18. Asymptotic expansions.

The distinction between the "infinitely small" (or "infinitely great") of various orders, appears implicitly from the first writings on differential Calculus, and for example in those of Fermat; it becomes fully conscious with Newton and Leibniz, with the theory of "differences of higher order"; and no time is lost in observing that, in the simplest cases, the limit (or "true value") of an expression of the form $f(x)/g(x)$, at a point where $f$ and $g$ both tend to 0, is given by the Taylor expansion of these functions in a neighbourhood of the point under consideration ("l'Hôpital's rule", due probably to Johann Bernoulli).

Apart from this elementary case, the main problem of "asymptotic expansion" which presents itself to mathematicians already at the end of the XVIIth century is the calculation, exact or approximate, of sums of the form $\sum_{k=1}^n f(k)$, when $n$ is very large; such a calculation is in effect necessary as much for interpolation and the numerical evaluation of the sum of a series, as in the Calculus of probabilities, where the "functions of large numbers" such as $n!$ or $\binom{a}{n}$ play a preponderant role. Already Newton, in order to obtain approximate values of $\sum_{k=1}^n \frac{1}{a+k}$ when $n$ is large, indicates a method that reduces (in this particular case) to calculating the first terms of the Euler-Maclaurin formula ([262], v. II, pp. 309-310). Towards the end of the century, Jakob Bernoulli, in the course of his research on the Calculus of probabilities, proposes determining the sums $S_k(n) = \sum_{p=1}^n p^k$, polynomials$^1$ in $n$ for which he discovers the general formation law (without giving a proof of it), introducing thus for the first time, in the expression for the coefficients of these polynomials, the numbers that bear his name, and the recurrence relation that enables their calculation ([19 b], p. 97). In 1730, Stirling obtains an asymptotic expansion for $\sum_{k=1}^n \log(x + ka)$, $n$ growing indefinitely, by a procedure for the calculation of the coefficients by recurrence.

From 1730 to 1745 is dated the decisive work of Euler on series and the relevant questions. Setting $S(n) = \sum_{k=1}^n f(k)$, he applies Taylor's formula to the function $S(n)$, which gives him

$^1$ They are the primitives of the "Bernoulli polynomials" $B_k(x)$.

an equation that he "inverts" by the method of indeterminate coefficients, while looking for a solution of the form

he thus obtains step by step

without at first being able to determine the formation law for the coefficients ([108 a], (1), v. XIV, pp. 42-72 and 108-123). But around 1735, by analogy with the decomposition of a polynomial into factors of degree one, he does not hesitate to write the formula

and in putting the coefficients of the expansions of the two members of the whole series equal, he obtains in particular (for $\alpha = \frac{\pi}{2}$) the famous expressions for the series $\sum_{n=1}^{\infty} \frac{1}{n^{2k}}$ by means of the powers$^2$ of $\pi$ (loc. cit., pp. 73-86). A few years later, he realises at last that the coefficients of these powers of $\pi$ are given by the same equations as those of his summation formula, and recognises their link with the numbers introduced by Bernoulli, and with the coefficients of the expansion in series of $z/(e^z - 1)$ (loc. cit., pp. 407-462).

Independently of Euler, Maclaurin had arrived at about the same time at the same summing formula, by a somewhat less hazardous path, close to that followed today: he iterates in effect the "Taylorian" formula which expresses $f(x)$ by means of the differences $f^{(2k+1)}(x+1) - f^{(2k+1)}(x)$, a formula that he obtains by "inverting" the Taylor expansions of these differences by the method of indeterminate coefficients ([214], v. II, pp. 672-675); he does not notice besides the formation law for the coefficients, discovered by Euler.

But Maclaurin, like Euler and all the mathematicians of his time, presents all his formulae as expansions in series, whose convergence is not even studied. It is not that the notion of convergent series was totally neglected at this time: it is known from Jakob Bernoulli that the harmonic series is divergent, and Euler had himself recorded this result while evaluating the sum of the first $n$ terms of this series by means of his summing formula ([108 a], (1), v. XIV,

$^2$ In 1743, Euler, in order to reply to the various criticisms by his contemporaries gives a slightly more plausible derivation of the "Eulerian expansions" of the trigonometric functions; for example, the expansion as an infinite product of $\sin x$ is taken from the expression $\sin x = \frac{1}{2i}(e^{ix} - e^{-ix})$, and from the fact that $e^{ix}$ is the limit of the polynomial $(1 + \frac{ix}{n})^n$ (loc. cit., pp. 138-155).

pp. 87-100 and 108-123); it is also Euler who remarks that the ratio of two consecutive Bernoulli numbers grows indefinitely, and that it follows that an entire series having these numbers as coefficients can not converge (loc. cit., p. 357).$^3$ But the tendency to a formal calculation is the stronger, and the extraordinary intuition of Euler himself does not prevent him from lapsing sometimes into the absurd, when he writes for example $0 = \sum_{n=-\infty}^{+\infty} x^n$ (loc. cit., p. 362).$^4$

We have said elsewhere (see p. 153) how the mathematicians of the beginning of the XIXth century, tired of this formalism without restraint and without basis, brought Analysis back into the paths of rigour. Once the notion of a convergent series is made precise, the necessity appeared for simple criteria providing a proof of the convergence of integrals and series by comparison with known integrals or series; Cauchy gives a certain number of criteria in his Analyse algébrique ([56 a], (2), v. III), whereas Abel, in a posthumous memoir ([1], v. II, pp. 197-205), obtains logarithmic criteria of convergence. Cauchy, on the other hand ([56 a], (1), v. VIII, pp. 18-25), elucidates the paradox of series such as the Stirling series, obtained by the application of the Euler-Maclaurin formula (and often called "semi-convergent series"): he shows that if (because of the remark of Euler on the Bernoulli numbers) the general term $u_k(n)$ of such a series, for a fixed value of $n$, grows indefinitely with $k$, it remains none the less that, for a fixed value of $k$, the partial sum $s_k(n) = \sum_{h=1}^k u_h(n)$ gives an asymptotic expansion (for $n$ tending to $+\infty$) of the function "represented" by the series, all the more accurate as $k$ is larger.

In the majority of the calculations of classical Analysis, it is possible to obtain a general formation law for asymptotic expansions of a function, having an arbitrarily large number of terms; this fact contributed to creating a lasting confusion (at least in the language) between series and asymptotic expansions; so much so that H. Poincaré, when he takes the trouble, in 1886 ([251 a], v. I, pp. 290-296), of codifying the elementary rules of asymptotic expansions (following the integral powers of $1/x$ in the neighbourhood of $+\infty$), uses still the vocabulary of the theory of series. It is only with the appearance of asymptotic expansions coming from the analytic theory of numbers that there was a clear distinction operating between the notion of asymptotic expansion and that of series, by reason of the fact that, in the majority of problems that this theory deals with, only a very small number of terms (most often only one) in the sought for expansion can be obtained explicitly.

$^3$ As the series that Euler considers at this point is introduced in connection with numerical calculation, he only takes the sum of terms that are decreasing, and beginning with the index where the terms start to grow, he replaces them with a remainder of which he does not indicate the origin (the remainder of the Euler-Maclaurin formula in its general form does not appear before Cauchy).

$^4$ It is ironic that this formula should follow, at a distance of a page, a passage where Euler warns against the unconsidered use of divergent series!

These problems also made mathematicians familiar with the use of scales of comparison other than those of powers of the variable (real or integral). This extension goes back to the work of P. du Bois-Reymond [94 a and b] who, first, tackled systematically the problems of comparing functions in the neighbourhood of a point, and, in very original works, recognised the "non-Archimedean" character of the scales of comparison, at the same time as he studied in a general way the integration and differentiation of the comparison relations, and drew from this a large number of interesting consequences [94 b]. His proofs however lacked clarity and rigour, and it is to G. H. Hardy [147] that the correct presentation of the results of du Bois-Reymond is due: his main contribution consisted in recognising and proving the existence of a set of "elementary functions", the functions $(H)$, where the usual operations of Analysis (notably differentiation) are applicable to comparison relations.