Chapter I: Functions of Real Variables (2)

With a view to the establishment of the notion of integration through a domain, we must define the "extent" of the domain. Take first a domain consisting of the point a and all the points x for which |x - a| < 1/2h, where h is a chosen positive number; the extent of this domain is h^n, n being the number of variables; such a domain may be described as "square," and the number h may be called its "breadth"; it is a homogeneous part of the numerical continuum of n dimensions, and its boundary consists of all the points for which |x - a| = 1/2h. Now the points of any domain, which does not extend to an infinite distance, may be assigned to a finite number m of square domains of finite breadths, so that every point of the domain is either within one of these square domains or on its boundary, and so that no point is within two of the square domains; also we may devise a rule by which, as the number m increases indefinitely, the breadths of all the square domains are diminished indefinitely. When this process is applied to a homogeneous part, H, of the numerical continuum C_n, then, at any stage of the process, there will be some square domains of which all the points belong to H, and there will generally be others of which some, but not all, of the points belong to H. As the number m is increased indefinitely the sums of the extents of both these categories of square domains will tend to definite limits, which cannot be negative; when the second of these limits is zero the domain H is said to be "measurable," and the first of these limits is its "extent"; it is independent of the rule adopted for constructing the square domains and contracting their breadths. The notion thus introduced may be adapted by suitable modifications to continua of lower dimensions in C_n.

The integral of a function f(x) through a measurable domain H, which
is a homogeneous part of the numerical continuum of n dimensions, is
defined in just the same way as the integral through an interval, the
extent of a square domain taking the place of the difference of the
end-values of a partial interval; and the condition of integrability
takes the same form as in the simple case. In particular, the
condition is satisfied when the function is continuous throughout the
domain. The definition of an integral through a domain may be adapted
to any domain of measurable extent. The extensions to "improper"
definite integrals may be made in the same way as for a function of
one variable; in the particular case of a function which tends to
become infinite at a point in the domain of integration, the point is
enclosed in a partial domain which is omitted from the integration,
and a limit is taken when the extent of the omitted partial domain is
diminished indefinitely; a divergent integral may have different
(principal) values for different modes of contracting the extent of
the omitted partial domain. In applications to mathematical physics
great importance attaches to convergent integrals and to principal
values of divergent integrals. For example, any component of magnetic
force at a point within a magnet, and the corresponding component of
magnetic induction at the same point are expressed by different
principal values of the same divergent integral. Delicate questions
arise as to the possibility of representing the integral of a function
of n variables through a domain H_n, as a repeated integral, of
evaluating it by successive integrations with respect to the variables
one at a time and of interchanging the order of such integrations.
These questions have been discussed very completely by C. Jordan, and
we may quote the result that all the transformations in question are
valid when the function is continuous throughout the domain.

20. _Representation of Functions in General._--We have seen that the notion of a function is wider than the notion of an analytical expression, and that the same function may be "represented" by one expression in one part of the domain of the argument and by some other expression in another part of the domain (S 5). Thus there arises the general problem of the representation of functions. The function may be given by specifying the domain of the argument and the rule of calculation, or else the function may have to be determined in accordance with certain conditions; for example, it may have to satisfy in a prescribed domain an assigned differential equation. In either case the problem is to determine, when possible, a single analytical expression which shall have the same value as the function at all points in the domain of the argument. For the representation of most functions for which the problem can be solved recourse must be had to limiting processes. Thus we may utilize infinite series, or infinite products, or definite integrals; or again we may represent a function of one variable as the limit of an expression containing two variables in a domain in which one variable remains constant and another varies. An example of this process is afforded by the expression Lt_y = [oo]xy/(x^2y + 1), which represents a function of x vanishing at x = 0 and at all other values of x having the value of 1/x. The method of series falls under this more general process (cf. S 6). When the terms u1, u2, ... of a series are functions of a variable x, the sum s_n of the first n terms of the series is a function of x and n; and, when the series is convergent, its sum, which is Lt_n = [oo]s_n, can represent a function of x. In most cases the series converges for some values of x and not for others, and the values for which it converges form the "domain of convergence." The sum of the series represents a function in this domain.

The apparently more general method of representation of a function of
one variable as the limit of a function of two variables has been
shown by R. Baire to be identical in scope with the method of series,
and it has been developed by him so as to give a very complete account
of the possibility of representing functions by analytical
expressions. For example, he has shown that Riemann's totally
discontinuous function, which is equal to 1 when x is rational and to
0 when x is irrational, can be represented by an analytical
expression. An infinite process of a different kind has been adapted
to the problem of the representation of a continuous function by T.
Broden. He begins with a function having a graph in the form of a
regular polygon, and interpolates additional angular points in an
ordered sequence without limit. The representation of a function by
means of an infinite product falls clearly under Baire's method, while
the representation by means of a definite integral is analogous to
Broden's method. As an example of these two latter processes we may
cite the Gamma function [[Gamma](x)] defined for positive values of x
by the definite integral
_
/ [oo]
| e^(-t)t^(x - 1)dt,
_/0

or by the infinite product

/ x \
L t_(n = [oo]) n^x/x (1 + x)(1 + 1/2x) ... ( 1 + ----- ).
\ n - 1 /

The second of these expressions avails for the representation of the
function at all points at which x is not a negative integer.

21. _Power Series._--Taylor's theorem leads in certain cases to a representation of a function by an infinite series. We have under certain conditions (S 13)

_n-1
\ (x - a)^r
[f](x) = [f](a) + /_ --------- [f]^(r) (a) + R_n;
r=1 r!

and this becomes

_[oo]
\ (x - a)^r
[f](x) = [f](a) + /_ --------- [f]^(r) (a),
r=1 r!

provided that ([alpha]) a positive number k can be found so that at all points in the interval between a and a + k (except these points) [f](x) has continuous differential coefficients of all finite orders, and at a has progressive differential coefficients of all finite orders; ([beta]) Cauchy's form of the remainder R_n, viz. [(x - a)^n / (n - 1)!] (1 - [theta])^(n - 1)[f]^n {a + [theta](x - a)}, has the limit zero when n increases indefinitely, for all values of [theta] between 0 and 1, and for all values of x in the interval between a and a + k, except possibly a + k. When these conditions are satisfied, the series (1) represents the function at all points of the interval between a and a + k, except possibly a + k, and the function is "analytic" (S 13) in this domain. Obvious modifications admit of extension to an interval between a and a - k, or between a - k and a + k. When a series of the form (1) represents a function it is called "the Taylor's series for the function."

Taylor's series is a power series, i.e. a series of the form

_[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.)

II.--FUNCTIONS OF COMPLEX VARIABLES

In the preceding section the doctrine of functionality is discussed with respect to real quantities; in this section the theory when complex or imaginary quantities are involved receives treatment. The following abstract explains the arrangement of the subject matter: (S 1), _Complex numbers_, states what a complex number is; (S 2), _Plotting of simple expressions involving complex numbers_, illustrates the meaning in some simple cases, introducing the notion of conformal representation and proving that an algebraic equation has complex, if not real, roots; (S 3), _Limiting operations_, defines certain simple functions of a complex variable which are obtained by passing to a limit, in particular the exponential function, and the generalized logarithm, here denoted by [lambda](z); (S 4), _Functions of a complex variable in general_, after explaining briefly what is to be understood by a region of the complex plane and by a path, and expounding a logical principle of some importance, gives the accepted definition of a function of a complex variable, establishes the existence of a complex integral, and proves Cauchy's theorem relating thereto; (S 5), _Applications_, considers the differentiation and integration of series of functions of a complex variable, proves Laurent's theorem, and establishes the expansion of a function of a complex variable as a power series, leading, in (S 6), _Singular points_, to a definition of the region of existence and singular points of a function of a complex variable, and thence, in (S 7), _Monogenic Functions_, to what the writer believes to be the simplest definition of a function of a complex variable, that of Weierstrass; (S 8), _Some elementary properties of single valued functions_, first discusses the meaning of a pole, proves that a single valued function with only poles is rational, gives Mittag-Leffler's theorem, and Weierstrass's theorem for the primary factors of an integral function, stating generalized forms for these, leading to the theorem of (S 9), _The construction of a monogenic function with a given region of existence_, with which is connected (S10), _Expression of a monogenic function by rational functions in a given region_, of which the method is applied in (S 11), _Expression of_ (1 - z)^(-1) _by polynomials_, to a definite example, used here to obtain (S 12), _An expansion of an arbitrary function by means of a series of polynomials, over a star region_, also obtained in the original manner of Mittag-Leffler; (S 13), _Application of Cauchy's theorem to the determination of definite integrals_, gives two examples of this method; (S 14), _Doubly Periodic Functions_, is introduced at this stage as furnishing an excellent example of the preceding principles. The reader who wishes to approach the matter from the point of view of Integral Calculus should first consult the section (S 20) below, dealing with _Elliptic Integrals_; (S 15), _Potential Functions, Conformal representation in general_, gives a sketch of the connexion of the theory of potential functions with the theory of conformal representation, enunciating the Schwarz-Christoffel theorem for the representation of a polygon, with the application to the case of an equilateral triangle; (S 16), _Multiple-valued Functions, Algebraic Functions_, deals for the most part with algebraic functions, proving the residue theorem, and establishing that an algebraic function has a definite Order; (S 17), _Integrals of Algebraic Functions_, enunciating Abel's theorem; (S 18), _Indeterminateness of Algebraic Integrals_, deals with the periods associated with an algebraic integral, establishing that for an elliptic integral the number of these is two; (S 19), _Reversion of an algebraic integral_, mentions a problem considered below in detail for an elliptic integral; (S 20), _Elliptic Integrals_, considers the algebraic reduction of any elliptic integral to one of three standard forms, and proves that the function obtained by reversion is single-valued; (S 21), _Modular Functions_, gives a statement of some of the more elementary properties of some functions of great importance, with a definition of Automorphic Functions, and a hint of the connexion with the theory of linear differential equations; (S 22), _A property of integral functions, deduced from the theory of modular functions_, proves that there cannot be more than one value not assumed by an integral function, and gives the basis of the well-known expression of the modulus of the elliptic functions in terms of the ratio of the periods; (S 23), _Geometrical applications of Elliptic Functions_, shows that any plane curve of deficiency unity can be expressed by elliptic functions, and gives a geometrical proof of the addition theorem for the function RN(u); (S 24), _Integrals of Algebraic Functions in connexion with the theory of plane curves_, discusses the generalization to curves of any deficiency; (S 25), _Monogenic Functions of several independent variables_, describes briefly the beginnings of this theory, with a mention of some fundamental theorems: (S 26), _Multiply-Periodic Functions and the Theory of Surfaces_, attempts to show the nature of some problems now being actively pursued.

Beside the brevity necessarily attaching to the account here given of advanced parts of the subject, some of the more elementary results are stated only, without proof, as, for instance: the monogeneity of an algebraic function, no reference being made, moreover, to the cases of differential equations whose integrals are monogenic; that a function possessing an algebraic addition theorem is necessarily an elliptic function (or a particular case of such); that any area can be conformally represented on a half plane, a theorem requiring further much more detailed consideration of the meaning of _area_ than we have given; while the character and properties, including the connectivity, of a Riemann surface have not been referred to. The theta functions are referred to only once, and the principles of the theory of Abelian Functions have been illustrated only by the developments given for elliptic functions.

S 1. _Complex Numbers._--Complex numbers are numbers of the form x + iy, where x, y are ordinary real numbers, and i is a symbol imagined capable of combination with itself and the ordinary real numbers, by way of addition, subtraction, multiplication and division, according to the ordinary commutative, associative and distributive laws; the symbol i is further such that i^2 = -1.

Taking in a plane two rectangular axes Ox, Oy, we assume that every
point of the plane is definitely associated with two real numbers x, y
(its co-ordinates) and conversely; thus any point of the plane is
associated with a single complex number; in particular, for every
point of the axis Ox, for which y = O, the associated number is an
ordinary real number; the complex numbers thus include the real
numbers. The axis Ox is often called the real axis, and the axis Oy
the imaginary axis. If P be the point associated with the complex
variable z = x + iy, the distance OP be called r, and the positive
angle less than 2[pi] between Ox and OP be called [theta], we may
write z = r(cos[theta] + i sin[theta]); then r is called the modulus
or absolute value of z and often denoted by |z| and [theta] is called
the phase or amplitude of z, and often denoted by ph (z); strictly the
phase is ambiguous by additive multiples of 2[pi]. If z' = x' + iy' be
represented by P', the complex argument z' + z is represented by a
point P" obtained by drawing from P' a line equal to and parallel to
OP; the geometrical representation involves for its validity certain
properties of the plane; as, for instance, the equation z' + z = z +
z' involves the possibility of constructing a parallelogram (with OP"
as diagonal). It is important constantly to bear in mind, what is
capable of easy algebraic proof (and geometrically is Euclid's
proposition III. 7), that the modulus of a sum or difference of two
complex numbers is generally less than (and is never greater than) the
sum of their moduli, and is greater than (or equal to) the difference
of their moduli; the former statement thus holds for the sum of any
number of complex numbers. We shall write E(i[theta]) for cos[theta] +
i sin [theta]; it is at once verified that E(i[alpha]). E(i[beta]) =
E[i([alpha] + [beta])], so that the phase of a product of complex
quantities is obtained by addition of their respective phases.

S 2. _Plotting and Properties of Simple Expressions involving a Complex Number._--If we put [zeta] = (z-i)/(z + i), and, putting [zeta] = [xi] + i[eta], take a new plane upon which [xi], [eta] are rectangular co-ordinates, the equations [xi] = (x^2 + y^2-1)/[x^2 + (y + 1)^2], [eta] = -2xy/[x^2 + (y + i)^2] will determine, corresponding to any point of the first plane, a point of the second plane. There is the one exception of z = -i, that is, x = 0, y = -1, of which the corresponding point is at infinity. It can now be easily proved that as z describes the real axis in its plane the point [zeta] describes once a circle of radius unity, with centre at [zeta] = 0, and that there is a definite correspondence of point to point between points in the z-plane which are above the real axis and points of the [zeta]-plane which are interior to this circle; in particular z = i corresponds to [zeta] = 0.

Moreover, [zeta] being a rational function of z, both [xi] and [eta]
are continuous differentiable functions of x and y, save when [zeta]
is infinite; writing [zeta] = [f](x, y) = [f](z - iy, y), the fact
that this is really independent of y leads at once to (Pd)f/(Pd)x +
i(Pd)[f]/(Pd)y = 0, and hence to

(Pd)[xi] (Pd)[eta] (Pd)[xi] (Pd)[eta] (Pd)^2[xi] (Pd)^2[xi]
-------- = ---------, -------- = - ---------, ---------- + ---------- = 0;
(Pd)x (Pd)y' (Pd)y (Pd)x' (Pd)x^2 (Pd)y^2

so that [xi] is not any arbitrary function of x, y, and when [xi] is
known [eta] is determinate save for an additive constant. Also, in
virtue of these equations, if [zeta], [zeta]' be the values of [zeta]
corresponding to two near values of z, say z and z', the ratio
([zeta]'-[zeta])/(z'- z) has a definite limit when z' = z, independent
of the ultimate phase of z'- z, this limit being therefore equal to
(Pd)[zeta]/(Pd)x, that is, (Pd)[xi]/(Pd)x + i(Pd)[eta])/(Pd)x.
Geometrically this fact is interpreted by saying that if two curves in
the z-plane intersect at a point P, at which both the differential
coefficients (Pd)[xi]/(Pd)x, (Pd)[eta]/(Pd)x are not zero, and P', P"
be two points near to P on these curves respectively, and the
corresponding points of the [zeta]-plane be Q, Q', Q", then (1) the
ratios PP"/PP', QQ"/QQ' are ultimately equal, (2) the angle P'PP" is
equal to Q'QQ", (3) the rotation from PP' to PP" is in the same sense
as from QQ' to QQ", it being understood that the axes of [xi], [eta]
in the one plane are related as are the axes of x, y. Thus any diagram
of the z-plane becomes a diagram of the [zeta]-plane with the same
angles; the magnification, however, which is equal to
_ _
| /(Pd)[xi]\^2 /(Pd)[xi]\^2 | 1/2
| ( -------- ) + ( -------- ) |
|_ \ (Pd)x / \ (Pd)y / _|

varies from point to point. Conversely, it appears subsequently that
the expression of any copy of a diagram (say, a map) which preserves
angles requires the intervention of the complex variable.

As another illustration consider the case when [zeta] is a polynomial
in z,

[zeta] = p0 z^n + p1 z^(n - 1) + ... + p_n;

H being an arbitrary real positive number, it can be shown that a
radius R can be found such for every |z| > R we have |[zeta]| > H;
consider the lower limit of |[zeta]| for |z| < R; as [xi]^2 + [eta]^2
is a real continuous function of x, y for |z| < R, there is a point
(x, y), say (x0, y0), at which |[zeta]| is least, say equal to [rho],
and therefore within a circle in the [zeta]-plane whose centre is the
origin, of radius [rho], there are no points [zeta] representing
values corresponding to |z| < R. But if [zeta]0 be the value of [zeta]
corresponding to (x0, y0), and the expression of [zeta] - [zeta]0 near
z0 = x0 + iy0, in terms of z - z0, be A(z - z0)^m + B(z - z0)^(m + 1)
+ ..., where A is not zero, to two points near to (x0, y0), say (x1,
y1) or z1 and z2 = z0 + (z1 - z0)(cos [pi]/m + i sin [pi]/m), will
correspond two points near to [zeta]0, say [zeta]1, and 2[zeta]0
-[zeta]'1, situated so that [zeta]0 is between them. One of these must
be within the circle ([rho]). We infer then that [rho] = 0, and have
proved that every polynomial in z vanishes for some value of z, and
can therefore be written as a product of factors of the form z -
[alpha], where [alpha] denotes a complex number. This proposition
alone suffices to suggest the importance of complex numbers.

S 3. _Limiting Operations._--In order that a complex number [zeta] = [xi] + i[eta] may have a limit it is necessary and sufficient that each of [xi] and [eta] has a limit. Thus an infinite series w0 + w1 + w2 + ..., whose terms are complex numbers, is convergent if the real series formed by taking the real parts of its terms and that formed by the imaginary terms are both convergent. The series is also convergent if the real series formed by the moduli of its terms is convergent; in that case the series is said to be absolutely convergent, and it can be shown that its sum is unaltered by taking the terms in any other order. Generally the necessary and sufficient condition of convergence is that, for a given real positive [epsilon], a number m exists such that for every n > m, and every positive p, the batch of terms w_n + w_(n + 1) + ... + w_(n + p) is less than [epsilon] in absolute value. If the terms depend upon a complex variable z, the convergence is called _uniform_ for a range of values of z, when the inequality holds, for the same [epsilon] and m, for all the points z of this range.

The infinite series of most importance are those of which the general
term is a_nz^n, wherein a_n is a constant, and z is regarded as
variable, n = 0, 1, 2, 3, ... Such a series is called a power series,
if a real and positive number M exists such that for z = z0 and every
n, |a_n z0^n| < M, a condition which is satisfied, for instance, if
the series converges for z = z0, then it is at once proved that the
series converges absolutely for every z for which |z| < |z0|, and
converges uniformly over every range |z| < r' for which r' < |z0|. To
every power series there belongs then a circle of convergence within
which it converges absolutely and uniformly; the function of z
represented by it is thus continuous within the circle (this being the
result of a general property of uniformly convergent series of
continuous functions); the sum for an interior point z is, however,
continuous with the sum for a point z0 on the circumference, as z
approaches to z0 provided the series converges for z = z0, as can be
shown without much difficulty. Within a common circle of convergence
two power series [Sigma] a_n z^n, [Sigma] b_n z^n can be multiplied
together according to the ordinary rule, this being a consequence of a
theorem for absolutely convergent series. If r1 be less than the
radius of convergence of a series [Sigma] a_nz^n and for |z| = r1, the
sum of the series be in absolute value less than a real positive
quantity M, it can be shown that for |z| = r1 every term is also less
than M in absolute value, namely, |a_n| < Mr1^(-n). If in every
arbitrarily small neighbourhood of z = 0 there be a point for which
two converging power series [Sigma]a_nz^n, [Sigma]b_n z^n agree in
value, then the series are identical, or a_n = b_n; thus also if
[Sigma]a_nz^n vanish at z = 0 there is a circle of finite radius about
z = 0 as centre within which no other points are found for which the
sum of the series is zero. Considering a power series [f](z) =
[Sigma]a_nz^n of radius of convergence R, if |z0| < R and we put z =
z0 + t with |t| < R-|z0|, the resulting series [Sigma]a_n (z0 + t)^n
may be regarded as a double series in z0 and t, which, since |z0| + t
< R, is absolutely convergent; it may then be arranged according to
powers of t. Thus we may write [f](z) = [Sigma]A_n t^n; hence A0 =
[f](z0), and we have [[f](z0 + t) - [f](z0)]/t = [Sigma](n=1) A_n
t^(n-l), wherein the continuous series on the right reduces to A1 for
t = 0; thus the ratio on the left has a definite limit when t = 0,
equal namely to A1 or [Sigma]na_nz0^(n - 1). In other words, the
original series may legitimately be differentiated at any interior
point z0 of its circle of convergence. Repeating this process we find
[f](z0 + t) = [Sigma]t^n [f]^(n) (z0)/n!, where [f]^(n) (z0) is the
nth differential coefficient. Repeating for this power series, in t,
the argument applied about z = 0 for [Sigma]a_n z^n, we infer that for
the series [f](z) every point which reduces it to zero is an isolated
point, and of such points only a finite number lie within a circle
which is within the circle of convergence of [f](z).

Perhaps the simplest possible power series is e^z = exp(z) = 1 +
z^2/2! + z^3/3! + ... of which the radius of convergence is infinite.
By multiplication we have exp(z).exp(z^1) = exp(z + z^1). In
particular when x, y are real, and z = x + iy, exp(z) = exp(x)exp(iy).
Now the functions

U0 = sin y, V0 = 1 - cos y, U1 = y - sin y,
V1 = 1/2y^2 - 1 + cos y, U2 = (1/6)y^3 - y + sin y,
V2 = (1/24)y^4 - 1/2y^2 + 1 - cos y, ...

all vanish for y = 0, and the differential coefficient of any one
after the first is the preceding one; as a function (of a real
variable) is increasing when its differential coefficient is positive,
we infer, for y positive, that each of these functions is positive;
proceeding to a limit we hence infer that

cos y = 1 - 1/2y^2 + (1/24)y^4 - ..., sin y = y - (1/6)y^3 +
(1/120)y^5 - ...,

for positive, and hence, for all values of y. We thus have exp(iy) =
cos y + i sin y, and exp (z) = exp (x).(cos y + i sin y). In other
words, the modulus of exp (z) is exp (x) and the phase is y. Hence
also

exp(z + 2[pi]i) = exp(x) [cos (y + 2[pi]) + i sin(y + 2[pi])],

which we express by saying that exp (z) has the period 2[pi]i, and
hence also the period 2k[pi]i, where k is an arbitrary integer. From
the fact that the constantly increasing function exp (x) can vanish
only for x = 0, we at once prove that exp (z) has no other periods.

Taking in the plane of z an infinite strip lying between the lines y =
0, y = 2[pi] and plotting the function [zeta] = exp (z) upon a new
plane, it follows at once from what has been said that every complex
value of [zeta] arises when z takes in turn all positions in this
strip, and that no value arises twice over. The equation [zeta] =
exp(z) thus defines z, regarded as depending upon [zeta], with only an
additive ambiguity 2k[pi]i, where k is an integer. We write z =
[lambda]([zeta]); when [zeta] is real this becomes the logarithm of
[zeta]; in general [lambda]([zeta]) = log |[zeta]| + i ph ([zeta]) +
2k[pi]i, where k is an integer; and when [zeta] describes a closed
circuit surrounding the origin the phase of [zeta] increases by 2[pi],
or k increases by unity. Differentiating the series for [zeta] we have
d[zeta]/dz = [zeta], so that z, regarded as depending upon [zeta], is
also differentiable, with dz/d[zeta] = [zeta]^(-1). On the other hand,
consider the series [zeta] - 1 - 1/2([zeta] - 1)^2 + 1/3([zeta] - 1)^3
- ...; it converges when [zeta] = 2 and hence converges for |[zeta] -
1| < 1; its differential coefficient is, however, 1 - ([zeta] - 1) +
([zeta] - 1)^2 - ..., that is, (1 + [zeta] - 1)^(-1). Wherefore if
[phi]([zeta]) denote this series, for |[zeta] - 1| < 1, the difference
[lambda]([zeta]) - [phi]([zeta]), regarded as a function of [xi] and
[eta], has vanishing differential coefficients; if we take the value
of [lambda]([zeta]) which vanishes when [zeta] = 1 we infer thence
that for |[zeta] - 1| < 1, [lambda]([zeta]) = [Sigma][n = 1] [(-1)^(n
- 1)]/n ([zeta] - 1)^n. It is to be remarked that it is impossible for
[zeta] while subject to |[zeta] - 1| < 1 to make a circuit about the
origin. For values of [zeta] for which |[zeta] - 1| [not less than] 1,
we can also calculate [lambda]([zeta]) with the help of infinite
series, utilizing the fact that [lambda]([zeta][zeta]') =
[lambda]([zeta]) + [lambda]([zeta]').

The function [lambda]([zeta]) is required to define [zeta]^a when
[zeta] and a are complex numbers; this is defined as exp
[a[lambda]([zeta])], that is as [Sigma] (n=0) a^n[[lambda]
([zeta])]^n/n!. When a is a real integer the ambiguity of
[lambda]([zeta]) is immaterial here, since exp [a[lambda]([zeta]) +
2ka[pi]i] = exp[a[lambda]([zeta])]; when a is of the form 1/q, where q
is a positive integer, there are q values possible for [zeta]^(1/q),
of the form exp [1/q [lambda]([zeta])] exp(2k[pi]i/q), with k = 0, 1,
... q - 1, all other values of k leading to one of these; the qth
power of any one of these values is [zeta]; when a = p/q, where p, q
are integers without common factor, q being positive, we have
[zeta]^(p/q) = ([zeta]^(1/q))^p. The definition of the symbol [zeta]^a
is thus a generalization of the ordinary definition of a power, when
the numbers are real. As an example, let it be required to find the
meaning of i^i; the number i is of modulus unity and phase 1/2[pi];
thus [lambda](i) = i(1/2[pi] + 2k[pi]); thus

i^i = exp(-1/2[pi] - 2k[pi]) = exp(-1/2[pi]) exp(-2k[pi]),

is always real, but has an infinite number of values.

The function exp (z) is used also to define a generalized form of the
cosine and sine functions when z is complex; we write, namely, cos z =
1/2[exp(iz) + exp(-iz)] and sin z = -1/2i[exp(iz) - exp(-iz)]. It will
be found that these obey the ordinary relations holding when z is
real, except that their moduli are not inferior to unity. For example,
cos i = 1 + 1/2! + 1/4! + ... is obviously greater than unity.