ dt/t² is convergent and equal to 1, for ∫₁ˣ dt/t² = 1 - 1/x. 088G
Proposition 1 (Cauchy’s criterion for integrals) Let f be a regulated function on an interval I ⊂ R having endpoints a and b (a < b). For the integral ∫ₐᵇ f(t) dt to exist it is necessary and sufficient that for every ε > 0 there… 088H
Lemma Let J₀ = [α, β] be a compact interval contained in I. In order that | ∫_J f(t) dt - ∫_J' f(t) dt | ≤ ε for every pair of compact intervals J, J' contained in I and containing J' it… 088I
Example If the interval I is bounded, and if f is bounded on I, then the integral ∫_I f(t) dt always exists, for, by the mean value theorem, one has, for y ≤ α ≤ β ≤ z, | ∫_y^α f(t) dt | ≤… 088J
Proposition 2 Let f be a real regulated function ≥ 0 on an interval I ⊂ R with endpoints a and b (a < b). For the integral ∫ₐᵇ f(t) dt to exist it is necessary and sufficient that the set of num… 088K
Definition 2 One says that the integral of a regulated function f over an interval I ⊂ R is absolutely convergent if the integral of the positive function | f(x) | is convergent. 088L
Proposition 4 If the integral of f over I is absolutely convergent then it is convergent, and one has 088M
Corollary Let E, F, G be three complete normed spaces over R, and (x, y) ↦ x.y a continuous bilinear map from E × F into G. Let f, g be two regulated functions on I, with values in E and F r… 088N
Remark An integral can be convergent without being absolutely convergent; this is what is shown by Example 3 of II, p. 64, where the series with general term uₙ is convergent without bein… 088O

§ 3. Derivatives and integrals of functions depending on a parameter

Proposition 1 Let A be a set filtered by a filter 𝔉, and (f_α)_α ∈ A a family of regulated functions on a compact interval I = [a, b]; if the functions f_α converge uniformly on I to a (regulate… 088Y
Corollary 1 Let (fₙ) be a sequence of regulated functions on a compact interval I = [a, b]. If the sequence (fₙ) converges uniformly on I to a (regulated) function f, one has 088Z
Corollary 2 Let A be a subset of a topological space F, and f a map from I × A into a complete normed space E over R, such that, for each α ∈ A, the function x ↦ f(x, α) is regulated on I. If… 0890
Proposition 2 ("continuity of an integral with respect to a parameter") Let F be a compact space, let I = [a,\ b] be a compact interval in R, and let f be a continuous map of I × F into a complete normed space E over R; then the function h(α) = ∫ₐᵇ f(x… 0891
Definition 1 One says that the integral ∫_I f_α(t) dt is uniformly convergent for α ∈ A (or uniformly convergent over A) if, for every ε > 0 there exists a compact interval J₀ ⊂ J such that, fo… 0892
Proposition 3 Let (f_α ) be a family of regulated functions on an interval I such that: 1 with respect to the filter 𝔉 the family (f_α ) converges uniformly to a function f (regulated on I) on e… 0893
Corollary 1 Let (fₙ) be a sequence of regulated functions on an arbitrary interval I, converging uniformly to a function f on each compact interval contained in I; if the integral ∫_I fₙ(t) dt… 0894
Remark The hypotheses imposed in this corollary are sufficient, but not necessary, for the validity of formula (9); we shall generalize this formula later, at the same time as the concept… 0895
Corollary 2 Let A be a subset of a topological space F, and f a map of I × A into a complete normed space E over R, such that, for every α ∈ A, the function x ↦ f(x, α) is regulated on I. If… 0896
Proposition 4 ("continuity of an improper integral with respect to a parameter") Let F be a compact space, let I be any interval in R, and f a continuous map from I × F into a complete normed space E over R; if the integral h(α) = ∫_I f(x, α) dx is uniformly co… 0897
Proposition 5 Suppose that the partial derivative f'_α(x, α) exists for all x ∈ I and all α in an open neighbourhood V of α₀, and that, for all α ∈ V, the map x ↦ f'_α(x, α) is regulated on I. U… 0898
Corollary If the partial derivative f'_α(x, α) exists on I × V and is a continuous function of (x, α) on this set, then the function g admits a derivative given by the formula (12) at the po… 0899
Proposition 6 Supposing that the hypotheses of prop. 5 are satisfied, let a(α), b(α) be two functions defined on V, with values in I; if the derivatives a'(α₀),\ b'(α₀) exist and are finite then… 089A
Proposition 7 Let I be an arbitrary interval in R, and f a continuous function on I × V. Suppose that: the partial derivative f'_α(x, α) exists for all x ∈ I and all α ∈ V, and, for all α ∈ V, t… 089B
Remark 1 Conditions 1 and 2 of prop. 7 are satisfied a fortiori when f'_α(x, α) is a continuous function of (x, α) on I × V. 089C
Remark 2 When, in an integral ∫_a(α)^b(α) f(t, α) dt, the endpoints of the interval are finite functions of the parameter, the study of this integral as a function of α can be related to th… 089D
Proposition 8 If f is continuous on I × A one has 089E
Proposition 9 If the function f is continuous on I × A, and if the integral ∫_I f(t, α) dt is uniformly convergent on A, then the integral ∫_I dt ∫_cᵈ f(t, α) dα is convergent, and one has 089F

III. Elementary functions

§ 1. Derivatives of the exponential and circular functions

Theorem 1 Let E be a complete normed algebra over the field R, with a unit element e, and let f be a continuous group homomorphism of the additive group R into the multiplicative group G of… 089Q
Proposition 1 For every number a > 0 and ≠ 1 the exponential function a^λ admits at every point x ∈ R a derivative equal to (logₑ a)a^λ where e is a number > 1 (independent of a). 089R
Definition 1 Logarithms to the base e are called Naperian logarithms (or natural logarithms). 089S
Proposition 2 ("geometric mean inequality") For any numbers zᵢ > 0 (1 ≤ i ≤ n) and numbers pᵢ > 0 such that ∑ᵢ₌₁ⁿ pᵢ = 1, one has z₁^p₁ z₂^p₂ ⋯ zₙ^pₙ ≤ p₁ z₁ + p₂ z₂ + ⋯ + pₙ zₙ. (6) Moreover, the two sides of (6) are equal… 089T
Example If u is a function ≠ 0 on an interval I, and v is any real function, then log(|u|ᵛ) = v · log |u|, so if u and v are differentiable 1/|u|ᵛ D(|u|ᵛ) = v' log |u| + v u'/u. 089U
Proposition 3 The function e(x) has a derivative equal to 2π i e(x) at every point of R, where π is a constant > 0. 089V
Definition 2 The homomorphism x + iy ↦ eᶻ e(y/2π) of C onto C^* is called the complex exponential; its value at an arbitrary complex number z is denoted by eᶻ or exp z. 089W
Remark If, in formula (27), one restricts the function e^a z (a complex) to the real axis, one again obtains, for x real, D(e^a x) = a e^a x. This formula allows us to calculate a primiti… 089X
Example One has 089Y
Remark For complex z one sometimes writes 089Z

§ 2. Expansions of the exponential and circular functions and of the functions associated with them

Remark Since the remainder in formula (1) is > 0 for x > 0 one has, for x > 0 08B3

IV. Differential equations

§ 1. Existence theorems

Lemma 1 Let f be a map from I × H into E such that, on writing fₓ (for every x ∈ H) for the map t ↦ f(t, x) of I into E, the following conditions are satisfied: 1° fₓ is regulated on I for… 08BD
Proposition 1 Let t₀ be a point of I and x₀ a point of H; for a continuous function u to be a solution of the equation (1) on I and take the value x₀ at the point t₀, it is necessary and suffici… 08BE
Corollary At every point of I distinct from the left (resp. right) endpoint of this interval, every solution u of (1) on I admits a left (resp. right) derivative equal to the left (right) li… 08BF
Proposition 2 If f is a continuous map from I × H into E, then every solution of (1) on I is a strict solution. 08BG
Proposition 3 On every compact interval with left (or right) endpoint t0 contained in J, with length less than r/(M + ε), there exists an approximate solution to within ε of equation (1), with v… 08BH
Proposition 4 The set of approximate solutions of (1) to within ε, defined on the same interval K ⊂ J and taking values in S, is uniformly equicontinuous. 08BI
Corollary (Peano's theorem) If E is of finite dimension over R then there exists a solution of (1) with values in S and equal to x₀ at t₀, on every compact interval K with left (or right) endpoint t₀ containe… 08BJ
Remark 1 There can be infinitely many integrals of a differential equation (1) taking the same value at a given point. For example, the scalar differential equation x' = 2sqrt|x| has all th… 08BK
Remark 2 Peano’s theorem no longer holds when E is an arbitrary complete normed space of infinite dimension (IV, p. 204, exerc. 18). 08BL
Definition 1 Given a positive real function t ↦ k(t) defined on I, one says that a map f from I × H into E is Lipschitz with respect to the function k(t) if, for every x ∈ H, the function t ↦ f… 08BM
Example When E = R and H is an interval in R, if the function f(t, x) admits a partial derivative f'ₓ (II, p. 74) at every point (t, x) of I × H, such that |f'ₓ(t, x)| ≤ k(t) on I × H, the… 08BN
Proposition 5 Let k(t) be a real regulated function and > 0 on I, and let f(t, x) be a function that is defined and Lipschitz with respect to k(t) on I × H. If u and v are two approximate soluti… 08BO
Lemma 2 If w is a continuous real function on the interval [t0, t1] and satisfies the inequality 08BP
Corollary Let f be a Lipschitz function for the constant k > 0, defined on I × H. If u and v are two approximate solutions of (1) to within ε₁ and ε₂ respectively, defined on I and taking th… 08BQ
Remark 1 When k = 0 the inequality (15) is replaced by the inequality | u(t) - v(t) | ≤ | u(t₀) - v(t₀) | + (ε₁ + ε₂) |t - t₀| whose proof is immediate. 08BR
Remark 2 When E is of finite dimension, and f is Lipschitz on I × H, one can show the existence of approximate solutions of (1) (IV, p. 166, prop. 3) without using the axiom of choice (IV… 08BS
Proposition 6 Let f and g be two functions defined on I × H, satisfying the hypotheses of lemma 1 of IV, p. 165, and such that, on I × H, | f(t, x) - g(t, x) | ≤ α. (16) ↩ 1 08BT
Theorem 1 (Cauchy) Let f be a Lipschitz function on I × H, let J be a compact subinterval of I, not reducing to a single point, t₀ a point of J, S an open ball with centre x₀ and radius r, contained… 08BU
Theorem 2 Let I ⊂ R be an interval (not reducing to a single point) with left-hand endpoint t₀ ∈ I, let H be a nonempty open set in E, and f a locally Lipschitz function on I × H. 08BV
Corollary 1 If H = E and J ≠ I then f(t, u(t)) is not bounded on J, if, further, E is finite dimensional, then |u(t)| has left limit +∞ at the right-hand endpoint of J. 08BW
Corollary 2 If, on I × H, the function f is Lipschitz with respect to a regulated function k(t), and if the right-hand endpoint β of J belongs to I, then u has a left limit at β; if H = E and… 08BX
Example 1 For a differential equation of the form x' = g(t), where g is regulated on I, every integral u is clearly defined on all of I. One should note that u can be bounded on I without g(… 08BY
Example 2 For the scalar equation x' = sqrt1 - x² one has I = R,\ H = ] - 1, +1[. If one takes t₀ = x₀ = 0 the corresponding integral is sin t on the largest interval containing 0 where the… 08BZ
Example 3 For the scalar equation x' = 1 + x² one has I = H = R: the integral that vanishes at t = 0 is tan t, and the largest interval containing 0 where this function is continuous is J =… 08C0
Example 4 For the scalar equation x' = sin tx one has I = H = R and the right-hand side is bounded on I × H, so (IV, p. 173, cor. 1) every integral is defined on all of R. ↩ 1 08C1
Proposition 7 Let f and g be two functions defined on I × H, satisfying the hypotheses of lemma 1 of IV. p. 165, and such that, on I × H 08C2
Theorem 3 Let F be a topological space and let f be a map from I × H × F into E such that, for every ξ ∈ F, the function (t, x) ↦ f(t, x, ξ) is Lipschitz on I × H, and such that, when ξ tend… 08C3
Remark When H = E and the condition (16) of IV, p. 174 is satisfied on I × E, prop. 7 of IV, p. 174 applies to every solution u of x' = g(t, x) on any interval where this solution is defi… 08C4
Theorem 4 Let f be a locally Lipschitz function on I × H and (a, b) an arbitrary point of I × H. 08C5

§ 2. Linear differential equations

Example 1 When E is of finite dimension n over R one can identify the endomorphism A(t) with its matrix (aᵢⱼ(t)) with respect to any basis of E (Alg., II, p. 343); when one identifies a vect… 08CO
Example 2 Let G be a complete normed algebra over R, and a(t), b(t) and c(t) three maps from J into G; the equation 08CP
Theorem 1 Let t ↦ A(t) be a regulated map of J into L(E), and t ↦ b(t) be a regulated map of J into E. For every point (t₀, x₀) of J × E the linear equation (2) admits one and only one solut… 08CQ
Proposition 1 The set I of integrals of the homogeneous linear equation (4), defined on J, is a vector subspace of the space C(J; E) of continuous maps from J into E. 08CR
Theorem 2 For every point (t₀, x₀) of J × E let u(t, t₀, x₀) be the integral of the homogeneous equation (4) defined on J and equal to x₀ at t₀. For every point t ∈ J the map x₀ ↦ u(t, t₀, x… 08CS
Corollary 1 The map which to every point x₀ ∈ E associates the continuous function t ↦ C(t, t₀).x₀, defined on J, is an isomorphism of the normed space E onto the vector space I of integrals o… 08CT
Corollary 2 The map (s, t) ↦ C(s, t) of J × J into L(E) is continuous. 08CU
Corollary 3 Let K be a compact interval contained in J, and let k = sup_t ∈ K |A(t)|. For all t and t₀ in K |C(t, t₀) - I| ≤ e^k|t-t₀| - 1 . (7) 08CV
Proposition 2 Let B be a continuous endomorphism of E, independent of t, and commuting with A(t) for all t ∈ J; then B commutes with C(t, t₀) for all t and t₀ in J. 08CW
Proposition 3 With the notation of th. 2 (IV, p. 179), for every point (t₀, x₀) of J × E the integral of the linear equation (2) defined on J and equal to x₀ at t₀ is given by the formula 08CX
Proposition 4 For the n integrals uⱼ (1 ≤ j ≤ n) of equation (4) to form a fundamental system it is necessary and sufficient that their values uⱼ(t₀) at a point t₀ ∈ J be linearly independent ve… 08CY
Proposition 5 The determinant of the matrix C(t, t₀) is given by the formula det ( C(t, t₀) ) = exp ( ∫_t₀ᵗ Tr(A(s)) ds ). (15) 08CZ
Remark 1 All the above applies also to the case where E is of dimension n over the field R and A(t) is an endomorphism of E for every t ∈ J: one has only to replace C by R throughout. 08D0
Remark 2 Let A(t) = (aᵢⱼ(t)) be a square matrix of order n whose entries are real (resp. complex) regulated functions of t on J, and let C(t, t₀) = (cᵢⱼ(t, t₀)) be the resolvent matrix of t… 08D1
Proposition 6 The map t ↦ v(t) of J into E^*, a primitive of a regulated function on J, is such that ⟨ u(t), v(t) ⟩ is constant on J for every solution u of the equation (4) of IV, p. 183 if and… 08D2
Remark Let E and F be two complete normed spaces over R (or over C), and (x, y) ↦ ⟨ x, y ⟩ a continuous bilinear form on E × F, such that the relation “⟨ x, y ⟩ = 0 for all y ∈ F” (resp… 08D3
Definition 1 Given a continuous endomorphism A of E we denote by eᴬ or exp A the automorphism of E equal to the value at the point t = 1 of the integral of the equation (20) which takes the val… 08D4
Proposition 7 1' The map X ↦ e^X is a continuous map of L(E) into the group of automorphisms of E (invertible elements of L(E)). 08D5
Remark When there exists a basis of E such that the matrix of A with respect to this basis has real elements (cf. IV, p. 186, Remark 2), the characteristic equation of A has real coeffici… 08D6
Proposition 8 The set of integrals of the homogeneous linear equation (37), defined on J, is a vector space of dimension n over the field C. 08D7
Proposition 9 Let rⱼ (1 ≤ j ≤ q) be the distinct roots of the characteristic equation (40), and let nⱼ be the multiplicity of the root rⱼ (1 ≤ j ≤ q). Then the n functions tᵏ e^rⱼ t (1 ≤ k ≤ nⱼ… 08D8
Proposition 10 Let g and h be two relatively prime polynomials such that f = gh. The subspace of solutions of (42) is the direct sum of the subspaces of solutions of the two equations 08D9
Remark 1 Some of the polynomials fⱼ(D) may reduce to nonzero constants; the corresponding yⱼ are then completely determined. 08DA
Remark 2 When the bⱼ are all zero, that is, if the system (47) is homogeneous, the conditions (52) are always satisfied; if, further, r = n, one sees that the set of solutions of (47) is a… 08DB
Remark 3 Given the polynomials pⱼₖ(D), a system (47) which admits solutions when the right-hand sides are indefinitely differentiable (or differentiable of a certain order) may not admit th… 08DC

V. Local study of functions

§ 1. Comparison of functions on a filtered set

Example If f and g are two functions in H(𝔉, R), the relation "there is an X ∈ 𝔉 such that f and g are defined on X and f(t) ≤ g(t) for every t ∈ X" is of local character relative to f and… 08DN
Remarks 1) Let Vᵢ (1 ≤ i ≤ n) be n normed vector spaces over K, and φ a function defined on V₁ × V₂ × ⋯ × Vₙ, with values in V; by passing to the quotient along R_∞ the function φ defines… 0E5L
Definition 1 Given two real functions f, g belonging to H(𝔉, R), which are ≥ 0 on a set in 𝔉, we say that f is dominated by g, or that g dominates f (along 𝔉), and write f preceq g or g succeq… 08DO
Examples 1) The relation f preccurlyeq 1 means that f is bounded on a set of 𝔉. 0E5I
Proposition 1 If f, g, h are three functions in H(𝔉, R) then the relations f preccurlyeq g and g preccurlyeq h imply f preccurlyeq h. 08DP
Proposition 2 Let f₁, f₂ be two functions in H(𝔉, V) and g a function in H(𝔉, R). The relations f₁ preccurlyeq g and f₂ preccurlyeq g imply f₁ + f₂ preccurlyeq g. 08DQ
Proposition 3 Let V₁, V₂, V be three normed spaces over the same valued field, and (x, y) ↦ [x.y] a bilinear map from V₁ × V₂ into V. If f₁ and f₂ are functions in H(𝔉, V₁) and H(𝔉, V₂) respecti… 08DR
Corollary If V is a normed algebra, if f₁, f₂ are two functions in H(𝔉, V), and g₁, g₂ are two functions in H(𝔉, R), then the relations f₁ preccurlyeq g₁, f₂ preccurlyeq g₂ imply f₁ f₂ precc… 08DS
Definition 2 Given two functions f, g of H(𝔉, V) we say that f and g are similar (along 𝔉), and write f ≃ g, if f preccurlyeq g and g preccurlyeq f. 08DT
Example 1 For a real function f ∈ H(𝔉, R) the relation f ≃ 1 means that there are two numbers a > 0,\ b > 0 such that a ≤ |f(x)| ≤ b on a set in 𝔉, or that the function log |f| is bounded on… 08DU
Example 2 Let V be a normed space over a non-discrete valued field K, and let f(x) = a₀ xⁿ + a₁ xⁿ⁻¹ + ⋯ + aₙ be a polynomial in the variable x ∈ K, with coefficients in V, such that a₀ ≠ 0… 08DV
Example 3 We have seen that sin² x preccurlyeq sin x as x tends to +∞, but we do not have sin² x ≃ sin x, even though these functions vanish at the same points. 08DW
Example 4 One has x² + xy + y² ≃ x² + y² when (x,\ y) tends to (0,\ 0) in R², but not xy ≃ x² + y². 08DX
Remark 1 Two functions in H(𝔉, R) need not be weakly comparable, as is shown by the example of the functions 1 and x sin x as x tends to +∞. 08DY
Remark 2 Denote by R₀ the relation f ≃ g on H(𝔉, V), and by H₀(𝔉, V) the quotient set H(𝔉, V)/R₀; note that the relation R implies R₀. Passing to the quotient the relation f preccurlyeq g g… 08DZ
Definition 3 Given two real functions f,\ g belonging to H(𝔉, R), which are ≥ 0 on a set in 𝔉, one says that f is negligible relative to g, or that g is preponderant over f (along 𝔉), and one w… 08E0
Examples 1) The relation f ll 1 means that f tends to 0 along 𝔉. 0E5N
Proposition 4 If f, g, h are three functions in H(𝔉, R) then the relations f preccurlyeq g and g ll h (resp. f ll g and g preccurlyeq h) imply f ll h. ↩ 1 08E1
Proposition 5 Let f₁, f₂ be two functions in H(𝔉, V) and g a function in H(𝔉, R). Then the relations f₁ ll g and f₂ ll g imply f₁ + f₂ ll g. 08E2
Proposition 6 With the notation of prop. 3, the relations f₁ ll g₁ and f₂ preccurlyeq g₂ (resp. f₁ preccurlyeq g₁ and f₂ ll g₂) imply [f₁.f₂] ll g₁g₂. 08E3
Proposition 7 If f and g are two functions in H(𝔉, V) then the relation f - g ll f is equivalent to f - g ll g. 08E4
Corollary The relation f - g ll f is an equivalence relation on H(𝔉, V). 08E5
Definition 4 Given two functions f, g of H(𝔉, V) one says that f and g are equivalent (along 𝔉), and writes f ∼ g, if f - g ll f. 08E6
Example 1 If a is a constant function, ≠ 0 on E, the relation f ∼ a means that f tends to a along 𝔉. 08E7
Example 2 Let V be a normed space over a non-discrete valued field K, and let f(x) = a₀ xⁿ + a₁ xⁿ⁻¹ + ⋯ + aₙ be a polynomial in the variable x ∈ K, with coefficients in V, such that a₀ ≠ 0… 08E8
Example 3 When the real number x tends to +∞ one has ( 1 + 1/x ) sin x ∼ sin x. 08E9
Example 4 As the complex variable z tends to 0 one has eᶻ - 1 ∼ z. More generally, if V is a normed space over a valued field K, and f is a function defined on a neighbourhood of x₀ ∈ K, wit… 08EA
Example 5 As (x, y) approaches (0, 0) in R² one has sqrtsin² x + sin² y ∼ sqrtx² + y². 08EB
Example 6 Let f(x, y) be a polynomial with real coefficients in two real variables x, y, not having a constant term. If, as x approaches 0 while remaining > 0, there is a function φ(x), cont… 08EC
Example 7 For every x > 0 let π(x) denote the number of prime numbers which are ≤ x; it has been proved that π(x) ∼ x / log x ² as x tends to +∞. 08ED
Remark Note that the relation f ∼ g by no means implies that the difference f - g tends to 0 along 𝔉; this difference can even be unbounded, as is shown by the example x² + x ∼ x² as x te… 08EE
Proposition 8 Let K be a valued field and f₁, f₂, g₁, g₂ four functions in H(𝔉, K); then the relations f₁ ∼ g₁ and f₂ ∼ g₂ imply f₁ f₂ ∼ g₁ g₂. 08EF
Remark 1 Two functions can be weakly comparable though not strongly comparable, for example the functions 1 and sin x as x tends to +∞. 08EG
Remark 2 In the definitions of the comparison relations f₁ preccurlyeq f₂ and f₁ ll f₂ the norms on the spaces V₁, V₂ where f₁ and f₂ respectively take their values, are involved only appar… 08EH
Proposition 9 Let f and g be two functions in H(𝔉, R), strictly positive on a set in 𝔉. For f and g to be comparable it is necessary and sufficient that for every number t geqslant 0, with at mo… 08EI
Proposition 10 Let f and g be two functions in H(𝔉, R) which are strictly positive on a set in 𝔉; for every α real and ≠ 0 the relation f asymp g (resp. f ∼ g) is equivalent to f^α asymp g^α (res… 08EJ
Proposition 11 Let g be a function in H(𝔉, R) such that lim_𝔉 g = +∞; the relation f ll g implies eᶠ ll eᵍ; the relation f ∼ g implies log f ∼ log g. 08EK
Definition 5 Let g be a function in H(𝔉, R), strictly positive on a set in 𝔉, and such that lim_𝔉 g = 0 or lim_𝔉 g = +∞. One says that a function f ∈ H(𝔉, R) is of order ρ (finite or infinite)… 08EL
Proposition 12 Let g be a function in H(𝔉, R) such that lim_𝔉 g = +∞; let f be a function in H(𝔉, R). 08EM
Remark 1 Observe that though f may be of finite order ρ relative to g nevertheless the ratio f/g^ρ need not tend to a limit; for example, every logarithmically bounded function is of order… 08EN
Remark 2 A function defined on a set in 𝔉 need not have a determinate order (finite or not) relative to g, for the functions having a determinate order relative to g are comparable to all t… 08EO

§ 2. Asymptotic expansions

Definition 1 One says that a subset E of H(𝔉, K) formed of functions not equivalent to 0 modulo R_∞ is a comparison scale when E is totally ordered by the relation " f ll g or f = g". 08ES
Example 1 For x real and tending to +∞ the set of functions x^α (α an arbitrary real number) is a comparison scale. The same is true for the functions (x-a)^α when 𝔉 is the set of open inter… 08ET
Example 2 For z complex tending to ∞ the set of functions zⁿ (n a rational integer) is a comparison scale; so are the functions (z-a)ⁿ when 𝔉 is the trace on the complement of the point a ∈… ↩ 1 08EU
Example 3 Let F be normed space; the family of functions |x-a|^α (α an arbitrary real number) is a comparison scale when 𝔉 is the trace on the complement of a of the filter of neighbourhoods… 08EV
Example 4 For x real tending to +∞ the family E of functions of the form exp(p(x)), where p runs through the set of polynomials without constant term (with real coefficients), is a compariso… 08EW
Example 5 For x real tending to +∞ the set E of functions of the form x^α (log x)^β (defined for x > 1), where α and β are arbitrary real numbers, is a comparison scale. Indeed, here again t… 08EX
Example 1 For x real (resp. complex) tending to +∞ (resp. ∞), every polynomial a₀ xⁿ + a₁ xⁿ⁻¹ + … + aₙ with coefficients in V, such that a₀ ≠ 0, has principal part a₀ xⁿ with respect to the… 08EY
Example 2 A function may be comparable to all the functions of a scale and yet not admit a principal part with respect to this scale. For example, for x real tending to +∞, sqrtx has no prin… 08EZ
Definition 2 One says that a function f ∈ H(𝔉, V) admits an asymptotic expansion to precision g_α (relative to the scale E) if there exists a family (a_λ)_λ ≤ α of elements of V, all but a fini… 08F0
Example We have (1 + x)^1/x = exp ( 1/x log (1 + x) ): when x tends to +∞ we have log (1 + x) = log x + log ( 1 + 1/x ), whence the asymptotic expansion of 1/x log (1 + x) with respect to… 08F1
Example We have x^1/1 = exp ( log x · exp ( 1/x log x ) ); when x tends to +∞ one has log x ll x, whence the asymptotic expansion of log x · exp ( 1/x log x ) with respect to the scale x^α… 08F2
Example For x real tending to +∞ the periodic bounded functions on R, having the same period τ, satisfy conditions (CO₁), (CO₂) and (CO₃): if lim_x → +∞ a(x) = 0 then for every ε > 0 there… 08F3

§ 3. Asymptotic expansions of functions of a real variable

Proposition 1 Let f be a regulated vector function, g a regulated function geqslant 0 on an interval [a, +∞[, such that ∫ₐ^+∞ g(t) dt > 0. The relation f preccurlyeq g as x tends to +∞ implies t… 08F4
Corollary 1 If f and g are regulated functions and geqslant 0 on the interval [a, +∞[, and such that f succcurlyeq g, and if ∫ₐ^+∞ g(t) dt = +∞, then ∫ₐ^+∞ f(t) dt = +∞. 08F5
Corollary 2 If f and g are geqslant 0 and not identically zero on [a, +∞[, and such that f succeq g, then ∫ₐˣ f(t) dt succeq ∫ₐˣ g(t) dt. 08F6
Proposition 2 ("logarithmic criterion of order 0") Let f be a regulated function ≥ 0 on an interval [a, +∞[; if f(x) preccurlyeq x^μ for some μ < -1 then the integral ∫ₐ^+∞ f(t) dt converges; if f(x) succcurlyeq x^μ for some μ ≥ -1… 08F7
Proposition 3 ("logarithmic criterion of order 1") Let f be a regulated function ≥ 0 on an interval [a, +∞[; if f(x) preccurlyeq 1/x(log x)^μ for some μ > 1 then the integral ∫ₐ^+∞ f(t) dt converges; if f(x) succcurlyeq 1/x(log x)^… 08F8
Proposition 4 ("logarithmic criterion of order n") Let f be a regulated function ≥ 0 on an interval [a, +∞[; if, for some μ > 1, f(x) preccurlyeq frac1x l₁(x) l₂(x) … lₙ₋₁(x) (lₙ(x))^μ, then the integral ∫ₐ^+∞ f(t) dt converges; if… 08F9
Proposition 5 ("logarithmic criterion of order 0") If on a neighbourhood of α one has f(x) ≤ 1/(x-α)^μ for some μ < 1 then the integral ∫_αᵃ f(t) dt converges; if f(x) ≥ 1/(x-α)^μ for some μ ≥ 1 then the integral ∫_αᵃ f(t) dt is in… 08FA
Example 1 The function tᵖ(1-t)^q is not bounded on ]0, 1[ when p < 0 or q < 0; by the logarithmic criterion of order 0 applied on a neighbourhood of the points 0 and 1 the integral ∫₀¹ tᵖ(1-… 08FB
Example 2 Consider the integral ∫₀^+∞ t^λ-1 e⁻ᵗ dt. Since e⁻ᵗ ∼ 1 on a neighbourhood of 0, one must have λ > 0 if this integral is to converge; this condition is also sufficient since on a n… 08FC
Proposition 6 Let f be a regulated vector function, and g a regulated function ≥ 0 on [a, +∞[. 08FD
Example 1 On applying prop. 6 of V, p. 230, to the relation 1/x ll x^α-1 where α > 0, one again obtains the relation log x ll x^α for every α > 0, which is equivalent to the relation y^1/α l… 08FE
Example 2 We have ( eᵗ/x )' = eᵗ/x ( 1 - 1/x ) ∼ eᵗ / x; since eᵗ / x tends to +∞ with x, one deduces from prop. 6 of V, p. 230, that ∫₁ˣ eᵗ/t dt ∼ eˣ / x. 08FF
Remark When g is not assumed to remain ≥ 0 on an interval [a, +∞[ (or to remain ≤ 0 on such an interval), and ∫ₐ^+∞ g(t) dt is not convergent, the relation f ∼ g does not necessarily impl… 08FG
Proposition 7 If two real functions f,\ g, are comparable of order 1, then they are comparable; further, the relation f ll g (resp. f ∼ cg, c constant) implies f' ll g' (resp. f' ∼ cg') unless g… 08FH
Corollary If two real functions f, g, are comparable of order k geqslant 1, then they are comparable of order p for 0 leqslant p leqslant k; further, the relation f ll g (resp. f ∼ cg) impli… 08FI
Remark 1 The restriction on g in prop. 7 is essential. For example, one has 1/x ll 1 + 1/x even though the derivatives of the two sides are equivalent; similarly 1 + 1/x ∼ 1 + 1/x², but 1/x… 08FJ
Remark 2 Though f and g are comparable of order k a function f₁ equivalent to f need not be comparable of order k to g; however it will be so if one assumes that f₁ is comparable of order k… 08FK
Remark 3 If f and g are comparable of order k this need not be so for hf and hg even for a monotone function h as simple as h(x) = x (V, p. 260, exerc. 3); likewise, 1/f and 1/g are not nec… 08FL
Proposition 8 Put F(x) = ∫ₓ^+∞ f(t) dt if ∫ₐ^+∞ f(t) dt converges, and F(x) = ∫ₐˣ f(t) dt if ∫ₐ^+∞ f(t) dt is infinite. Suppose that log|f| and log x are comparable of order 1. ↩ 3 08FM
Example 1 The function 1/log x is of order 0 relative to x and satisfies the conditions of prop. 8 of V, p. 233; thus ∫ₐˣ dt/(log t) ∼ x/(log x). 08FN
Example 2 The function e^t² is of order +∞ relative to x and satisfies the conditions of prop. 8, so ∫ₐˣ e^t² dt ∼ 1/2x e^x². 08FO
Remark Prop. 8 does not apply directly to a function f of order -1 relative to x. But then one can write f(x) = f₁(x)/x, with f₁ of order 0 relative to x. Suppose, for example, that ∫ₐ^+∞… 08FP
Example 1 Let f(x) = 1/log x (x > 1); we have seen that ∫ₐ^λ dt / log t ∼ x / log x, and that the difference ∫ₐ^λ dt/(log t) - λ/(log λ) is a primitive of 1/(log x)²; one can apply prop. 8 a… 08FQ
Example 2 Let f(x) = eˣ/(x² + 1); one can write f(x) = eˣ/x² - eˣ/x⁴ + o₁ ( eˣ/x⁴ ). Prop. 8 gives the expansions 08FR

§ 4. Application to series with positive terms

Proposition 1 Let (uₙ) and (vₙ) be two series with positive terms. If uₙ preccurlyeq vₙ and if the series (vₙ) converges, then (uₙ) converges; if uₙ succ vₙ and if ∑ₙ₌₁^∞ vₙ = +∞, then ∑ₙ₌₁^∞ uₙ… 08FZ
Proposition 2 Let (uₙ) and (vₙ) be two series with positive terms: If the series (vₙ) converges then the relation uₙ ll vₙ (resp. uₙ ∼ vₙ) implies ∑ₚ₌ₙ^∞ uₚ ll ∑ₚ₌ₙ^∞ vₚ (resp. ∑ₚ₌ₙ^∞ uₚ ∼ ∑ₚ₌ₙ^… 08G0
Proposition 3 (Cauchy-Maclaurin criterion) If f is a function ≥ 0 and decreasing on [x₀, +∞[, then the series with general term vₙ = f(n) converges if and only if the integral ∫_x₀^+∞ f(t) dt converges. 08G1
Proposition 4 ("logarithmic criterion of order 0") Let (uₙ) be a series with positive terms; if uₙ preccurlyeq n^μ for some μ < -1 then the series (uₙ) converges; if uₙ succcurlyeq n^μ for some μ ≥ -1 then the series (uₙ) has an in… 08G2
Proposition 5 ("logarithmic criterion of order $p$") Let (uₙ) be a series with positive terms. If uₙ preccurlyeq frac1nl₁(n)l₂(n)… lₚ₋₁(n)(lₚ(n))^μ for some μ > 1 then the series (uₙ) converges; if uₙ succcurlyeq frac1nl₁(n)l₂(n)… lₚ… 08G3
Proposition 6 Let g be a real function, 0 and monotone, defined on an interval [x0, +\infty[ (where x0 \leq 1 ), and such that log g and x are comparable to order 1. 08G4
Example Let g(x) = x^λ = exp(λ log x), of order +∞ relative to e^λ. Taking p = 2 one has 08G5
Example Let g(n) = 1/n; then sₙ = ∑ₘ₌₁ⁿ 1/m ∼ ∫₁ⁿ dt/t = log n then 1/n - (log n - log(n-1)) ∼ -1/2n² whence sₙ = log n + γ + 1/2n + o(1/n). The constant γ which appears in this formula pl… 08G6
Example Stirling’s formula. Let us seek an asymptotic expansion for n!; this leads us to expand sₙ = ∑ₚ₌₁ⁿ log p, and then exp(sₙ). The method of n 2 gives successively 08G7
Proposition 7 ("Raabe's test") Let (uₙ) be a series with terms > 0 from some point on. If, from a certain stage, uₙ₊₁/uₙ leqslant 1 - α/n for some α > 1, then the series (uₙ) converges; if, from a certain point… 08G8
Proposition 8 Let (uₙ) be a series with terms > 0 from a certain point on. If, from a certain point, one has 08G9
Example Consider the hypergeometric series, with general term 08GA
Remark 1 As a particular instance of Raabe's test one sees that if limsup_n → ∞ uₙ₊₁/uₙ < 1 the series (uₙ) converges; if, on the other hand, liminf_n → ∞ uₙ₊₁/uₙ > 1, the series (uₙ) has a… 08GB
Remark 2 The criteria of the second kind can be applied only to series whose general term behaves in a very regular way as n tends to +∞; in other words, their scope is much more restricted… 08GC

App. 0. Hardy Fields. (H) Functions

Definition 1 Given a subset 𝔎 of H(𝔉, R) one says that 𝔎/R_∞ (the canonical image of 𝔎 in H(𝔉, R)/R_∞) is a Hardy field, if 𝔎 satisfies the following conditions: 08GO
Example 1 Every Hardy field contains the field of rational constants (the smallest field of characteristic 0, cf. Alg., V, §1), which one can identify with the field Q; moreover, since two c… 08GP
Example 2 A very important example of a Hardy field is the set of rational functions with real coefficients, which we shall denote by R(x), by an abuse of language; if f(x) = p(x)/q(x) is a… 08GQ
Lemma 1 Let a(x), b(x) be two continuous real functions that do not change sign on an interval [x₀, +∞[. If, on this interval, the function y(x) is continuous and differentiable, and satis… 08GR
Remark This very elementary property does not extend to linear differential equations of order > 1; for example, the function y = sin x satisfies y'' + y = 0, but changes sign on every ne… 08GS
Lemma 2 Let a(x) and b(x) be two functions belonging to a given Hardy field 𝔎 and y(x) a function satisfying the identity (1) on an interval [x₀, +∞[ where a and b are defined and continuo… 08GT
Theorem 1 Let a(x) and b(x) be two functions belonging to a given Hardy field 𝔎 and y(x) a function satisfying (1) on an interval [ x₀, +∞[. When r(u) = p(u)/q(u) runs through the set of rat… 08GU
Corollary 1 If y is a function in 𝔎, not identically zero on a neighbourhood of +∞, then 𝔎(log|y|) is a Hardy field. 08GV
Proposition 1 Two functions in the same Hardy field are comparable to any order (V, p. 232). 08GW
Proposition 2 Let f be a function belonging to a Hardy field K. 08GX
Proposition 3 If K₀ is a Hardy field there exists a Hardy field K containing K₀ and such that, for every function z ∈ K, not identically zero on a neighbourhood of +∞, both eᶻ and log |z| belong… 08GY
Definition 2 The field of (H) functions is called the (H) extension of the Hardy field R(x) of rational functions with real coefficients. Any function belonging to this extension is called an (… 08GZ
Remark Despite the practical usefulness of the scale E which we have just defined, it is easy to give examples of (H) functions which have no principal part with respect to E. Indeed, if… 08H0
Proposition 4 Let p and q be two (H) functions which are strictly positive on an interval [x₀, +∞[. ↩ 1 08H1
Remark The conditions imposed on q(x) may not be improved, as the following examples show: 08H2
Proposition 5 Let g be an (H) function not equivalent to a constant ≠ 0, and such that g(x) ll x, and let u(x) be the inverse function of x - g(x), defined on a neighbourhood of +∞. Let (uₙ) be… 08H3
Remark 1 If g is of order < 1 relative to x, the function u(x) - uₙ(x) tends to 0 with x once n is sufficiently large. Indeed, in the opposite case one would have gg''ⁿ gg 1 for every n, so… 08H4
Remark 2 When g(x) is an (H) function equivalent to a constant k ≠ 0 one has g(x) = k + g₁(x), with g₁ ll 1; the function u₁(x) = u(x) - k is the inverse of the function x - g₁(x + k), and… 08H5
Example 1 Let us seek an asymptotic expansion for the inverse function v(x) of x⁵ + x (for x tending to +∞); putting x⁵ = t one is led to seek an expansion of the inverse function u(t) of t… 08H6
Example 2 Let us seek an asymptotic expansion for the inverse function of the function x / log x; from the identity x = y / log y, where y = v(x), one deduces log x = log y - log log y; on p… 08H7
Remark Note that two equivalent (H) functions can have non-equivalent inverses, as the example of the two functions log x and 1 + log x shows. 08H8

VI. Generalized Taylor expansions euler-maclaurin summation formula

§ 1. Generalized Taylor expansions

Definition 1 One says that an operator U on K[X] is a composition operator if it commutes with the translation operator, that is, if U_X T_Y = T_Y U. 08HO
Example 1 For every λ ∈ K the operator which to every polynomial f(X) associates the polynomial f(X + λ) is a composition operator. ↩ 1 08HP
Example 2 The derivation D on K[X] is a composition operator (cf. prop. 1). 08HQ
Remark Since K is an infinite field the operator U on K[X] is a composition operator if and only if for every polynomial f ∈ K[X] and every element α ∈ K one has, putting g = U(f), that g… 08HR
Proposition 1 For an operator U on K[X] to be a composition operator it is necessary and sufficient that it commute with the derivation D on K[X]. 08HS
Theorem 1 The map which to every formal series u(S) = ∑ₖ₌₀^∞ αₖ Sᵏ in one indeterminate over K associates the composition operator u(D) = ∑ₖ₌₀^∞ αₖ Dᵏ on K[X] is an isomorphism of the algebr… 08HT
Corollary The algebra Γ of composition operators in K[X] is commutative. 08HU
Example If U is the operator which to each polynomial f(X) associates f(X + λ) (where λ ∈ K), one has U₀(Xᵏ) = λᵏ, and so U = ∑ₖ₌₀^∞ 1/k! (λ D)ᵏ. By analogy with the series expansion of e^… 08HV
Scholium The isomorphism of the algebra K[[S]] of formal series and the algebra Γ of composition operators on K[X] sometimes allows one to prove propositions on formal series most simply, b… 08HW
Proposition 2 Every composition operator of order 0 is invertible in the algebra Γ of composition operators on K[X]. 08HX
Proposition 3 Let U be a composition operator of order p; then U(f) = 0 for every polynomial f of degree < p; for every polynomial f ≠ 0 of degree n ≥ p, U(f) is a polynomial ≠ 0 of degree n - p… 08HY
Definition 2 Let U = Dᵖ V be a composition operator of order p on K[X]. The polynomial uₙ(X) = V⁻¹(Xⁿ) is called the Appell polynomial of index n attached to the operator U. 08HZ
Proposition 4 The Appell polynomials attached to U satisfy the relations (d uₙ)/(d X) = n uₙ₋₁ (7) uₙ(X + Y) = ∑ₖ₌₀ⁿ binomnk uₙ₋ₖ(X) Yᵏ (8) U(uₙ(X)) = n!/((n-p)!) Xⁿ⁻ᵖ. (9) 08I0
Proposition 5 For every polynomial f ∈ K[X] and every composition operator of order p one has 08I1
Proposition 6 Let U = Dᵖ V = u(D) be a composition operator of order p on K[X], u(S) being a formal power series of order p in K[[S]]. Then 08I2
Definition 3 One says that an operator U on E is a composition operator if, for every a geqslant 0, it is permutable with the translation operator by a. 08I3
Theorem 2 Let f be a function admitting a continuous (n + 1)ᵗʰ derivative on I, and belonging, with all its derivatives f⁽ᵐ⁾ for 1 ≤ m ≤ n, to E. If U is a composition operator of order p ≤… 08I4
Remark One can use formula (33) (resp. (34)) to calculate the βₙ (resp. the uₙ(x)) by using the following lemma on entire series: 08I5
Example 1 If U is the identity map one has u(λ) = 1 and the operator U is clearly singular; since uₙ(x) = xⁿ the formula (27) of VI, p. 278, can be written, putting ξ = ξ - η 08I6
Example 2 Let us take for U the composition operator which, to every function f defined on R₊, associates the function x ↦ f(x + 1) - f(x); then 08I7
Example 3 Let E be the vector space of functions f defined and continuous on R, and such that the integral ∫_-∞^+∞ f(x + ξ) e^-ξ²/2 dξ converges for all x ≥ 0. The operator U defined by 08I8

§ 2. Eulerian expansions of the trigonometric functions and Bernoulli numbers

Proposition 1 For every complex number z and every integer n one has 08IC
Corollary 1 For every integer n one has sin π/n sin 2π/n … sin ((n-1)π)/n = fracn2ⁿ⁻¹. (3) 08ID
Corollary 2 For every odd integer n = 2m + 1, and every complex number z such that nz is not an integral multiple of π, one has cot nz = (-1)ᵐ cot z cot ( z + π/n ) … cot ( z + ((n-1)π)/n ). (… 08IE
Theorem 1 For every complex number z not an integral multiple of π one has ↩ 1 08IF
Theorem 2 For every complex number z one has 08IG

VII. The gamma function

§ 1. The gamma function in the real domain

Proposition 1 There exists one and only one convex function g defined on ]0, +∞[ that satisfies the equation (3) and takes the value 0 for x = 1. 08IX
Definition 1 We denote by Γ(x) the function > 0 which is defined on the interval ]0, +∞[, that satisfies the equation Γ(x+1) = x Γ(x), (7) and is such that Γ(1) = 1 and that log Γ(x) is convex… 08IY
Proposition 2 For every x > 0 one has Γ(x) = lim_n → ∞ (nˣ n!)/(x(x+1)…(x+n)) (Gauss' formula), and Γ(x) = e^-γ x 1/x ∏ₙ₌₁^∞ frace^x/n1 + x/n (9) where γ denotes Euler’s constant, and the infini… 08IZ
Lemma 1 Let f and g be two functions > 0 and twice differentiable on an open interval I. If f and g are logarithmically convex functions on I, then f + g is logarithmically convex on I. 08J0
Lemma 2 Let f be a finite real function, > 0, defined and continuous on the product I × J of two open intervals of R, and such that, for every t ∈ J the function x ↦ f(x, t) is logarithmic… 08J1
Lemma 3 Let φ be a continuous function and > 0 on an open interval J contained in [0, +∞[. If I is an open interval such that the integral g(x) = ∫_J tˣ⁻¹ φ(t) dt converges for all x ∈ I… 08J2
Proposition 3 For all x > 0 08J3
Proposition 4 For x > 0 and y > 0 the integral 08J4

§ 2. The gamma function in the complex domain

Proposition 1 For every complex z one has 08JB
Corollary For every real t one has 08JC
Proposition 2 For every complex number z not an integer leqslant 0 and for every integer p > 0 one has Γ ( z/p ) Γ ( (z+1)/p ) … Γ ( (z+p-1)/p ) = (2π)^(p-1)/2\ p^1/2-z\ Γ(z) (Legendre-Gauss mul… 08JD
Proposition 3 For every real x > 0 one has ∫₁ˣ⁺¹ log Γ(t)\ dt = x(log x - 1) + 1/2 log 2π (Raabe’s integral). 08JE
Proposition 4 Along the filter 𝔉 one has (for every integer p geqslant 1) the asymptotic expansion 08JF
Corollary Along the filter 𝔉 one has 08JG
Proposition 5 Along the filter 𝔉 one has (for every integer p geqslant 1), the asymptotic expansion 08JH