Part 23
22. _Uniform Convergence._--We shall suppose that the domain of convergence of an infinite series of functions is an interval with the possible exception of isolated points. Let [f](x) be the sum of the series at any point x of the domain, and [f]_n(x) the sum of the first n + 1 terms. The condition of convergence at a point a is that, after any positive number [epsilon], however small, has been specified, it must be possible to find a number n so that |[f]_m(a) - [f]_p(a)| < [epsilon] for all values of m and p which exceed n. The sum, [f](a), is the limit of the sequence of numbers [f]_n(a) at n = [oo]. The convergence is said to be "uniform" in an interval if, after specification of [epsilon], the same number n suffices at all points of the interval to make |[f](x) - [f]_m(x)| < [epsilon] for all values of m which exceed n. The numbers n corresponding to any [epsilon], however small, are all finite, but, when [epsilon] is less than some fixed finite number, they may have an infinite superior limit (S 7); when this is the case there must be at least one point, a, of the interval which has the property that, whatever number N we take, [epsilon] can be taken so small that, at some point in the neighbourhood of a, n must be taken > N to make |[f](x) - f_m(x)| < [epsilon] when m > n; then the series does not converge uniformly in the neighbourhood of a. The distinction may be otherwise expressed thus: Choose a first and [epsilon] afterwards, then the number n is finite; choose [epsilon] first and allow a to vary, then the number n becomes a function of a, which may tend to become infinite, or may remain below a fixed number; if such a fixed number exists, ho wever small [epsilon] may be, the convergence is uniform.
For example, the series sin x - 1/2 sin 2x + {1/3} sin 3x - ... is convergent for all real values of x, and, when [pi] > x > -[pi] its sum is 1/2x; but, when x is but a little less than [pi], the number of terms which must be taken in order to bring the sum at all near to the value of 1/2x is very large, and this number tends to increase indefinitely as x approaches [pi]. This series does not converge uniformly in the neighbourhood of x = [pi]. Another example is afforded by the series
_[oo] nx (n + 1)x \ ---------- - ---------------- , /_ n^2x^2 + 1 (n + 1)^2x^2 + 1 n=0
of which the remainder after n terms is nx/(n^2x^2 + 1). If we put x = 1/n, for any value of n, however great, the remainder is 1/2; and the number of terms required to be taken to make the remainder tend to zero depends upon the value of x when x is near to zero--it must, in fact, be large compared with 1/x. The series does not converge uniformly in the neighbourhood of x = 0.
As regards series whose terms represent continuous functions we have the following theorems:
(1) If the series converges uniformly in an interval it represents a function which is continuous throughout the interval.
(2) If the series represents a function which is discontinuous in an interval it cannot converge uniformly in the interval.
(3) A series which does not converge uniformly in an interval may nevertheless represent a function which is continuous throughout the interval.
(4) A power series converges uniformly in any interval contained within its domain of convergence, the end-points being excluded.
(5) If [Sigma] (r=0 to [oo]) [f]_r(x) = [f](x) converges uniformly in the interval between a and b
_ _[oo] _ / b \ / b | [f](x)dx = /_ | [f]_r(x)dx, _/ a r=0 _/a
or a series which converges uniformly may be integrated term by term.
(6) If [Signa] (r=0 to [oo]) [f]'_r(x) converges uniformly in an interval, then [Signa] (r=o to [oo]) [f]_r(x) converges in the interval, and represents a continuous differentiable function, [phi](x); in fact we have
_[oo] [phi]'(x) = \ [f]'_r(x), /_ r=0
or a series can be differentiated term by term if the series of derived functions converges uniformly.
A series whose terms represent functions which are not continuous throughout an interval may converge uniformly in the interval. If [Signa] (r=0 to [oo]) [f]_r(x) = [f](x), is such a series, and if all the functions [f]_r(x) have limits at a, then [f](x) has a limit at a, which is [Signa] (r=0 a=0 to [oo]) Lt [f]_r(x). A similar theorem holds for limits on the left or on the right.
23. _Fourier's Series._--An extensive class of functions admit of being represented by series of the form
_[oo] / n[pi]x n[pi]x \ a0 + \ ( a_n cos ------ + b_n sin ------ ), (i.) /_ \ c c / n=1
and the rule for determining the coefficients a_n, b_n of such a series, in order that it may represent a given function [f](x) in the interval between -c and c, was given by Fourier, viz. we have _ _ 1 / c 1 / c n[pi]x a0 = --- | [f](x)dx, a_n = -- | [f](x)cos ------ dx, 2c _/-c c _/-c c _ / c 1 n[pi]x b_n = | -- [f](x)sin ------ dx. _/-c c c
The interval between -c and c may be called the "periodic interval," and we may replace it by any other interval, e.g. that between 0 and 1, without any restriction of generality. When this is done the sum of the series takes the form
_ _r=n / 1 \ [f](z)cos {2r[pi](z - x)} dz, Lt | /_ n=[oo] _/0 r=-n
and this is _ / 1 sin {(2n + 1)(z - x)[pi]} Lt | [f](z) ------------------------ dz. (ii.) n=[oo] _/0 sin {(z - x)[pi]}
Fourier's theorem is that, if the periodic interval can be divided into a finite number of partial intervals within each of which the function is ordinary (S 14), the series represents the function within each of those partial intervals. In Fourier's time a function of this character was regarded as completely arbitrary.
By a discussion of the integral (ii.) based on the Second Theorem of the Mean (S 15) it can be shown that, if [f](x) has restricted oscillation in the interval (S 11), the sum of the series is equal to 1/2{[f](x + 0) + [f](x - 0)} at any point x within the interval, and that it is equal to 1/2{[f]( + 0) + [f](1 - 0} at each end of the interval. (See the article FOURIER'S SERIES.) It therefore represents the function at any point of the periodic interval at which the function is continuous (except possibly the end-points), and has a definite value at each point of discontinuity. The condition of restricted oscillation includes all the functions contemplated in the statement of the theorem and some others. Further, it can be shown that, in any partial interval throughout which [f](x) is continuous, the series converges uniformly, and that no series of the form (i), with coefficients other than those determined by Fourier's rule, can represent the function at all points, except points of discontinuity, in the same periodic interval. The result can be extended to a function [f](x) which tends to become infinite at a finite number of points a of the interval, provided (1) [f](x) tends to become determinately infinite at each of the points a, (2) the improper definite integral of [f](x) through the interval is convergent, (3) [f](x) has not an infinite number of discontinuities or of maxima or minima in the interval.
24. _Representation of Continuous Functions by Series._--If the series for [f](x) formed by Fourier's rule converges at the point a of the periodic interval, and if [f](x) is continuous at a, the sum of the series is [f](a); but it has been proved by P. du Bois Reymond that the function may be continuous at a, and yet the series formed by Fourier's rule may be divergent at a. Thus some continuous functions do not admit of representation by Fourier's series. All continuous functions, however, admit of being represented with arbitrarily close approximation in either of two forms, which may be described as "terminated Fourier's series" and "terminated power series," according to the two following theorems:
(1) If [f](x) is continuous throughout the interval between 0 and 2[pi], and if any positive number [epsilon] however small is specified, it is possible to find an integer n, so that the difference between the value of [f](x) and the sum of the first n terms of the series for [f](x), formed by Fourier's rule with periodic interval from 0 to 2[pi], shall be less than [epsilon] at all points of the interval. This result can be extended to a function which is continuous in any given interval.
(2) If [f](x) is continuous throughout an interval, and any positive number [epsilon] however small is specified, it is possible to find an integer n and a polynomial in x of the nth degree, so that the difference between the value of [f](x) and the value of the polynomial shall be less than [epsilon] at all points of the interval.
Again it can be proved that, if [f](x) is continuous throughout a given interval, polynomials in x of finite degrees can be found, so as to form an infinite series of polynomials whose sum is equal to [f](x) at all points of the interval. Methods of representation of continuous functions by infinite series of rational fractional functions have also been devised.
## Particular interest attaches to continuous functions which are not
differentiable. Weierstrass gave as an example the function represented by the series [Sigma] (n=0 to [oo]) a^n cos(b^[n] x[pi]), where a is positive and less than unity, and b is an odd integer exceeding (1 + (3/2)[pi]) / a. It can be shown that this series is uniformly convergent in every interval, and that the continuous function [f](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 {[f](x') - [f](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 [f](x) through the interval between a fixed point and a variable point x, is a continuous differentiable function F(x), for which F'(x) = [f](x); and, if [f](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 [Sigma][a]_n [phi]_n (x), where [Sigma][a]_n is an absolutely convergent series of numbers, and [phi]_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 S 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 - 1/2 + 1/3 - ...
In general, let [f]0(x) + [f]1(x) + ... be a series of functions which does not converge in a certain domain. It may happen that, if any number [epsilon], however small, is first specified, a number n can afterwards be found so that, at a point a of the domain, the value [f](a) of a certain function [f](x) is connected with the sum of the first n + 1 terms of the series by the relation |[f](a) - [Sigma] (r=0 to n) [f]_r(a) | < [epsilon]. 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', | [Sigma](r=0 to m) [f]_r(a) | > N. The divergent series [f]0(x) + [f]1(x) + ... is then an asymptotic expansion for the function f(x) in the domain.
The best known example of an asymptotic expansion is Stirling's formula for n! when n is large, viz. _____ / n! = \/(2[pi]) 1/2n^(n + 1/2) e^(-n + [theta] / 12n),
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 - 1/2) log x - x + 1/2 log 2[pi] + [~omega](x),
where [~omega](x) is the function defined by the definite integral _ / [oo] ~[omega](x) = | {[1 - e^(-t)]^(-1) - t^(-1) - 1/2} t^(-1) e^(-tx)dt. _/0
The multiplier of e^(-tx) under the sign of integration can be expanded in the power series
B1 B2 B3 ---- - ---- t^2 + ---- t^4 - ..., 2! 4! 6!
where B1, B2, ... are "Bernoulli's numbers" given by the formula
_[oo] \ B_m = 2.2m! (2[pi])^(-2m) /_ [r^(-2m)]. r=1
When the series is integrated term by term, the right-hand member of the equation for [~omega](x) takes the form
B1 1 B2 1 B3 1 ---- --- - ---- --- + ---- --- - ..., 1.2 x 3.4 x^3 5.6 x^5
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 [~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. Poincare. 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 E. 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 (S 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 (S 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 [Sigma] (r=0 to [oo]) [f]_r(x) represents a function ([f]x) in an interval containing a point a, and that each of the functions [f]_r(x) has a limit at a. If we first put x = a, and then sum the series, we have the value [f](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 [f](x) at a; if we first replace each function [f]_r(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 [f](x) is continuous at a, the first and second results are equal; if the functions [f]_r(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); E. Borel, _Theorie des fonctions_ (Paris, 1898) (containing an introductory account of the Theory of Aggregates), and _Series divergentes_ (Paris, 1901), also _Fonctions de variables reelles_ (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 Grosse_ (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, _Lecons sur l'integration_ (Paris, 1904); M. Pasch, _Diff.- u. Int.-Rechnung_ (Leipzig, 1882); E. Picard, _Traite d'analyse_ (Paris, 1891); O. Stolz, _Allgemeine Arithmetik_ (Leipzig, 1885), and _Diff.- u. Int.-Rechnung_ (Leipzig, 1893-1899); J. Tannery, _Theorie des fonctions_ (Paris, 1886); W.H. and G.C. Young, _The Theory of Sets of Points_ (Cambridge, 1906); Broden, "Stetige Functionen e. reellen Veranderlichen," _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. Ecole 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 series trigonometriques," _Crelle_, Bd. iv.; P. Du Bois Reymond, _Allgemeine Functionentheorie_ (Tubingen, 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 _Encyclopadie der math. Wissenschaften_, Bde. i., ii. (Leipzig, 1898, 1899). (A. E. H. L.)
II.--FUNCTIONS OF COMPLEX VARIABLES