_[oo]
\ a_n (x - a)^n.
/_
n=0
As regards power series we have the following theorems:
1. If the power series converges at any point except a there is a
number k which has the property that the series converges absolutely
in the interval between a - k and a + k, with the possible exception
of one or both end-points.
2. The power series represents a continuous function in its domain of
convergence (the end-points may have to be excluded).
3. This function is analytic in the domain, and the power series
representing it is the Taylor's series for the function.
The theory of power series has been developed chiefly from the point
of view of the theory of functions of complex variables.
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.)