The multiplier of e−tx under the sign of integration can be expanded in the power series
| B1 | − | B2 | t2 + | B3 | t4 − ..., |
| 2! | 4! | 6! |
where B1, B2, ... are “Bernoulli’s numbers” given by the formula
Bm = 2.2m! (2π)−2m Σ ∞r = 1 (r−2m).
When the series is integrated term by term, the right-hand member of the equation for ω(x) takes the form
| B1 | 1 | − | B2 | 1 | + | B3 | 1 | − ..., | |||
| 1·2 | x | 3·4 | x3 | 5·6 | x5 |
This series is divergent; but, if it is stopped at any term, the difference between the sum of the series so terminated and the value of ω(x) is less than the last of the retained terms. Stirling’s formula is obtained by retaining the first term only. Other well-known examples of asymptotic expansions are afforded by the descending series for Bessel’s functions. Methods of obtaining such expansions for the solutions of linear differential equations of the second order were investigated by G.G. Stokes (Math. and Phys. Papers, vol. ii. p. 329), and a general theory of asymptotic expansions has been developed by H. Poincaré. A still more general theory of divergent series, and of the conditions in which they can be used, as above, for the purposes of approximate calculation has been worked out by É. Borel. The great merit of asymptotic expansions is that they admit of addition, subtraction, multiplication and division, term by term, in the same way as absolutely convergent series, and they admit also of integration term by term; that is to say, the results of such operations are asymptotic expansions for the sum, difference, product, quotient, or integral, as the case may be.
26. Interchange of the Order of Limiting Operations.—When we require to perform any limiting operation upon a function which is itself represented by the result of a limiting process, the question of the possibility of interchanging the order of the two processes always arises. In the more elementary problems of analysis it generally happens that such an interchange is possible; but in general it is not possible. In other words, the performance of the two processes in different orders may lead to two different results; or the performance of them in one of the two orders may lead to no result. The fact that the interchange is possible under suitable restrictions for a particular class of operations is a theorem to be proved.
Among examples of such interchanges we have the differentiation and integration of an infinite series term by term (§ 22), and the differentiation and integration of a definite integral with respect to a parameter by performing the like processes upon the subject of integration (§ 19). As a last example we may take the limit of the sum of an infinite series of functions at a point in the domain of convergence. Suppose that the series Σ ∞0 ƒr(x) represents a function (ƒx) in an interval containing a point a, and that each of the functions ƒr(x) has a limit at a. If we first put x=a, and then sum the series, we have the value ƒ(a); if we first sum the series for any x, and afterwards take the limit of the sum at x = a, we have the limit of ƒ(x) at a; if we first replace each function ƒr(x) by its limit at a, and then sum the series, we may arrive at a value different from either of the foregoing. If the function ƒ(x) is continuous at a, the first and second results are equal; if the functions ƒr(x) are all continuous at a, the first and third results are equal; if the series is uniformly convergent, the second and third results are equal. This last case is an example of the interchange of the order of two limiting operations, and a sufficient, though not always a necessary, condition, for the validity of such an interchange will usually be found in some suitable extension of the notion of uniform convergence.
Authorities.—Among the more important treatises and memoirs connected with the subject are: R. Baire, Fonctions discontinues (Paris, 1905); O. Biermann, Analytische Functionen (Leipzig, 1887); É. Borel, Théorie des fonctions (Paris, 1898) (containing an introductory account of the Theory of Aggregates), and Séries divergentes (Paris, 1901), also Fonctions de variables réelles (Paris, 1905); T.J. I’A. Bromwich, Introduction to the Theory of Infinite Series (London, 1908); H.S. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals (London, 1906); U. Dini, Functionen e. reellen Grösse (Leipzig, 1892), and Serie di Fourier (Pisa, 1880); A. Genocchi u. G. Peano, Diff.- u. Int.-Rechnung (Leipzig, 1899); J. Harkness and F. Morley, Introduction to the Theory of Analytic Functions (London, 1898); A. Harnack, Diff. and Int. Calculus (London, 1891); E.W. Hobson, The Theory of Functions of a real Variable and the Theory of Fourier’s Series (Cambridge, 1907); C. Jordan, Cours d’analyse (Paris, 1893-1896); L. Kronecker, Theorie d. einfachen u. vielfachen Integrale (Leipzig, 1894); H. Lebesgue, Leçons sur l’intégration (Paris, 1904); M. Pasch, Diff.- u. Int.-Rechnung (Leipzig, 1882); E. Picard, Traité d’analyse (Paris, 1891); O. Stolz, Allgemeine Arithmetik (Leipzig, 1885), and Diff.- u. Int.-Rechnung (Leipzig, 1893-1899); J. Tannery, Théorie des fonctions (Paris, 1886); W.H. and G.C. Young, The Theory of Sets of Points (Cambridge, 1906); Brodén, “Stetige Functionen e. reellen Veränderlichen,” Crelle, Bd. cxviii.; G. Cantor, A series of memoirs on the “Theory of Aggregates” and on “Trigonometric series” in Acta Math. tt. ii., vii., and Math. Ann. Bde. iv.-xxiii.; Darboux, “Fonctions discontinues,” Ann. Sci. École normale sup. (2), t. iv.; Dedekind, Was sind u. was sollen d. Zahlen? (Brunswick, 1887), and Stetigkeit u. irrationale Zahlen (Brunswick, 1872); Dirichlet, “Convergence des séries trigonométriques,” Crelle, Bd. iv.; P. Du Bois Reymond, Allgemeine Functionentheorie (Tübingen, 1882), and many memoirs in Crelle and in Math. Ann.; Heine, “Functionenlehre,” Crelle, Bd. lxxiv.; J. Pierpont, The Theory of Functions of a real Variable (Boston, 1905); F. Klein, “Allgemeine Functionsbegriff,” Math. Ann. Bd. xxii.; W.F. Osgood, “On Uniform Convergence,” Amer. J. of Math. vol. xix.; Pincherle, “Funzioni analitiche secondo Weierstrass,” Giorn. di mat. t. xviii.; Pringsheim, “Bedingungen d. Taylorschen Lehrsatzes,” Math. Ann. Bd. xliv.; Riemann, “Trigonometrische Reihe,” Ges. Werke (Leipzig, 1876); Schoenflies, “Entwickelung d. Lehre v. d. Punktmannigfaltigkeiten,” Jahresber. d. deutschen Math.-Vereinigung, Bd. viii.; Study, Memoir on “Functions with Restricted Oscillation,” Math. Ann. Bd. xlvii.; Weierstrass, Memoir on “Continuous Functions that are not Differentiable,” Ges. math. Werke, Bd. ii. p. 71 (Berlin, 1895), and on the “Representation of Arbitrary Functions,” ibid. Bd. iii. p. 1; W.H. Young, “On Uniform and Non-uniform Convergence,” Proc. London Math. Soc. (Ser. 2) t. 6. Further information and very full references will be found in the articles by Pringsheim, Schoenflies and Voss in the Encyclopädie der math. Wissenschaften, Bde. i., ii. (Leipzig, 1898, 1899).