Chapter I: Functions of Real Variables (3)

S4. _Of Functions of a Complex Variable in General._--We have in what precedes shown how to generalize the ordinary rational, algebraic and logarithmic functions, and considered more general cases, of functions expressible by power series in z. With the suggestions furnished by these cases we can frame a general definition. So far our use of the plane upon which z is represented has been only illustrative, the results being capable of analytical statement. In what follows this representation is vital to the mode of expression we adopt; as then the properties of numbers cannot be ultimately based upon spatial intuitions, it is necessary to indicate what are the geometrical ideas requiring elucidation.

Consider a square of side a, to whose perimeter is attached a definite
direction of description, which we take to be counter-clockwise;
another square, also of side a, may be added to this, so that there is
a side common; this common side being erased we have a composite
region with a definite direction of perimeter; to this a third square
of the same size may be attached, so that there is a side common to it
and one of the former squares, and this common side may be erased. If
this process be continued any number of times we obtain a region of
the plane bounded by one or more polygonal closed lines, no two of
which intersect; and at each portion of the perimeter there is a
definite direction of description, which is such that the region is on
the left of the describing point. Similarly we may construct a region
by piecing together triangles, so that every consecutive two have a
side in common, it being understood that there is assigned an upper
limit for the greatest side of a triangle, and a lower limit for the
smallest angle. In the former method, each square may be divided into
four others by lines through its centre parallel to its sides; in the
latter method each triangle may be divided into four others by lines
joining the middle points of its sides; this halves the sides and
preserves the angles. When we speak of a _region_ of the plane in
general, unless the contrary is stated, we shall suppose it capable of
being generated in this latter way by means of a finite number of
triangles, there being an upper limit to the length of a side of the
triangle and a lower limit to the size of an angle of the triangle. We
shall also require to speak of a _path_ in the plane; this is to be
understood as capable of arising as a limit of a polygonal path of
finite length, there being a definite direction or sense of
description at every point of the path, which therefore never meets
itself. From this the meaning of a closed path is clear. The boundary
points of a region form one or more closed paths, but, in general, it
is only in a limiting sense that the interior points of a closed path
are a region.

There is a logical principle also which must be referred to. We
frequently have cases where, about every interior or boundary, point
z0 of a certain region a circle can be put, say of radius r0, such
that for all points z of the region which are interior to this circle,
for which, that is, |z - z0| < r0, a certain property holds. Assuming
that to r0 is given the value which is the upper limit for z0, of the
possible values, we may call the points |z - z0| < r0, the
neighbourhood belonging to or _proper_ to z0, and may speak of the
property as the property (z, z0). The value of r0 will in general vary
with z0; what is in most cases of importance is the question whether
the lower limit of r0 for all positions is zero or greater than zero.
(A) This lower limit is certainly greater than zero provided the
property (z, z0) is of a kind which we may call extensive; such,
namely, that if it holds, for some position of z0 and all positions of
z, within a certain region, then the property (z, z1) holds within a
circle of radius R about any interior point z1 of this region for all
points z for which the circle |z - z1| = R is within the region. Also
in this case r0 varies continuously with z0. (B) Whether the property
is of this extensive character or not we can prove that the region can
be divided into a finite number of sub-regions such that, for every
one of these, the property holds, (1) for _some_ point z0 within or
upon the boundary of the sub-region, (2) for _every_ point z within or
upon the boundary of the sub-region.

We prove these statements (A), (B) in reverse order. To prove (B) let
a region for which the property (z, z0) holds for all points z and
some point z0 of the region, be called _suitable_: if each of the
triangles of which the region is built up be suitable, what is desired
is proved; if not let an unsuitable triangle be subdivided into four,
as before explained; if one of these subdivisions is unsuitable let it
be again subdivided; and so on. Either the process terminates and then
what is required is proved; or else we obtain an indefinitely
continued sequence of unsuitable triangles, each contained in the
preceding, which converge to a point, say [zeta]; after a certain
stage all these will be interior to the proper region of [zeta]; this,
however, is contrary to the supposition that they are all unsuitable.

We now make some applications of this result (B). Suppose a definite
finite real value attached to every interior or boundary point of the
region, say [f](x, y). It may have a finite upper limit H for the
region, so that no point (x, y) exists for which [f](x, y) > H, but
points (x, y) exist for which [f](x, y) > H - [epsilon], however small
[epsilon] may be; if not we say that its upper limit is infinite.
There is then at least one point of the region such that, for points
of the region within a circle about this point, the upper limit of
[f](x, y) is H, however small the radius of the circle be taken; for
if not we can put about every point of the region a circle within
which the upper limit of [f](x, y) is less than H; then by the result
(B) above the region consists of a finite number of sub-regions within
each of which the upper limit is less than H; this is inconsistent
with the hypothesis that the upper limit for the whole region is H. A
similar statement holds for the lower limit. A case of such a function
[f](x, y) is the radius r0 of the neighbourhood proper to any point
z0, spoken of above. We can hence prove the statement (A) above.

Suppose the property (z, z0) extensive, and, if possible, that the
lower limit of r0 is zero. Let then [zeta] be a point such that the
lower limit of r0 is zero for points z0 within a circle about [zeta]
however small; let r be the radius of the neighbourhood proper to
[zeta]; take z0 so that |z0 - [zeta]| < 1/2r; the property (z, z0),
being extensive, holds within a circle, centre z0, of radius r - |z0 -
[zeta]|, which is greater than |z0 - [zeta]|, and increases to r as
|z0 - [zeta]| diminishes; this being true for all points z0 near
[zeta], the lower limit of r0 is not zero for the neighbourhood of
[zeta], contrary to what was supposed. This proves (A). Also, as is
here shown that r0 [ = >] r - |z0-[zeta]|, may similarly be shown that
r [=>] r0 - |z0 - [zeta]|. Thus r0 differs arbitrarily little from r
when |z0-[zeta]| is sufficiently small; that is, r0 varies
continuously with z0. Next suppose the function [f](x, y), which has a
definite finite value at every point of the region considered, to be
continuous but not necessarily real, so that about every point z0,
within or upon the boundary of the region, [eta] being an arbitrary
real positive quantity assigned beforehand, a circle is possible, so
that for all points z of the region interior to this circle, we have
|[f](x, y) - [f](x0, y0)| < 1/2[eta], and therefore (x', y') being any
other point interior to this circle, |[f](x', y') - [f](x, y)| <
[eta]. We can then apply the result (A) obtained above, taking for the
neighbourhood proper to any point z0 the circular area within which,
for any two points (x, y), (x', y'), we have |[f](x', y') - [f](x, y)|
< [eta]. This is clearly an extensive property. Thus, a number r is
assignable, greater than zero, such that, for any two points (x, y),
(x', y') within a circle |z - z0| = r about any point z0, we have
|[f](x', y') - [f](x, y)| < [eta], and, in particular, |[f](x, y) -
[f](x0, y0)| < [eta], where [eta] is an arbitrary real positive
quantity agreed upon beforehand.

Take now any path in the region, whose extreme points are z0, z, and
let z1, ... z_(n - 1) be intermediate points of the path, in order;
denote the continuous function [f](x, y) by [f](z), and let [f]_r
denote any quantity such that |[f]_r - [f](z_r)| [=<] |[f](z_(r + 1))
- [f](z_r)|; consider the sum

(z1 - z0)[f]0 + (z2 - z1)[f]1 + ... + (z - z_(n - 1))[f](n - 1).

By the definition of a path we can suppose, n being large enough, that
the intermediate points z1, ... z_(n - 1) are so taken that if z_i,
z_(i + 1) be any two points intermediate, in order, to z_r and z_(r +
1), we have |z_(i + i) - z_i| < |z_(r + 1) - z_r|; we can thus suppose
|z1 - z0|, |z2 - z1|, ... |z - z_(n - 1)|all to converge constantly
to zero. This being so, we can show that the sum above has a definite
limit. For this it is sufficient, as in the case of an integral of a
function of one real variable, to prove this to be so when the
convergence is obtained by taking new points of division intermediate
to the former ones. If, however, z_(r, 1), z_(r, 2), ... z_(r, m - 1)
be intermediate in order to z_r and z_(r + 1), and |[f]_(r, i) -
[f](z_(r, i))| < |[f](z_(r, i + 1)) - [f](z_(r, i))|, the difference
between [Sigma](z_(r + 1) - z_r)[f]_r and

[Sigma]{(z_(r, 1) - z_r)[f]_(r, 0) + (z_(r, 2) - z{r, 1})[f]_(r, 1)
+ ... + (z_(r + 1) - z_(r, m - 1))[f]_(r, m - 1)},

which is equal to

[Sigma]_r [Sigma]_i (z_(r, i + 1) - z_(r, i))([f]_(r, i) - [f]_r),

is, when |z_(r + 1) - z_r| is small enough, to ensure |[f](z_(r +
1)) - [f](z_r)| < [eta], less in absolute value than

[Sigma]2[eta] [Sigma] |z_(r, i + 1) - z{r, i}|,

which, if S be the upper limit of the perimeter of the polygon from
which the path is generated, is < 2[eta]S, and is therefore
arbitrarily small.

The limit in question is called [int](z_0 to z) [f](z)dz. In
particular when [f](z) = 1, it is obvious from the definition that its
value is z - z0; when [f](z) = z, by taking [f]_r = 1/2(z_(r + 1) -
z_r), it is equally clear that its value is 1/2(z^2 - z0^2); these
results will be applied immediately.

Suppose now that to every interior and boundary point z0 of a certain
region there belong two definite finite numbers [f](z0), F(z0), such
that, whatever real positive quantity [eta] may be, a real positive
number [epsilon] exists for which the condition

| [f](z) - [f](z0) |
| ---------------- - F(z0) | < [eta],
| z-z0 |

which we describe as the condition (z, z0), is satisfied for every
point z, within or upon the boundary of the region, satisfying the
limitation |z - z0| < [epsilon]. Then [f](z0) is called a
differentiable function of the complex variable z0 over this region,
its differential coefficient being F(z0). The function [f](z0) is thus
a continuous function of the real variables x0, y0, where z0 = x0 +
iy0, over the region; it will appear that F(z0) is also continuous and
in fact also a differentiable function of z0.

Supposing [eta] to be retained the same for all points z0 of the
region, and [sigma]0 to be the upper limit of the possible values of
[epsilon] for the point z0, it is to be presumed that [sigma]0 will
vary with z0, and it is not obvious as yet that the lower limit of the
values of [sigma]0 as z0 varies over the region may not be zero. We
can, however, show that the region can be divided into a finite number
of sub-regions for each of which the condition (z, z0), above, is
satisfied for all points z, within or upon the boundary of this
sub-region, for an appropriate position of z0, within or upon the
boundary of this sub-region. This is proved above as result (B).

Hence it can be proved that, for a differentiable function [f](z), the
integral [int](z_1 to z) [f](z)dz has the same value by whatever path
within the region we pass from z1 to z. This we prove by showing that
when taken round a closed path in the region the integral
[int][f](z)dz vanishes. Consider first a triangle over which the
condition (z, z0) holds, for some position of z0 and every position of
z, within or upon the boundary of the triangle. Then as

[f](z) = [f](z0) + (z - z0)F(z0) + [eta][theta](z - z0), where |[theta]| < 1,

we have
_ _ _ _
/ / / /
|[f](z)dz = [[f](z0) - z0F(z0)] |dz + F(z0) |zdz + [eta] |[theta](z - z0)dz,
_/ _/ _/ _/

which, as the path is closed, is [eta] [int][theta](z-z0)dz. Now, from
the theorem that the absolute value of a sum is less than the sum of
the absolute values of the terms, this last is less, in absolute
value, than [eta]ap, where a is the greatest side of the triangle and
p is its perimeter; if [Delta] be the area of the triangle, we have
[Delta] = 1/2ab sin C > ([alpha]/[pi])ba, where [alpha] is the least
angle of the triangle, and hence a(a + b + c) < 2a(b + c) <
4[pi][Delta]/[alpha]; the integral [int][f](z)dz round the perimeter
of the triangle is thus < 4[pi][eta][Delta]/[alpha]. Now consider any
region made up of triangles, as before explained, in each of which the
condition (z, z0) holds, as in the triangle just taken. The integral
[int][f](z)dz round the boundary of the region is equal to the sum of
the values of the integral round the component triangles, and thus
less in absolute value than 4[pi][eta]K/[alpha], where K is the whole
area of the region, and [alpha] is the smallest angle of the component
triangles. However small [eta] be taken, such a division of the region
into a finite number of component triangles has been shown possible;
the integral round the perimeter of the region is thus arbitrarily
small. Thus it is actually zero, which it was desired to prove. Two
remarks should be added: (1) The theorem is proved only on condition
that the closed path of integration belongs to the region at every
point of which the conditions are satisfied. (2) The theorem, though
proved only when the region consists of triangles, holds also when the
boundary points of the region consist of one or more closed paths, no
two of which meet.

Hence we can deduce the remarkable result that the value of [f](z) at
any interior point of a region is expressible in terms of the value of
[f](z) at the boundary points. For consider in the original region the
function [f](z)/(z - z0), where z0 is an interior point: this
satisfies the same conditions as [f](z) except in the immediate
neighbourhood of z0. Taking out then from the original region a small
regular polygonal region with z0 as centre, the theorem holds for the
remaining portion. Proceeding to the limit when the polygon becomes a
circle, it appears that the integral [int] dz[f](z)/(z - z0) round the
boundary of the original region is equal to the same integral taken
counter-clockwise round a small circle having z0 as centre; on this
circle, however, if z - z0 = rE(i[theta]), dz/(z - z0) = id[theta],
and [f](z) differs arbitrarily little from f(z0) if r is sufficiently
small; the value of the integral round this circle is therefore,
ultimately, when r vanishes, equal to 2[pi]i[f](z0). Hence [f](z0) = 1
/ 2[pi]i [int] [dt[f](t)/(t - z0)], where this integral is round the
boundary of the original region. From this it appears that
_
[f](z) - [f](z0) 1 / dt[f](t)
F(z0) = lim. ---------------- = ------ | ---------
z - z0 2[pi]i _/ (t-z0)^2

also round the boundary of the original region. This form shows,
however, that F(z0) is a continuous, finite, differentiable function
of z0 over the whole interior of the original region.

S 5. _Applications._--The previous results have manifold applications.

(1) If an infinite series of differentiable functions of z be
uniformly convergent along a certain path lying with the region of
definition of the functions, so that S(2) = u0(z) + u1(z) + ... + u_(n
- 1)(z) + R_n(z), where |R_n(z)| < [epsilon] for all points of the
path, we have
_ _ _ _ _
/z /z /z /z /z
| S(z)dz = | u0(z)dz + | u1(z)dz + ... + | u_(n - 1)(z)dz + | R_n(z)dz,
_/z0 _/z0 _/z0 _/z0 _/z0

wherein, in absolute value, [int](z_0 to z) R_n(z)dz < [epsilon]L, if
L be the length of the path. Thus the series may be integrated, and
the resulting series is also uniformly convergent.

(2) If [f](x, y) be definite, finite and continuous at every point of
a region, and over any closed path in the region [int][f](x, y)dz = 0,
then [psi](z) = [int](z_0 to z) [f](x, y)dz, for interior points z0,
z, is a differentiable function of z, having for its differential
coefficient the function [f](x, y), which is therefore also a
differentiable function of z at interior points.

(3) Hence if the series u0(z) + u1(z) + ... to [oo] be uniformly
convergent over a region, its terms being differentiable functions of
z, then its sum S(z) is a differentiable function of z, whose
differential coefficient, given by (1 / 2[pi]i) [int] (2[pi]i /
(t - z)^2), is obtainable by differentiating the series. This theorem,
unlike (1), does not hold for functions of a real variable.

(4) If the region of definition of a differentiable function [f](z)
include the region bounded by two concentric circles of radii r, R,
with centre at the origin, and z0 be an interior point of this region,
_ _
1 / [f](t)dt 1 / [f](t)dt
[f](z0) = ------ | -------- - ------ | --------,
2[pi]i _/R t - z0 2[pi]i _/r t - z0

where the integrals are both counter-clockwise round the two
circumferences respectively; putting in the first (t - z0)^(-1) =
[Sigma]_(n=0) z0^n/t^(n + 1), and in the second (t - z0)^(-1) =
[Sigma]_(n=0) t^n/z0^(n + 1), we find [f](z0) = [Sigma] (-[oo] to
[oo]) A_nz0^n, wherein A_n = (1 / 2[pi]i) [int] [f(t) / t^(n + 1)] dt,
taken round any circle, centre the origin, of radius intermediate
between r and R. Particular cases are: ([alpha]) when the region of
definition of the function includes the whole interior of the outer
circle; then we may take r = 0, the coefficients A_n for which n < 0
all vanish, and the function [f](z0) is expressed for the whole
interior |z0| < R by a power series [Sigma] (0 to [oo]) A_n z0^n. In
other words, _about every interior point c of the region of definition
a differentiable function of z is expressible by a power series in z -
c; a very important result.

([beta]) If the region of definition, though not including the origin,
extends to within arbitrary nearness of this on all sides, and at the
same time the product z^m [f](z) has a finite limit when |z|
diminishes to zero, all the coefficients A_n for which n < -m vanish,
and we have

f(z0) = A_(-m) z0^(-m) + A_(-m + 1) z0^(-m + 1)
+ ... + A_(-1) z0^(-1) + A0 + A1z0 ... to [oo].

Such a case occurs, for instance, when [f](z) = cosec z, the number m
being unity.

S 6. _Singular Points._--The _region of existence_ of a differentiable function of z is an unclosed aggregate of points, each of which is an interior point of a neighbourhood consisting wholly of points of the aggregate, at every point of which the function is definite and finite and possesses a unique finite differential coefficient. Every point of the plane, not belonging to the aggregate, which is a limiting point of points of the aggregate, such, that is, that points of the aggregate lie in every neighbourhood of this, is called a _singular point_ of the function.

About every interior point z0 of the region of existence the function
may be represented by a power series in z-z0, and the series converges
and represents the function over any circle centre at z0 which
contains no singular point in its interior. This has been proved
above. And it can be similarly proved, putting z = 1/[zeta], that if
the region of existence of the function contains all points of the
plane for which |z| > R, then the function is representable for all
such points by a power series in z^(-1) or [zeta]; in such case we say
that the region of existence of the function contains the point z =
[oo]. A series in z^(-1) has a finite limit when |z| = [oo]; a series
in z cannot remain finite for all points z for which |z| > R; for if,
for |z| = R, the sum of a power series [Sigma]a_n z^n in z is in
absolute value less than M, we have |a_n| < Mr(-n), and therefore, if
M remains finite for all values of r however great, a_n = 0. Thus the
region of existence of a function if it contains all finite points of
the plane cannot contain the point z = [oo]; such is, for instance,
the case of the function exp (z) = [Sigma]z^n/n!. This may be regarded
as a particular case of a well-known result (S 7), that the
circumference of convergence of any power series representing the
function contains at least one singular point. As an extreme case
functions exist whose region of existence is circular, there being a
singular point in every arc of the circumference, however small; for
instance, this is the case for the functions represented for |z| < 1
by the series [Sigma]_(n=0) z^m, where m = n^2, the series
[Sigma]_(n=0) z^m where m = n!, and the series [Sigma](n=1 to 0)
z^m/(m + 1)(m + 2) where m = a^n, a being a positive integer, although
in the last case the series actually converges for every point of the
circle of convergence |z| = 1. If z be a point interior to the circle
of convergence of a series representing the function, the series may
be rearranged in powers of z - z0; as z0 approaches to a singular
point of the function, lying on the circle of convergence, the radii
of convergence of these derived series in z - z0 diminish to zero;
when, however, a circle can be put about z0, not containing any
singular point of the function, but containing points outside the
circle of convergence of the original series, then the series in z -
z0 gives the value of the function for these external points. If the
function be supposed to be given only for the interior of the original
circle, by the original power series, the series in z - z0 converging
beyond the original circle gives what is known as an _analytical
continuation_ of the function. It appears from what has been proved
that the value of the function at all points of its region of
existence can be obtained from its value, supposed given by a series
in one original circle, by a succession of such processes of
analytical continuation.

S 7. _Monogenic Functions._--This suggests an entirely different way of formulating the fundamental parts of the theory of functions of a complex variable, which appears to be preferable to that so far followed here.

Starting with a convergent power series, say in powers of z, this
series can be arranged in powers of z - z0, about any point z0
interior to its circle of convergence, and the new series converges
certainly for |z - z0| < r - |z0|, if r be the original radius of
convergence. If for every position of z0 this is the greatest radius
of convergence of the derived series, then the original series
represents a function existing only within its circle of convergence.
If for some position of z0 the derived series converges for |z - z0| <
r - |z0| + D, then it can be shown that for points z, interior to the
original circle, lying in the annulus r - |z0| < |z - z0| < r - |z0| +
D, the value represented by the derived series agrees with that
represented by the original series. If for another point z1 interior
to the original circle the derived series converges for |z - z1| < r -
|z1| + E, and the two circles |z - z0| = r - |z0| + D, |z - z1| = r -
|z1| + E have interior points common, lying beyond |z| = r, then it
can be shown that the values represented by these series at these
common points agree. Either series then can be used to furnish an
analytical continuation of the function as originally defined.
Continuing this process of continuation as far as possible, we arrive
at the conception of the function as defined by an aggregate of power
series of which every one has points of convergence common with some
one or more others; the whole aggregate of points of the plane which
can be so reached constitutes the region of existence of the function;
the limiting points of this region are the points in whose
neighbourhood the derived series have radii of convergence diminishing
indefinitely to zero; these are the singular points. The circle of
convergence of any of the series has at least one such singular point
upon its circumference. So regarded the function is called a
_monogenic_ function, the epithet having reference to the single
origin, by one power series, of the expressions representing the
function; it is also sometimes called a _monogenic analytical_
function, or simply an _analytical_ function; all that is necessary to
define it is the value of the function and of all its differential
coefficients, at some one point of the plane; in the method previously
followed here it was necessary to suppose the function differentiable
at every point of its region of existence. The theory of the
integration of a monogenic function, and Cauchy's theorem, that
[int][f](z)dz = 0 over a closed path, are at once deducible from the
corresponding results applied to a single power series for the
interior of its circle of convergence. There is another advantage
belonging to the theory of monogenic functions: the theory as
originally given here applies in the first instance only to single
valued functions; a monogenic function is by no means necessarily
single valued--it may quite well happen that starting from a
particular power series, converging over a certain circle, and
applying the process of analytical continuation over a closed path
back to an interior point of this circle, the value obtained does not
agree with the initial value. The notion of basing the theory of
functions on the theory of power series is, after Newton, largely due
to Lagrange, who has some interesting remarks in this regard at the
beginning of his _Theorie des fonctions analytiques_. He applies the
idea, however, primarily to functions of a real variable for which the
expression by power series is only of very limited validity; for
functions of a complex variable probably the systematization of the
theory owes most to Weierstrass, whose use of the word monogenic is
that adopted above. In what follows we generally suppose this point of
view to be regarded as fundamental.

S 8. _Some Elementary Properties of Single Valued Functions._--A _pole_ is a singular point of the function [f](z) which is not a singularity of the function 1/[f](z); this latter function is therefore, by the definition, capable of representation about this point, z0, by a series [[f](z)]^(-1 ) = [Sigma]a_n (z - z0)^n. If herein a0 is not zero we can hence derive a representation for [f](z) as a power series about z0, contrary to the hypothesis that z0 is a singular point for this function. Hence a0 = 0; suppose also a1 = 0, a2 = 0, ... a_(m - 1) = 0, but a_m [+-] 0. Then [[f](z)]^(-1) = (z - z0)^m [a_m + a_(m + 1)(z - z0) + ...], and hence (z - z0)^m [f](z) = a_m^(-1) + [Sigma]b_n (z - z0)^n, namely, the expression of [f](z) about z = z0 contains a finite number of negative powers of z - z0 and a (finite or) infinite number of positive powers. Thus a pole is always an isolated singularity.

The integral [int][f](z)dz taken by a closed circuit about the pole
not containing any other singularity is at once seen to be 2[pi]iA1,
where A1 is the coefficient of (z - z0)^(-1) in the expansion of
[f](z) at the pole; this coefficient has therefore a certain
uniqueness, and it is called the _residue of [f](z) at the pole_.
Considering a region in which there are no other singularities than
poles, all these being interior points, _the integral (1 / 2[pi]i)
[int][f](z)dz round the boundary of this region is equal to the sum
of the residues at the included poles_, a very important result. Any
singular point of a function which is not a pole is called an
_essential singularity_; if it be isolated the function is capable, in
the neighbourhood of this point, of approaching arbitrarily near to
any assigned value. For, the point being isolated, the function can be
represented, in its neighbourhood, as we have proved, by a series
[Sigma] (-[oo] to [oo]) a_n(z - z0)^n; it thus cannot remain finite in
the immediate neighbourhood of the point. The point is necessarily an
isolated essential singularity also of the function {[f](z) - A}^(-1)
for if this were expressible by a power series about the point, so
would also the function [f](z) be; as {[f](z) - A}^(-1) approaches
infinity, so does [f](z) approach the arbitrary value A. Similar
remarks apply to the point z = [oo], the function being regarded as a
function of [zeta] = z^(-1). In the neighbourhood of an essential
singularity, which is a limiting point also of poles, the function
clearly becomes infinite. For an essential singularity which is not
isolated the same result does not necessarily hold.

A single valued function is said to be an _integral_ function when it has no singular points except z = [oo]. Such is, for instance, an integral polynomial, which has z = [oo] for a pole, and the functions exp (z) which has z = [oo] as an essential singularity. A function which has no singular points for finite values of z other than poles is called a _meromorphic_ function. If it also have a pole at z = [oo] it is a _rational_ function; for then, if a1, ... a_s be its finite poles, of orders m1; m2, ... m_s, the product (z - a1)^m1 ... (z - a_s)^m_s[f](z) is an integral function with a pole at infinity, capable therefore, for large values of z, of an expression (z^ - 1)^(-m) [Sigma]_(r=0) a_r(z^ - 1)^r; thus (z - a1)^m1 ... (z - a_s)^m_s[f](z) is capable of a form [Sigma]_(r=0) b_r z^r, but z^(-m) [Sigma]_(r=0) b_r z^r remains finite for z = [oo]. Therefore b_(r + 1) = b_(r + 2) = ... = 0, and[f](z) is a rational function.

If for a single valued function F(z) every singular point in the
finite part of the plane is isolated there can only be a finite number
of these in any finite part of the plane, and they can be taken to be
a1, a2, a3, ... with |a1| [=<] |a2| [=<] |a3| ... and limit |a_n| =
[oo]. About a_s the function is expressible as [Sigma] (-[oo] to [oo])
A_n(z - a_s)^n; let [f]_s(z) = [Sigma] (-[oo] to 1) A^n(z - a_s)^n be
the sum of the negative powers in this expansion. Assuming z = 0 not
to be a singular point, let [f]_s(z) be expanded in powers of z, in
the form [Sigma]_(n=0) C_n z^n, and [mu]_s be chosen so that F_s(z) =
[f]_s(z) - [Sigma] (1 to [mu]_s-1) C_nz^n = [Sigma] ([mu]_s to [oo])
C_n z^n is, for |z| < r_s < |a_s|, less in absolute value than the
general term [epsilon]_s of a fore-agreed convergent series of real
positive terms. Then the series [phi](z) = [Sigma] (s=1 to [oo])
F_s(z) converges uniformly in any finite region of the plane, other
than at the points a_s, and is expressible about any point by a power
series, and near a_s, [phi](z) - f_s(z) is expressible by a power
series in z-a_s. Thus F(z) - [phi](z) is an integral function. In
particular when all the finite singularities of F(z) are poles, F(z)
is hereby expressed as the sum of an integral function and a series of
rational functions. The condition |F_s(z)| < [epsilon]_s is imposed
only to render the series [Sigma]F_s(z) uniformly convergent; this
condition may in particular cases be satisfied by a series [Sigma]
G_s(z) where G_s(z) = [f]_s(z) - [Sigma] (1 to [nu]_s-1) C_nz^n and
[nu]_s < [mu]_s. An example of the theorem is the function [pi] cot
[pi]z - z^(-1) for which, taking at first only half the poles,
[f]_s(z) = 1/(z-s); in this case the series [Sigma]F_s(z) where F_s(z)
= (z - s)^-1 + s^-1 is uniformly convergent; thus [pi]cot[pi]z - z^-1
- [Sigma] (-[oo] to [oo]) [(z - s)^-1 + s^-1], where s = 0 is excluded
from the summation, is an integral function. It can be proved that
this integral function vanishes.

Considering an integral function [f](z), if there be no finite
positions of z for which this function vanishes, the function
[lambda][[f](z)] is at once seen to be an integral function, [phi](z),
or [f](z) = exp[[phi](z)]; if however great R may be there be only a
finite number of values of z for which [f](z) vanishes, say z = a1,
... a_m, then it is at once seen that [f](z) = exp [[phi](z)].(z -
a1)^h1...(z - a_m)^h_m, where [phi](z) is an integral function, and
h1, ... h_m are positive integers. If, however, [f](z) vanish for z =
a1, a2 ... where |a1| [=<] |a2| [=<] ... and limit |a_n| = [oo], and
if for simplicity we assume that z - 0 is not a zero and all the zeros
a1, a2, ... are of the first order, we find, by applying the preceding
theorem to the function (1 / [f](z)) (d[f](z) / dz), that [f](z) =
exp[[phi](z)] [Pi] (n=1 to [oo]) {(1 - z/a_n) exp[phi]_n(z)}, where
[phi](z) is an integral function, and [phi]_n(z) is an integral
polynomial of the form

z z^2 z^s
[phi]_n(z) = --- + ------ + ... + ------.
a_n 2a^2_n sa_n^s

The number s may be the same for all values of n, or it may increase
indefinitely with n; it is sufficient in any case to take s = n. In
particular for the function sin[pi]x / [pi]x, we have
_ _
sin[pi]x [oo] | / x \ / x \ |
-------- = [Pi] | ( 1 - -- ) exp ( -- ) |,
[pi]x -[oo] |_ \ n / \ n / _|

where n = 0 is excluded from the product. Or again we have
_ _
1 [oo] | / x \ / x \ |
---------- = xe^C_x [Pi] | ( 1 - -- ) exp ( - -- ) |,
[Gamma](x) n=1 |_ \ n / \ n / _|

where C is a constant, and [Gamma](x) is a function expressible when x
is real and positive by the integral [int] (0 to [oo])
e^(-t) t^(x - 1)dt.

There exist interesting investigations as to the connexion of the
value of s above, the law of increase of the modulus of the integral
function [f](z), and the law of increase of the coefficients in the
series [f](z) = [Sigma] a_n z^n as n increases (see the bibliography
below under _Integral Functions_). It can be shown, moreover, that an
integral function actually assumes every finite complex value, save,
in exceptional cases, one value at most. For instance, the function
exp (z) assumes every finite value except zero (see below under S 21,
_Modular Functions_).

The two theorems given above, the one, known as Mittag-Leffler's theorem, relating to the expression as a sum of simpler functions of a function whose singular points have the point z = [oo] as their only limiting point, the other, Weierstrass's factor theorem, giving the expression of an integral function as a product of factors each with only one zero in the finite part of the plane, may be respectively generalized as follows:--

I. If a1, a2, a3, ... be an infinite series of isolated points having
the points of the aggregate (c) as their limiting points, so that in
any neighbourhood of a point of (c) there exists an infinite number of
the points a1, a2, ..., and with every point a_i there be associated a
polynomial in (z - a_i)^-1, say g_i; then there exists a single valued
function whose region of existence excludes only the points (a) and
the points (c), having in a point a_i a pole whereat the expansion
consists of the terms g_i, together with a power series in z - a_i;
the function is expressible as an infinite series of terms g_i -
[gamma]_i, where [gamma]_i is also a rational function.

II. With a similar aggregate (a), with limiting points (c), suppose
with every point a_i there is associated a positive integer r_i. Then
there exists a single valued function whose region of existence
excludes only the points (c), vanishing to order r_i at the point a_i,
but not elsewhere, expressible in the form

[oo] / a_n - c_n \^r_n
[Pi] ( 1 - --------- ) exp(g_n),
n=1 \ z - c_n /

where with every point a_n is associated a proper point c_n of (c),
and

_[mu]_n
\ 1 / a_n - c_n \^s
g_n = r_n /_ -- ( --------- ),
s=1 s \ z - c_n /

[mu]_n being a properly chosen positive integer.

If it should happen that the points (c) determine a path dividing the
plane into separated regions, as, for instance, if a_n = R(1 - n^-1)
exp(i[pi] [root]2.n), when (c) consists of the points of the circle
|z| = R, the product expression above denotes different monogenic
functions in the different regions, not continuable into one another.

S 9. _Construction of a Monogenic Function with a given Region of Existence._--A series of isolated points interior to a given region can be constructed in infinitely many ways whose limiting points are the boundary points of the region, or are boundary points of the region of such denseness that one of them is found in the neighbourhood of every point of the boundary, however small. Then the application of the last enunciated theorem gives rise to a function having no singularities in the interior of the region, but having a singularity in a boundary point in every small neighbourhood of every boundary point; this function has the given region as region of existence.

S 10. _Expression of a Monogenic Function by means of Rational Functions in a given Region._--Suppose that we have a region R0 of the plane, as previously explained, for all the interior or boundary points of which z is finite, and let its boundary points, consisting of one or more closed polygonal paths, no two of which have a point in common, be called C0. Further suppose that all the points of this region, including the boundary points, are interior points of another region R, whose boundary is denoted by C. Let z be restricted to be within or upon the boundary of C0; let a, b, ... be finite points upon C or outside R. Then when b is near enough to a, the fraction (a - b)/(z - b) is arbitrarily small for all positions of z; say

| a - b |
| ----- | < [epsilon], for |a - b| < [eta];
| z - b |

the rational function of the complex variable t,
_ _
1 | / a - b \^n |
----- |1 - ( ----- ) |,
t - a |_ \ t - b / _|

in which n is a positive integer, is not infinite at t = a, but has a pole at t = b. By taking n large enough, the value of this function, for all positions z of t belonging to R0, differs as little as may be desired from (t - a)^-1. By taking a sum of terms such as

_ _ _
\ { 1 | / a - b \^n | }^p
F = /_ A_p { ----- |1 - ( ----- ) | },
{ t - a |_ \ t - b / _| }

we can thus build a rational function differing, in value, in R0, as
little as may be desired from a given rational function
_
\
[f] = /_ A_p (t - a)^(-p),

and differing, outside R or upon the boundary of R, from [f], in the fact that while [f] is infinite at t = a, F is infinite only at t = b. By a succession of steps of this kind we thus have the theorem that, given a rational function of t whose poles are outside R or upon the boundary of R, and an arbitrary point c outside R or upon the boundary of R, which can be reached by a finite continuous path outside R from all the poles of the rational function, we can build another rational function differing in R0 arbitrarily little from the former, whose poles are all at the point c.

Now any monogenic function [f](t) whose region of definition includes
C and the interior of R can be represented at all points z in R[0] by
_
1 / [f](t)dt
[f](z) = ------ | --------,
2[pi]i _/ t - z

where the path of integration is C. This integral is the limit of a
sum
_
1 \ [f](t_i) (t_(i + 1) - t_i)
S = ------ /_ --------------------------,
2[pi]i t_i - z

where the points t_i are upon C; and the proof we have given of the
existence of the limit shows that the sum S converges to [f](z)
uniformly in regard to z, when z is in R0, so that we can suppose,
when the subdivision of C into intervals t_(i + 1) - t_i, has been
carried sufficiently far, that

| S - [f](z) | < [epsilon],

for all points z of R0, where [epsilon] is arbitrary and agreed upon
beforehand. The function S is, however, a rational function of z with
poles upon C, that is external to R0. We can thus find a rational
function differing arbitrarily little from S, and therefore
arbitrarily little from [f](z), for all points z of R0, with poles at
arbitrary positions outside R0 which can be reached by finite
continuous curves lying outside R from the points of C.

In particular, to take the simplest case, if C0, C be simple closed
polygons, and [GAMMA] be a path to which C approximates by taking the
number of sides of C continually greater, we can find a rational
function differing arbitrarily little from [f](z) for all points of R0
whose poles are at one finite point c external to [GAMMA]. By a
transformation of the form t - c = r^-1, with the appropriate change
in the rational function, we can suppose this point c to be at
infinity, in which case the rational function becomes a polynomial.
Suppose [epsilon]1, [epsilon]2, ... to be an indefinitely continued
sequence of real positive numbers, converging to zero, and P_r to be
the polynomial such that, within C0, |P_r - [f](z)| < [epsilon]_r;
then the infinite series of polynomials

P1(z) + {P2(z) - P1(z)} + {P3(z) - P2(z)} + ...,

whose sum to n terms is P_n(z), converges for all finite values of z
and represents [f](z) within C0.

When C consists of a series of disconnected polygons, some of which
may include others, and, by increasing indefinitely the number of
sides of the polygons C, the points C become the boundary points
[Gamma] of a region, we can suppose the poles of the rational
function, constructed to approximate to [f](z) within R0, to be at
points of [Gamma]. A series of rational functions of the form

H1(z) + {H2(z) - H1(z)} + {H3(z) - H2(z)} + ...

then, as before, represents [f](z) within R0. And R0 may be taken to
coincide as nearly as desired with the interior of the region bounded
by [Gamma].

S 11. _Expression of (1 - z)^(-1) by means of Polynomials. Applications._--We pursue the ideas just cursorily explained in some further detail.

Let c be an arbitrary real positive quantity; putting the complex
variable [zeta] = [xi] + i[eta], enclose the points [zeta] = l, [zeta]
= 1 + c by means of (i.) the straight lines [eta] = [+-]a, from [xi] =
l to [xi] = 1 + c, (ii.) a semicircle convex to [zeta] = 0 of equation
([xi] - 1)^2 + [eta]^2 = a^2, (iii.) a semicircle concave to [zeta] =
0 of equation ([xi] - 1 - c)^2 + [eta]^2 = a^2. The quantities c and a
are to remain fixed. Take a positive integer r so that 1/r (c/a) is
less than unity, and put [sigma] = 1/r (c/a). Now take

c1 = 1 + c/r, c2 = 1 + 2c/r, ... c_r = 1 + c;

if n1, n2, ... n_r, be positive integers, the rational function
_ _
1 | / c1 - 1 \^n1 |
---------- |1 - ( ----------- ) |
1 - [zeta] |_ \ c1 - [zeta] / _|

is finite at [zeta] = 1, and has a pole of order n1 at [zeta] = c1;
the rational function
_ _ _ _
1 | / c1 - 1 \^n1 | | / c2 - c1 \^n2 |^n1
---------- |1 - ( ------------ ) | |1 - ( ----------- ) |
1 - [zeta] |_ \ c1 - [zeta] / _| |_ \ c2 - [zeta]/ _|

is thus finite except for [zeta] = c2, where it has a pole of order
n1n2; finally, writing

/ c_s - c_(s-1) \^n_s
x_s = ( ------------- ),
\ c_s - [zeta] /

the rational function

U = (1 - [zeta])^(-1) (1 - x1)(1 - x2)^n1 (1 - x3)^n1n2 ...
(1 - x_r)^(n1n2 ... n_(r - 1))

has a pole only at [zeta] = 1 + c, of order n1n2 ... n_r.

The difference (1 - [zeta])^(-1) - U is of the form (1 -
[zeta])^(-1)P, where P, of the form

1 - (1 - [rho]1)(1 - [rho]2)...(1 - [rho]_k),

in which there are equalities among [rho]1, [rho]2, ... [rho]_k, is of
the form

[Sigma][rho]1 - [Sigma][rho]1[rho]2 + [Sigma][rho]1[rho]2[rho]3 - ...;

therefore, if |r_i| = |[rho]_i|, we have

|P| < [Sigma]r1 + [Sigma]r1r2 + [Sigma]r1r2r3 + ... <
(1 + r1)(1 + r2)...(1 + r_k) - 1;

now, so long as [zeta] is without the closed curve above described
round [zeta] = 1, [zeta] = 1 + c, we have

| 1 | 1 |c_m - c_(m-1)| c/r
|----------| < ---, |-------------| < --- < [sigma],
|1 - [zeta]| a |c_m - [zeta] | a

and hence

|(1 - [zeta])^(-1) - U| < a^(-1) {(1 + [sigma]^n1) (1 + [sigma]^n2)^n1
(1 + [sigma]^n3)^n1n2 ... (1 + [sigma]^n_r)^(n1n2 ... n_(r-1)) - 1}.

Take an arbitrary real positive [epsilon], and [mu], a positive
number, so that [epsilon]^[mu] - 1 < [epsilon]a, then a value of n1
such that [sigma]^n1 < [mu]/(1 + [mu]) and therefore [sigma]^n1 / (1 -
[sigma]^n1 < [mu], and values for n2, n3 ... such that[sigma]^n2 <
1/n1 [sigma]^2n1, [sigma]^n3 < 1/n1n2 [sigma]^{3n1, ... [sigma]^n_r} <
1/(n1...n_(r - 1)} [sigma]^n_rn1; then, as 1 + x < e^x, we have
|(-[zeta])^(-1) - U| < a^-1 {exp([sigma]^n1 + n1[sigma]^n2 +
n1n2[sigma]^n3 + ... + n1n2...n_(r - 1)[sigma]^n_r) - 1}, and
therefore less than

a^(-1) {exp([sigma]^n1 + [sigma]^2n1 + ... + [sigma]^n_rn1) - 1},

which is less than
_ _
1 | / [sigma]^n1 \ |
-- |exp ( -------------- ) - 1 |
a |_ \1 - [sigma]^n1/ _|

and therefore less than [epsilon].

The rational function U, with a pole at [zeta] = 1 + c, differs
therefore from (1 - [zeta])^(-1), for all points outside the closed
region put about [zeta] = 1, [zeta] = l + c, by a quantity numerically
less than [epsilon]. So long as a remains the same, r and [sigma] will
remain the same, and a less value of [epsilon] will require at most an
increase of the numbers n1, n2, ... n_r; but if a be taken smaller it
may be necessary to increase r, and with this the complexity of the
function U.

Now put

c[zeta] (c + 1)z
z = --------------, [zeta] = --------;
c + 1 - [zeta] c + z

thereby the points [zeta] = 0, 1, 1 + c become the points z = 0, 1,
[oo], the function (1 - z)^(-1) being given by (1-z)^(-1) = c(c +
1)^(-1)(1 - [zeta])^(-1) + (c + 1)^(-1); the function U becomes a
rational function of z with a pole only at z = [oo], that is, it
becomes a polynomial in z, say [(c + 1)/c]H - 1/c, where H is also a
polynomial in z, and
_ _
1 c | 1 |
----- - H = ----- | ---------- - U |;
1 - z c + 1 |_1 + [zeta] _|

the lines [eta] = [+-]a become the two circles expressed, if z = x +
iy, by

c(c + 1)
(x + c)^2 + y^2 = [+-] -------- y,
a

the points ([eta] = 0, [xi] = 1 - a), ([eta] = 0, [xi] = 1 + c + a)
become respectively the points (y = 0, x = c(1 - a)/(c + a), (y = 0, x
= -c(l + c + a)/a), whose limiting positions for a = 0 are
respectively (y = 0, x = 1), (y = 0, x = -[oo]). The circle (x + c)^2 +
y^2 = c(c + 1)y/a can be written

(x + c)^2 (x + c)^4
y = --------- + --------- {[mu] + [root][[mu]^2 - (x + c)^2]}^(-2),
2[mu] 2[mu]

where [mu] = 1/2c(c + 1)/a; its ordinate y, for a given value of x,
can therefore be supposed arbitrarily small by taking a sufficiently
small.

We have thus proved the following result; taking in the plane of z any
finite region of which every interior and boundary point is at a
finite distance, however short, from the points of the real axis for
which 1 =< x =< [oo], we can take a quantity a, and hence, with an
arbitrary c, determine a number r; then corresponding to an arbitrary
[epsilon]_s, we can determine a polynomial P_s, such that, for all
points interior to the region, we have

|(1 - z^(-1)) - P_s| < [epsilon]_s;

thus the series of polynomials

P1 + (P2 - P1) + (P3 - P2) + ...,

constructed with an arbitrary aggregate of real positive numbers
[epsilon]1, [epsilon]2, [epsilon]3, ... with zero as their limit,
converges uniformly and represents (1-z)^(-1) for the whole region
considered.

S 12. _Expansion of a Monogenic Function in Polynomials, over a Star
Region._--Now consider any monogenic function [f](z) of which the
origin is not a singular point; joining the origin to any singular
point by a straight line, let the part of this straight line, produced
beyond the singular point, lying between the singular point and z =
[oo], be regarded as a barrier in the plane, the portion of this
straight line from the origin to the singular point being erased.
Consider next any finite region of the plane, whose boundary points
constitute a path of integration, in a sense previously explained, of
which every point is at a finite distance greater than zero from each
of the barriers before explained; we suppose this region to be such
that any line joining the origin to a boundary point, when produced,
does not meet the boundary again. For every point x in this region R
we can then write
_
/ dt [f](t)
2[pi]i[f](x) = | -- -----------,
_/ t 1 - xt^(-1)