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 and that the continuous function ƒ(x) represented by it has the property that there is, in the neighbourhood of any point x0, an infinite aggregate of points x′, having x0 as a limiting point, for which {ƒ(x′) − ƒ(x0)} / (x′ − x0) tends to become infinite with one sign when x′ − x0 approaches zero through positive values, and infinite with the opposite sign when x′−x0 approaches zero through negative values. Accordingly the function is not differentiable at any point. The definite integral of such a function ƒ(x) through the interval between a fixed point and a variable point x, is a continuous differentiable function F(x), for which F′(x)＝ƒ(x); and, if ƒ(x) is one-signed throughout any interval F(x) is monotonous throughout that interval, but yet F(x) cannot be represented by a curve. In any interval, however small, the tangent would have to take the same direction for infinitely many points, and yet there is no interval in which the tangent has everywhere the same direction. Further, it can be shown that all functions which are everywhere continuous and nowhere differentiable are capable of representation by series of the form ann(x), where an is an absolutely convergent series of numbers, and n(x) is an analytic function whose absolute value never exceeds unity.

25. Calculations with Divergent Series.—When the series described in (1) and (2) of § 24 diverge, they may, nevertheless, be used for the approximate numerical calculation of the values of the function, provided the calculation is not carried beyond a certain number of terms. Expansions in series which have the property of representing a function approximately when the expansion is not carried too far are called “asymptotic expansions.” Sometimes they are called “semi-convergent series”; but this term is avoided in the best modern usage, because it is often used to describe series whose convergence depends upon the order of the terms, such as the series 1−＋−

In general, let ƒ0(x)＋ƒ1(x)＋ be a series of functions which does not converge in a certain domain. It may happen that, if any number, however small, is first specified, a number n can afterwards be found so that, at a point a of the domain, the value ƒ(a) of a certain function ƒ(x) is connected with the sum of the first n＋1 terms of the series by the relation |ƒ(a) − $$\sum_{r=0}^n$$ ƒr(a)| <. It must also happen that, if any number N, however great, is specified, a number n′(>n) can be found so that, for all values of m which exceed n′, |$$\sum_{r=0}^m$$ƒr(a)|>N. The divergent series ƒ0(x)＋ƒ1(x)＋ is then an asymptotic expansion for the function ƒ(x) in the domain.

The best known example of an asymptotic expansion is Stirling’s formula for n! when n is large, viz.

$n! = \sqrt{2\pi)}\tfrac{1}{2}n^{n+\tfrac{1}{2}}e^{n+\theta/12n},$|undefined

where $$\theta$$ is some number lying between 0 and 1. This formula is included in the asymptotic expansion for the Gamma function. We have in fact

$\log \{\Gamma(x)\}=(x-\tfrac{1}{2}) \log x-x+\tfrac{1}{2} \log 2\pi+\overline\omega(x),$

where $$\overline\omega(x)$$ is the function defined by the definite integral

$\overline\omega(x) = \int_0^\infty \{(1-e^{-t})^{-1}- t^{-1}-\tfrac{1}{2}\} t^{-1} e^{-tx}dt.$

The multiplier of e−tx under the sign of integration can be expanded in the power series

where B1, B2, are “Bernoulli’s numbers” given by the formula

Bm＝2.2m! (2)−2m $\sum_{r=1}^\infty$ (r−2m).

When the series is integrated term by term, the right-hand member of the equation for $$\overline\omega(x)$$ takes the form

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 $$\overline\omega(x)$$ 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 $$\sum_{r=0}^\infty f_r(x)$$ 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.

—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).