Chapter I: Functions of Real Variables (5)

we then call it a monogenic potential function. From this, save for an
additive constant, there is defined another potential function by
means of the equation
_
/(x, y) /(Pd)P (Pd)P \
Q = | ( ----- dy - ----- dx ).
_/ \(Pd)x (Pd)y /

The functions P, Q, being given by a finite number of power series,
will be single valued in R0, and P + iQ will be a monogenic function
of z within R0. In drawing this inference it is supposed that the
region R0 is such that every closed path drawn in it is capable of
being deformed continuously to a point lying within R0, that is, is
_simply connected_.

Suppose in particular, c being any point interior to R0, that P
approaches continuously, as z approaches to the boundary of R, to the
value log r, where r is the distance of c to the points of the
perimeter of R. Then the function of z expressed by

[zeta] = (z - c) exp (-P - iQ)

will be developable by a power series in (z - z0) about every point z0
interior to R0, and will vanish at z = c; while on the boundary of R
it will be of constant modulus unity. Thus if it be plotted upon a
plane of [zeta] the boundary of R will become a circle of radius unity
with centre at [zeta] = 0, this latter point corresponding to z = c. A
closed path within R0, passing once round z = c, will lead to a closed
path passing once about [zeta] = 0. Thus every point of the interior
of R will give rise to one point of the interior of the circle. The
converse is also true, but is more difficult to prove; in fact, the
differential coefficient d[zeta]/dz does not vanish for any point
interior to R. This being assumed, we obtain a conformal
representation of the interior of the region R upon the interior of a
circle, in which the arbitrary interior point c of R corresponds to
the centre of the circle, and, by utilizing the arbitrary constant
arising in determining the function Q, an arbitrary point of the
boundary of R corresponds to an arbitrary point of the circumference
of the circle.

There thus arises the problem of the determination of a real monogenic
potential function, single valued and finite within a given arbitrary
region, with an assigned continuous value at all points of the
boundary of the region. When the region is circular this problem is
solved by the integral 1/[pi] [int] Ud[omega] - 1/[pi] [int] Ud[theta]
previously given. When the region is bounded by the outermost portions
of the circumferences of two overlapping circles, it can hence be
proved that the problem also has a solution; more generally, consider
a finite simply connected region, whose boundary we suppose to consist
of a single closed path in the sense previously explained, ABCD;
joining A to C by two non-intersecting paths AEC, AFC lying within the
region, so that the original region may be supposed to be generated by
the overlapping regions AECD, CFAB, of which the common part is AECF;
suppose now the problem of determining a single valued finite
monogenic potential function for the region AECD with a given
continuous boundary value can be solved, and also the same problem for
the region CFAB; then it can be shown that the same problem can be
solved for the original area. Taking indeed the values assigned for
the original perimeter ABCD, assume arbitrarily values for the path
AEC, continuous with one another and with the values at A and C; then
determine the potential function for the interior of AECD; this will
prescribe values for the path CFA which will be continuous at A and C
with the values originally proposed for ABC; we can then determine a
function for the interior of CFAB with the boundary values so
prescribed. This in its turn will give values for the path AEC, so
that we can determine a new function for the interior of AECD. With
the values which this assumes along CFA we can then again determine a
new function for the interior of CFAB. And so on. It can be shown that
these functions, so alternately determined, have a limit representing
such a potential function as is desired for the interior of the
original region ABCD. There cannot be two functions with the given
perimeter values, since their difference would be a monogenic
potential function with boundary value zero, which can easily be shown
to be everywhere zero. At least two other methods have been proposed
for the solution of the same problem.

A particular case of the problem is that of the conformal
representation of the interior of a closed polygon upon the upper half
of the plane of a complex variable t. It can be shown without much
difficulty that if a, b, c, ... be real values of t, and [alpha],
[beta], [gamma], ... be n real numbers, whose sum is n - 2, the
integral

_
/
z = | (t - a)^([alpha] - 1) (t - b)^([beta] - 1) ... dt,
_/

as t describes the real axis, describes in the plane of z a polygon of
n sides with internal angles equal to [alpha][pi], [beta][pi], ...,
and, a proper sign being given to the integral, points of the upper
half of the plane of t give rise to interior points of the polygon.
Herein the points a, b, ... of the real axis give rise to the corners
of the polygon; the condition [Sigma][alpha] = n - 2 ensures merely
that the point t = [oo] does not correspond to a corner; if this
condition be not regarded, an additional corner and side is introduced
in the polygon. Conversely it can be shown that the conformal
representation of a polygon upon the half plane can be effected in
this way; for a polygon of given position of more than three sides it
is necessary for this to determine the positions of all but three of
a, b, c, ...; three of them may always be supposed to be at arbitrary
positions, such as t = 0, t = 1, t = [oo].

As an illustration consider in the plane of z = x + iy, the portion of
the imaginary axis from the origin to z = ih, where h is positive and
less than unity; let C be this point z = ih; let BA be of length unity
along the positive real axis, B being the origin and A the point z =
1; let DE be of length unity along the negative real axis, D being
also the origin and E the point z = -1; let EFA be a semicircle of
radius unity, F being the point z = i. If we put [zeta] = [(z^2 +
h^2)/(1 + h^2z^2)]^1/2, with [zeta] = 1 when z = 1, the function is
single valued within the semicircle, in the plane of z, which is slit
along the imaginary axis from the origin to z = ih; if we plot the
value of [zeta] upon another plane, as z describes the continuous
curve ABCDE, [zeta] will describe the real axis from [zeta] = 1 to
[zeta] = -1, the point C giving [zeta] = 0, and the points B, D giving
the points [zeta] = [+-]h. Near z = 0 the expansion of [zeta] is
[zeta] - h = z^2 1 - h^4 / 2h + ..., or [zeta] + h = -z^2 (1 - h^4)/2h
+ ...; in either case an increase of 1/2[pi] in the phase of z gives
an increase of [pi] in the phase of [zeta] - h or [zeta] + h. Near z =
ih the expansion of [zeta] is [zeta] = (z - ih)^1/2 [2ih/(1 -
h^4)]^1/2 + ..., and an increase of 2[pi] in the phase of z - ih also
leads to an increase of [pi] in the phase of [zeta]. Then as z
describes the semicircle EFA, [zeta] also describes a semicircle of
radius unity, the point z = i becoming [zeta] = i. There is thus a
conformal representation of the interior of the slit semicircle in the
z-plane, upon the interior of the whole semicircle in the
[zeta]-plane, the function

z = [([zeta]^2 - h^2) / (1 - h^2[zeta]^2)]^1/2

being single valued in the latter semicircle. By means of a
transformation t = ([zeta] + 1)^2/([zeta] - 1)^2, the semicircle in
the plane of [zeta] can further be conformably represented upon the
upper half of the whole plane of t.

As another illustration we may take the conformal representation of an
equilateral triangle upon a half plane. Taking the elliptic function
RN(u) for which RN'^2(u) = 4RN^3(u) - 4, so that, with [epsilon] = exp
(2/3[pi]i), we have e1 = 1, e2 = [epsilon]^2, e3 = [epsilon], the half
periods may be taken to be
_ _
/ [oo] dt / [oo] dt
1/2[omega] = | --------------, 1/2[omega]' = | -------------- = 1/2[epsilon][omega];
_/ 1 2(t^3 - 1)^1/2 _/ e3 2(t^3 - 1)^1/2

drawing the equilateral triangle whose vertices are O, of argument O,
A of argument [omega], and B of argument [omega] + [omega]' =
-[epsilon]^2[omega], and the equilateral triangle whose angular points
are O, B and C, of argument [omega]', let E, of argument
{1/3}(2[omega] + [omega]'), and D, of argument 1/3([omega] +
2[omega]'), be the centroids of these triangles respectively, and let
BE, OE, AE cut OA, AB, BO in K, L, H respectively, and BD, OD, CD cut
OC, BC, OB in F, G, H respectively; then if u = [xi] + i[eta] be any
point of the interior of the triangle OEH and v = [epsilon]u0 =
[epsilon]([xi] - i[eta]) be any point of the interior of the triangle
OHD, the points respectively of the ten triangles OEK, EKA, EAL, ELB,
EBH, DHB, DBG, DGC, DCF, DFO are at once seen to be given by
-[epsilon]v, [omega] + [epsilon]u, [omega] - [eta]^2v, [omega] +
[omega]' + [epsilon]^2u, [omega] + [omega]' - v, [omega] + [omega]' -
u, [omega] + [omega]' + [epsilon]v, [omega]' - [epsilon]u, [omega]' +
[epsilon]^2v, -[epsilon]^2u. Further, when u is real, since the term
-2(u + m[omega] + m'[epsilon]^2[omega])^(-3), which is the conjugate
complex of -2(u + m[omega] + m'[epsilon]^2[omega])^3, arises in the
infinite sum which expresses RN'(u), namely as -2(u + [mu][omega] +
[mu]'[epsilon][omega])^(-3), where [mu] = m - m', [mu]' = -m', it
follows that RN'(u) is real; in a similar way we prove that RN'(u) is
pure imaginary when u is pure imaginary, and that RN'(u) =
RN'([epsilon]u) = RN'([epsilon]^2u), as also that for v = [epsilon]u0,
RN'(v) is the conjugate complex of RN'(u). Hence it follows that the
variable

t = 1/2iRN'(u)

takes each real value once as u passes along the perimeter of the
triangle ODE, being as can be shown respectively [oo], 1, 0, -1 at O,
D, H, E, and takes every complex value of imaginary part positive once
in the interior of this triangle. This leads to
_
/ [oo]
u = 1/3i | (t^2 - 1)^(-2/3) dt
_/ t

in accordance with the general theory.

It can be deduced that [tau] = t^2 represents the triangle ODH on the
upper half plane of [tau], and [zeta] = {i-[tau]^(-1)}^(1/2)
represents similarly the triangle OBD.

S 16. _Multiple valued Functions. Algebraic Functions._--The explanations and definitions of a monogenic function hitherto given have been framed for the most part with a view to single valued functions. But starting from a power series, say in z - c, which represents a single value at all points of its circle of convergence, suppose that, by means of a derived series in z - c', where c' is interior to the circle of convergence, we can continue the function beyond this, and then by means of a series derived from the first derived series we can make a further continuation, and so on; it may well be that when, after a closed circuit, we again consider points in the first circle of convergence, the value represented may not agree with the original value. One example is the case z^(1/2), for which two values exist for any value of z; another is the generalized logarithm [Lambda] (z), for which there is an infinite number of values. In such cases, as before, the region of existence of the function consists of all points which can be reached by such continuations with power series, and the singular points, which are the limiting points of the point-aggregate constituting the region of existence, are those points in whose neighbourhood the radii of convergence of derived series have zero for limit. In this description the point z = [oo] does not occupy an exceptional position, a power series in z - c being transformed to a series in 1/z when z is near enough to c by means of z - c = c(1 - cz^(-1)) [1 - (1 - cz^(-1))]^(-1), and a series in 1/z to a series in z - c, when z is near enough to c, by means of

1 1 / z - c \^(-1)
-- = -- ( 1 + ----- ).
z c \ c /

The commonest case of the occurrence of multiple valued functions is
that in which the function s satisfies an algebraic equation [f](s, z)
= p_0s^n + p1s^(n - 1) + ... + p_n = 0, wherein p0, p1, ... p_n are
integral polynomials in z. Assuming [f](s, z) incapable of being
written as a product of polynomials rational in s and z, and excepting
values of z for which the polynomial coefficient of s^n vanishes, as
also the values of z for which beside [f](s, z) = 0 we have also
(Pd)f(s, z)/(Pd)s = 0, and also in general the point z = [oo], the
roots of this equation about any point z = c are given by n power
series in z-c. About a finite point z = c for which the equation
(Pd)f(s, z)/(Pd)s = 0 is satisfied by one or more of the roots s of
[f](s, z) = 0, the n roots break up into a certain number of cycles,
the r roots of a cycle being given by a set of power series in a
radical (z - c)^(1/r), these series of the cycle being obtainable from
one another by replacing (z - c)^(1/r) by [omega](z - r)^(1/r), where
[omega], equal to exp (2[pi]ih/r), is one of the rth roots of unity.
Putting then z - c = t^r we may say that the r roots of a cycle are
given by a single power series in t, an increase of 2[pi] in the phase
of t giving an increase of 2[pi]r in the phase of z - c. This single
series in t, giving the values of s belonging to one cycle in the
neighbourhood of z = c when the phase of z-c varies through 2[pi]r, is
to be looked upon as defining a single _place_ among the aggregate of
values of z and s which satisfy [f](s, z) = 0; two such places may be
at the same _point_ (z = c, s = d) without coinciding, the
corresponding power series for the neighbouring points being
different. Thus for an ordinary value of z, z = c, there are n places
for which the neighbouring values of s are given by n power series in
z-c; for a value of z for which (Pd)f(s, z)/(Pd)s = 0 there are less
than n places. Similar remarks hold for the neighbourhood of z = [oo];
there may be n places whose neighbourhood is given by n power series
in z^(-1) or fewer, one of these being associated with a series in t,
where t = (z^(-1))^(1/r); the sum of the values of r which thus arise
is always n. In general, then, we may say, with t of one of the forms
(z-c), (z-c)^(1/r), z^(-1), (z^(-1))^(1/r). that the neighbourhood of
any place (c, d) for which [f](c, d) = 0 is given by a pair of
expressions z = c + P(t), s = d + Q(t), where P(t) is a (particular
case of a) power series vanishing for t = 0, and Q(t) is a power
series vanishing for t = 0, and t vanishes at (c, d), the expression
z-c being replaced by z^(-1) when c is infinite, and similarly the
expression s-d by s^(-1) when d is infinite. The last case arises when
we consider the finite values of z for which the polynomial
coefficient of s^n vanishes. Of such a pair of expressions we may
obtain a continuation by writing t = t0 + [lambda]1[tau] +
[lambda]2[tau]^2 + .. , where [tau] is a new variable and [lambda]1 is
not zero; in particular for an ordinary finite place this equation
simply becomes t = t0 + [tau]. It can be shown that all the pairs of
power series z = c + P(t), s = d + Q(t) which are necessary to
represent all pairs of values of z, s satisfying the equation [f](s,
z) = 0 can be obtained from one of them by this process of
continuation, a fact which we express by saying that the equation
[f](s, z) = 0 defines a _monogenic algebraic construct_. With less
accuracy we may say that an irreducible algebraic equation [f](s, z) =
0 determines a single monogenic function s of z.

Any rational function of z and s, where [f](s, z) = 0, may be
considered in the neighbourhood of any place (c, d) by substituting
therein z = c + P(t), s = d + Q(t); the result is necessarily of the
form t^m H(t), where H(t) is a power series in t not vanishing for t =
0 and m is an integer. If this integer is positive, the function is
said to vanish to order m at the place; if this integer is negative, =
-[mu], the function is infinite to order [mu] at the place. More
generally, if A be an arbitrary constant, and, near (c, d), R(s, z) -
A is of the form t^mH(t), where m is positive, we say that R(s, z)
becomes m times equal to A at the place; if R(s, z) is infinite of
order [mu] at the place, so also is R(s, z) - A. It can be shown that
the sum of the values of m at all the places, including the places z =
[oo], where R(s, z) vanishes, which we call the number of zeros of
R(s, z) on the algebraic construct, is finite, and equal to the sum of
the values of [mu] where R (s, z) is infinite, and more generally
equal to the sum of the values of m where R(s, z) = A; this we express
by saying that a rational function R(s, z) takes any value (including
[oo]) the same number of times on the algebraic construct; this number
is called the _order_ of the rational function.

That the total number of zeros of R (s, z) is finite is at once
obvious, these values being obtainable by rational elimination of s
between [f](s, z) = 0, R(s, z) = 0. That the number is equal to the
total number of infinities is best deduced by means of a theorem which
is also of more general utility. Let R(s, z) be any rational function
of s, z, which are connected by [f](s, z) = 0; about any place (c, d)
for which z = c + P(t), s = d + Q(t), expand the product

dz
R(s, z) --
dt

in powers of t and pick out the coefficient of t^(-1). There is only a
finite number of places of this kind. The theorem is that the sum of
these coefficients of t^(-1) is zero. This we express by
_ _
| dz |
|R(s, z) -- | = 0.
|_ dt _|t^(-1)

The theorem holds for the case n = 1, that is, for rational functions
of one variable z; in that case, about any finite point we have z - c
= t, and about z = [oo] we have z^(-1) = t, and therefore dz/dt =
-t^(-2); in that case, then, the theorem is that in any rational
function of z,

_ / A1 A2 A_m \
\ ( ----- + --------- + ... + --------- ) + Pz^h + Qz^(h - 1) + ... + R,
/_ \z - a (z - a)^2 (z - a)^m /

the sum [Sigma]A1 of the sum of the residues at the finite poles is
equal to the coefficient of 1/z in the expansion, in ascending powers
of 1/z, about z = [oo]; an obvious result. In general, if for a finite
place of the algebraic construct associated with [f](s, z) = 0, whose
neighbourhood is given by z = c + t^r, s = d + Q(t), there be a
coefficient of t^(-1) in R(s, z)dz/dt, this will be r times the
coefficient of t^(-r) in R(s, z) or R[d + Q(t), c + t^r], namely will
be the coefficient of t^(-r) in the sum of the r series obtainable
from R[d + Q(t), c + t^r] by replacing t by [omega]t, where [omega] is
an rth root of unity; thus the sum of the coefficients of t^(-1) in
R(s, z)dz/dt for all the places which arise for z = c, and the
corresponding values of s, is equal to the coefficient of (z - c)^(-1)
in R(s1, z) + R(s2z) + ... + R(s_n, z), where s1, ... s_n are the n
values of s for a value of z near to z = c; this latter sum [Sigma]
R(s_i, z) is, however, a rational function of z only. Similarly, near
z = [oo], for a place given by z^(-1) = t^r, s = d + Q(t), or s^(-1) =
Q(t), the coefficient of t^(-1) in R(s, z)dz/dt is equal to -r times
the coefficient of t^r in R[d + Q(t), t^(-r)], that is equal to the
negative coefficient of z^(-l) in the sum of the r series R[d +
Q([omega]t), t^(-r)], so that, as before, the sum of the coefficients
of t^(-1) in R(s, z)dz/dt at the various places which arise for z =
[oo] is equal to the negative coefficient of z^(-1) in the same
rational function of z, [Sigma] R(s_i, z). Thus, from the
corresponding theorem for rational functions of one variable, the
general theorem now being proved is seen to follow.

Apply this theorem now to the rational function of s and z,

1 dR(s, z)
------- -------;
R(s, z) dz

at a zero of R(s, z) near which R(s, z) = t^mH(t), we have

1 dR(s, z) dz d
------- ------- -- = -- {[lambda] [R(s, z)]}
R(s, z) dz dt dt

where [lambda] denotes the generalized logarithmic function, that is
equal to

mt^(-1) + power series in t;

similarly at a place for which R(s, z) = t^(-[mu]) K(t); the theorem
_ _
| 1 dR(s, z) dz |
| ------- -------- -- | t^(-1) = 0
|_ R(s, z) dz dt _|

thus gives [Sigma]m = [Sigma][mu], or, in words, the total number of
zeros of R(s, z) on the algebraic construct is equal to the total
number of its poles. The same is therefore true of the function R(s,
z) - A, where A is an arbitrary constant; thus the number in question,
being equal to the number of poles of R(s, z) - A, is equal also to
the number of times that R(s, z) = A on the algebraic construct.

We have seen above that all single valued doubly periodic meromorphic
functions, with the same periods, are rational functions of two
variables s, z connected by an equation of the form s^2 = 4z^3 + Az +
B. Taking account of the relation connecting these variables s, z with
the argument of the doubly periodic functions (which was above denoted
by z), it can then easily be seen that the theorem now proved is a
generalization of the theorem proved previously establishing for a
doubly periodic function a definite _order_. There exists a
generalization of another theorem also proved above for doubly
periodic functions, namely, that the sum of the values of the argument
in one parallelogram of periods for which a doubly periodic function
takes a given value is independent of that value; this generalization,
known as Abel's Theorem, is given S 17 below.

S 17. _Integrals of Algebraic Functions._--In treatises on Integral Calculus it is proved that if R(z) denote any rational function, an indefinite integral [int]R(z)dz can be evaluated in terms of rational and logarithmic functions, including the inverse trigonometrical functions. In generalization of this it was long ago discovered that if s^2 = az^2 + bz + c and R(s, z) be any rational function of s, z any integral [int]R(s, z)dz can be evaluated in terms of rational functions of s, z and logarithms of such functions; the simplest case is [int]s^(-1)dz or [int](az^2 + bz + c)^(-1/2)dz. More generally if f(s, z) = 0 be such a relation connecting s, z that when [theta] is an appropriate rational function of s and z both s and z are rationally expressible, in virtue of [f](s, z) = 0 in terms of [theta], the integral [int]R(s, z)dz is reducible to a form [int]H ([theta])d[theta], where H([theta]) is rational in [theta], and can therefore also be evaluated by rational functions and logarithms of rational functions of s and z. It was natural to inquire whether a similar theorem holds for integrals [int]R(s, z)dz wherein s^2 is a cubic polynomial in z. The answer is in the negative. For instance, no one of the three integrals

_ _ _
/ dz / zdz / dz
| --, | ---, | --------
_/ s _/ s _/ (z - c)s

can be expressed by rational and logarithms of rational functions of s and z; but it can be shown that every integral [int]R(s, z)dz can be expressed by means of integrals of these three types together with rational and logarithms of rational functions of s and z (see below under S 20, Elliptic Integrals). A similar theorem is true when s^2 = quartic polynomial in z; in fact when s^2 = A(z - a)(z - b)(z - c)(z - d), putting y = s(z - a)^(-2), x = (z - a)^(-1), we obtain y2 = cubic polynomial in x. Much less is the theorem true when the fundamental relation [f](s, z) = 0 is of more general type. There exists then, however, a very general theorem, known as _Abel's Theorem_, which may be enunciated as follows: Beside the rational function R(s, z) occurring in the integral [int]R(s, z)dz, consider another rational function H(s, z); let (a1), ... (a_m) denote the places of the construct associated with the fundamental equation [f](s, z) = 0, for which H(s, z) is equal to one value A, each taken with its proper multiplicity, and let (b1), ... (b_m) denote the places for which H(s, z) = B, where B is another value; then the sum of the m integrals [int] [(b_i) to (a_i)] R(s, z)dz is equal to the sum of the coefficients of t^(-1) in the expansions of the function

dz / H(s, z) - B \
R(s, z) -- [lambda] ( ----------- ),
dt \ H(s, z) - A /

where [lambda] denotes the generalized logarithmic function, at the
various places where the expansion of R(s, z)dz/dt contains negative
powers of t. This fact may be obtained at once from the equation
_ _
| 1 dz |
| -------------- R(s, z) -- | = 0,
|_ H(s, z) - [mu] dt _|t^(-1)

wherein [mu] is a constant. (For illustrations see below, under S 20, Elliptic Integrals.)

S 18. _Indeterminateness of Algebraic Integrals._--The theorem that the integral [int][a to x] [f](z)dz is independent of the path from a to z, holds only on the hypothesis that any two such paths are equivalent, that is, taken together from the complete boundary of a region of the plane within which [f](z) is finite and single valued, besides being differentiable. Suppose that these conditions fail only at a finite number of isolated points in the finite part of the plane. Then any path from a to z is equivalent, in the sense explained, to any other path together with closed ~~ paths beginning and ending at the arbitrary point a each enclosing one or more of the exceptional points, these closed paths being chosen, when [f](z) is not a single valued function, so that the final value of [f](z) at a is equal to its initial value. It is necessary for the statement that this condition may be capable of being satisfied.

For instance, the integral [int][1 to z] z^(-1)dz is liable to an
additive indeterminateness equal to the value obtained by a closed
path about z = 0, which is equal to 2[pi]i; if we put u = [int][1 to
z] z^(-1)dz and consider z as a function of u, then we must regard
this function as unaffected by the addition of 2[pi]i to its argument
u; we know in fact that z = exp (u) and is a single valued function of
u, with the period 2[pi]i. Or again the integral [int][0 to z] (1 +
z^2)^(-1)dz is liable to an additive indeterminateness equal to the
value obtained by a closed path about either of the points z = [+-]i;
thus if we put u = [int][0 to z] (1 + z^2)^(-1)dz, the function z of u
is periodic with period [pi], this being the function tan (u). Next we
take the integral u = [int][(0) to (z)] (1 - z^2)^(-1/2)dz, agreeing
that the upper and lower limits refer not only to definite values of
z, but to definite values of z each associated with a definite
determination of the sign of the associated radical (1 - z^2)^(-1/2).
We suppose 1 + z, 1 - z each to have phase zero for z = 0; then a
single closed circuit of z = -1 will lead back to z = 0 with (l -
z^2)^1/2 = -1; the additive indeterminateness of the integral,
obtained by a closed path which restores the initial value of the
subject of integration, may be obtained by a closed circuit containing
both the points [+-]1 in its interior; this gives, since the integral
taken about a vanishing circle whose centre is either of the points z
= [+-] 1 has ultimately the value zero, the sum

_ _ _ _
/ -1 dz / 0 dz / 1 dz / 0 dz
| ----------- + | -------------- + | -------------- + | -------------,
_/ 0 (1-z^2)^1/2 _/-1 -(1 - z^2)^1/2 _/ 0 -(1 - z^2)^1/2 _/ 1 (1 - z^2)^1/2

where, in each case, (1 - z^2)^1/2 is real and positive; that is, it
gives
_
/ 1 dz
-4 | -------------
_/ 0 (1 - z^2)^1/2

or 2[pi]. Thus the additive indeterminateness of the integral is of
the form 2k[pi], where k is an integer, and the function z of u, which
is sin (u), has 2[pi] for period. Take now the case
_
/ (z) dz
u = | ------------------------------------,
_/ (z0) [root]{(z - a)(z - b)(z - c)(z - d)}

adopting a definite determination for the phase of each of the factors
z - a, z - b, z - c, z - d at the arbitrary point z0, and supposing
the upper limit to refer, not only to a definite value of z, but also
to a definite determination of the radical under the sign of
integration. From z0 describe a closed loop about the point z = a,
consisting, suppose, of a straight path from z0 to a, followed by a
vanishing circle whose centre is at a, completed by the straight path
from a to z0. Let similar loops be imagined for each of the points b,
c, d, no two of these having a point in common. Let A denote the value
obtained by the positive circuit of the first loop; this will be in
fact equal to twice the integral taken from z0 along the straight path
to a; for the contribution due to the vanishing circle is ultimately
zero, and the effect of the circuit of this circle is to change the
sign of the subject of integration. After the circuit about a, we
arrive back at z0 with the subject of integration changed in sign; let
B, C, D denote the values of the integral taken by the loops enclosing
respectively b, c and d when in each case the initial determination of
the subject of integration is that adopted in calculating A. If then
we take a circuit from z0 enclosing both a and b but not either c or
d, the value obtained will be A - B, and on returning to z0 the
subject of integration will have its initial value. It appears thus
that the integral is subject to an additive indeterminateness equal to
any one of the six differences such as A - B. Of these there are only
two linearly independent; for clearly only A - B, A - C, A - D are
linearly independent, and in fact, as we see by taking a closed
circuit enclosing all of a, b, c, d, we have A - B + C - D = 0; for
there is no other point in the plane beside a, b, c, d about which the
subject of integration suffers a change of sign, and a circuit
enclosing all of a, b, c, d may by putting z = 1/[zeta] be reduced to
a circuit about [zeta] = 0 about which the value of the integral is
zero. The general value of the integral for any position of z and the
associated sign of the radical, when we start with a definite
determination of the subject of integration, is thus seen to be of the
form u0 + m(A - B) + n(A - C), where m and n are integers. The value
of A - B is independent of the position of z0, being obtainable by a
single closed positive circuit about a and b only; it is thus equal to
twice the integral taken once from a to b, with a proper initial
determination of the radical under the sign of integration. Similar
remarks to the above apply to any integral [int]H(z)dz, in which H(z)
is an algebraic function of z; in any such case H(z) is a rational
function of z and a quantity s connected therewith by an irreducible
rational algebraic equation [f](s, z) = 0. Such an integral [f]K(z,
s)dz is called an Abelian Integral.

S 19. _Reversion of an Algebraic Integral._--In a limited number of cases the equation u = [int] [z0 to z] H(z)dz, in which H(z) is an algebraic function of z, defines z as a single valued function of u. Several cases of this have been mentioned in the previous section; from what was previously proved under S 14, _Doubly Periodic Functions_, it appears that it is necessary for this that the integral should have at most two linearly independent additive constants of indeterminateness; for instance, for an integral _ / z u = | [(z - a)(z - b)(z - c)(z - d)(z - e)(z - f)]^(-1/2) dz, _/ z0

there are three such constants, of the form A - B, A - C, A - D, which are not connected by any linear equation with integral coefficients, and z is not a single valued function of u.

S 20. _Elliptic Integrals._--An integral of the form [int] R(z, s)dz, where s denotes the square root of a quartic polynomial in z, which may reduce to a cubic polynomial, and R denotes a rational function of z and s, is called an _elliptic integral_.

To each value of z belong two values of s, of opposite sign; starting,
for some particular value of z, with a definite one of these two
values, the sign to be attached to s for any other value of z will be
determined by the path of integration for z. When z is in the
neighbourhood of any finite value z0 for which the radical s is not
zero, if we put z - z0 = t, we can find s - s0 = a power series in t,
say s = s0 + Q(t); when z is in the neighbourhood of a value, a, for
which s vanishes, if we put z = a + t^2, we shall obtain s = tQ(t),
where Q(t) is a power series in t; when z is very large and s^2 is a
quartic polynomial in z, if we put z^(-1) = t, we shall find s^(-1) =
t^2Q(t); when z is very large and s^2 is a cubic polynomial in z, if we
put z^(-1) = t^2, we shall find s^(-l) = t^3Q(t). By means of
substitutions of these forms the character of the integral [int] R(z,
s)dz may be investigated for any position of z; in any case it takes a
form [int] [Ht^(-m) + Kt^(-m + 1) + ... + Pt^(-1) + R + St + ...]dt
involving only a finite number of negative powers of t in the subject
of integration. Consider first the particular case [int] s^(-1)dz; it
is easily seen that neither for any finite nor for infinite values of
z can negative powers of t enter; the integral is _everywhere finite_,
and is said to be of _the first kind_; it can, moreover, be shown
without difficulty that no integral [int] R(z, s)dz, save a constant
multiple of [int] s^(-1)dz, has this property. Consider next, s^2 being
of the form a0z^4 + 4a1z^3 + ..., wherein a0 may be zero, the integral
[int] {a0z^2 + 2a1z) s^(-1)dz; for any finite value of z this integral
is easily proved to be everywhere finite; but for infinite values of z
its value is of the form At^(-1) + Q(t), where Q(t) is a power series;
denoting by [root]a0 a particular square root of a0 when a0 is not
zero, the integral becomes infinite for z = [oo] for both signs of s,
the value of A being + [root]a0 or - [root]a0 according as s is
[root]a0.z^2 (1 + [2a1/a0] z^(-1) + ...) or is the negative of this;
hence the integral J1 = [int] ([a0z^2 + 2a1z / s] + [root]a0)dz becomes
infinite when z is infinite, for the former sign of s, its infinite
term being 2[root]a0 t^(-1) or 2a0.z, but does not become infinite for
z infinite for the other sign of s. When a0 = 0 the signs of s for z =
[oo] are not separated, being obtained one from the other by a circuit
of z about an infinitely large circle, and the form obtained
represents an integral becoming infinite as before for z = [oo], its
infinite part being 2[root]a1.t^(-1) or 2[root]a1.[root]z. Similarly
if z0 be any finite value of z which is not a root of the polynomial
[f](z) to which s^2 is equal, and s0 denotes a particular one of the
determinations of s for z = z0, the integral
_
/ / s0^2 + 1/2(z - z0) [f]'(z0) s0 \
J2 = | ( --------------------------- + --------- ) dz,
_/ \ (z - z0)^2s (z - z0)^2 /

wherein [f]'(z) = d[f](z)/dz, becomes infinite for z = z0, s = s0, but
not for z = z0, s = -s0. its infinite term in the former case being
the negative of 2s0(z - z0). For no other finite or infinite value of
z is the integral infinite. If z = [theta] be a root of [f](z), in
which case the corresponding value of s is zero, the integral
_
/ dz
J3 = 1/2[f]'([theta]) | --------------
_/ (z - [theta])s

becomes infinite for z = 0, its infinite part being, if z - [theta] =
t^2, equal to -[[f]'([theta])] 1/2t^(-1): and this integral is not
elsewhere infinite. In each of these cases, of the integrals J1, J2,
J3, the subject of integration has been chosen so that when the
integral is written near its point of infinity in the form
[int][At^(-2) + Bt^(-1) + Q(t)]dt, the coefficient B is zero, so that
the infinity is of algebraic kind, and so that, when there are two
signs distinguishable for the critical value of z, the integral
becomes infinite for only one of these. An integral having only
algebraic infinities, for finite or infinite values of z, is called an
integral of the _second kind_, and it appears that such an integral
can be formed with only one such infinity, that is, for an infinity
arising only for one particular, and arbitrary, pair of values (s, z)
satisfying the equation s^2 = [f](z), this infinity being of the first
order. A function having an algebraic infinity of the mth order (m >
1), only for one sign of s when these signs are separable, at (1) z =
[oo], (2) z = z0, (3) z = a, is given respectively by (s d/dz)^(m -
1)J1, (s d/dz)^(m - 1) J2, (s d/dz )^(m - 1) J3, as we easily see. If
then we have any elliptic integral having algebraic infinities we can,
by subtraction from it of an appropriate sum of constant multiples of
J1, J2, J3 and their differential coefficients just written down,
obtain, as the result, an integral without algebraic infinities. But,
in fact, if J, J^1 denote any two of the three integrals J1, J2, J3,
there exists an equation AJ + BJ' + C[f]s^(-1)dz = rational function
of s, z, where A, B, C are properly chosen constants. For the rational
function

s + s0
------ + z [root]a0
z - z0

is at once found to become infinite for (z0, s0), not for (z0, -s0),
its infinite part for the first point being 2s/(z - z0), and to become
infinite for z infinitely large, and one sign of s only when these are
separable, its infinite part there being 2z [root] a0 or 2 [root] a1
[root] z when a0 = 0. It does not become infinite for any other pair
(z, s) satisfying the relation s^2 = [f](z); this is in accordance
with the easily verified equation
_
s + s0 / dz
------- + z [root]a0 - J1 + J2 + (a0z0^2 + 2a1z0) | -- = 0;
z - z^0 _/ s

and there exists the analogous equation
_
s / dz
----------- + z [root]a0 - J1 + J3 + (a0[theta]^2 + 2a1[theta]) | -- = 0.
z - [theta] _/ s

Consider now the integral
_
/ /s + s0 \ dz
P = | ( --------- + z [root]a0 ) --;
_/ \z - z0 / 2s

this is at once found to be infinite, for finite values of z, only for
(z0, s0), its infinite part being log (z - z0), and for z = [oo], for
one sign of s only when these are separable, its infinite part being
-log t, that is -log z when a0 /= 0, and -log (z^1/2) when a0 = 0.
And, if [f]([theta]) = 0, the integral

_
/ / s \ dz
P1 = | ( ----------- + z [root]a0 ) --
_/ \z - [theta] / 2s

is infinite at z = [theta], s = 0 with an infinite part log t, that is
log (z - [theta])^1/2, is not infinite for any other finite value of z,
and is infinite like P for z = [oo]. An integral possessing such
logarithmic infinities is said to be of the third kind.

Hence it appears that any elliptic integral, by subtraction from it of
an appropriate sum formed with constant multiples of the integral J3
and the rational functions of the form (s d/dz)^(m - 1) J1 with
constant multiples of integrals such as P or P1, with constant
multiples of the integral u = [int]s^(-1)dz, and with rational
functions, can be reduced to an integral H becoming infinite only for
z = [oo], for one sign of s only when these are separable, its
infinite part being of the form A log t, that is, A log z or A log
(z^1/2). Such an integral H = [int]R(z, s)dz does not exist, however,
as we at once find by writing R(z, s) = P(z) + sQ(z), where P(z), Q(z)
are rational functions of z, and examining the forms possible for
these in order that the integral may have only the specified infinity.
An analogous theorem holds for rational functions of z and s; there
exists no rational function which is finite for finite values of z and
is infinite only for z = [oo] for one sign of s and to the first order
only; but there exists a rational function infinite in all to the
first order for each of two or more pairs (z, s), however they may be
situated, or infinite to the second order for an arbitrary pair (z,
s); and any rational function may be formed by a sum of constant
multiples of functions such as

s + s0 s
------ + z [root]a0 or ----------- + z [root]a0
z - z0 z - [theta]

and their differential coefficients.

The consideration of elliptic integrals is therefore reducible to that
of the three
_ _ _
/ dz / /a0z^2 + 2a1z \ / /s + s0 \ dz
u = | --, J = | ( ------------ + z [root]a0 ) dz, P = | ( ------ + z [root]a0 )--
_/ s _/ \ s / _/ \z - z0 / 2s

respectively of the first, second and third kind. Now the equation s^2
= a0z^4 + ... = a0(z - [theta]) (z - [phi]) (z - [psi]) (z - [chi]),
by putting

y = 2s(z - [theta])^(-2) [a0([theta] - [phi]) ([theta] - [psi]) ([theta] - [chi])]^(-1/2)

1 1 / 1 1 1 \
x = ----------- + -- ( --------------- + -------------- + --------------- )
z - [theta] 3 \[theta] - [phi] [theta] - [psi] [theta] - [chi]/

is at once reduced to the form y^2 = 4x^3- g2x - g3 = 4(x - e1)(x -
e2(x - e3), say; and these equations enable us to express s and z
rationally in terms of x and y. It is therefore sufficient to consider
three elliptic integrals
_ _ _
/ dx / xdx / y + y0 dx
u = | --, J = | ---, P = | ------ --.
_/ y _/ y _/ x - x0 2y

Of these consider the first, putting
_
/ ([oo]) dx
u = | --,
_/ (x) y

where the limits involve not only a value for x, but a definite sign
for the radical y. When x is very large, if we put x^(-1) = t^2, y^(-1)
= 2t^3(1 - 1/4 g2t^4 - 1/4 g3t^6)^(-1/2), we have
_
/ t / 1 \ 1
u = | ( 1 + -- g2t^4 + ... )dt = t + -- g2t^5 + ...,
_/ 0 \ 8 / 40

whereby a definite power series in u, valid for sufficiently small
value of u, is found for t, and hence a definite power series for x,
of the form

x = u^(-2) + (1/20)g2u^2 + ...

Let this expression be valid for 0 < |u| < R, and the function defined
thereby, which has a pole of the second order for u = 0, be denoted by
[phi](u). In the range in question it is single valued and satisfies
the differential equation

[[phi]'(u)]^2 = 4[[phi](u)]^3 - g2[phi](u) - g3;

in terms of it we can write x = [phi](u), y = -[phi]'(u), and,
[phi]'(u) being an odd function, the sign attached to y in the
original integral for x = [oo] is immaterial. Now for any two values
u, v in the range in question consider the function
_ _
| [phi]'(u) - [phi]'(v) |^2
F(u, v) = 1/4 | --------------------- | - [phi](u) - [phi](v);
|_ [phi](u) - [phi](v) _|

it is at once seen, from the differential equation, to be such that
(Pd)F/(Pd)u = (Pd)F/(Pd)v; it is therefore a function of u + v;
supposing |u + v| < R we infer therefore, by putting v = 0, that
_ _
| [phi]'(u) - [phi]'(v)] |^2
[phi](u + v) = 1/4 | --------------------- | - [phi](u) - [phi](v).
|_ [phi](u) - [phi](v) _|

By repetition of this equation we infer that if u1, ... u_n be any
arguments each of which is in absolute value less than R, whose sum is
also in absolute value less than R, then [phi](u1 + ... + u_n) is a
rational function of the 2n functions [phi](u_s), [phi]'(u_s); and
hence, if |u| < R, that

[phi](u) = H [[phi](u/n), [phi]'(u/n)],

where H is some rational function of the arguments [phi](u/n),
[phi]'(u/n). In fact, however, so long as |u/n| < R, each of the
functions [phi](u/n), [phi]'(u/n) is single valued and without
singularity save for the pole at u = 0; and a rational function of
single valued functions, each of which has no singularities other than
poles in a certain region, is also a single valued function without
singularities other than poles in this region. We infer, therefore,
that the function of u expressed by H[[phi](u/n), [phi]'(u/n)] is
single valued and without singularities other than poles so long as
|u| < nR; it agrees with [phi](u) when |u| < R, and hence furnishes a
continuation of this function over the extended range |u| < nR.
Moreover, from the method of its derivation, it satisfies the
differential equation [[phi]'(u)]^2 = 4[[phi](u)]^3 - g2[phi](u) - g3.
This equation has therefore one solution which is a single valued
monogenic function with no singularities other than poles for any
finite part of the plane, having in particular for u = 0, a pole of
the second order; and the method adopted for obtaining this near u = 0
shows that the differential equation has no other such solution. This,
however, is not the only solution which is a single valued meromorphic
function, a the functions [phi](u + [alpha]), wherein [alpha] is
arbitrary, being such. Taking now any range of values of u, from u =
0, and putting for any value of u, x = [phi](u), y = -[phi]'(u), so
that y^2 = 4x^3 - g2x - g3, we clearly have
_
/ ([oo]) dx
u = | --;
_/ (x, y) y

conversely if x0 = [phi](u0), y0 = -[phi]'(u0) and [xi], [eta] be any
values satisfying [eta]2 = 4[xi]^2 - g2[xi] - g3, which are
sufficiently near respectively to x0, y0, while v is defined by
_
/ ([xi], [eta]) d[xi]
v - u0 = -| -----,
_/ (x0, y0) [eta]

then [xi], [eta] are respectively [phi](v) and -[phi]'(v); for this
equation leads to an expansion for [xi]-x0 in terms of v = u0 and only
one such expansion, and this is obtained by the same work as would be
necessary to expand [phi](v) when v is near to u0; the function
[phi](u) can therefore be continued by the help of this equation, from
v = u0, provided the lower limit of |[xi] - x0| necessary for the
expansions is not zero in the neighbourhood of any value (x0, y0). In
fact the function [phi](u) can have only a finite number of poles in
any finite part of the plane of u; each of these can be surrounded by
a small circle, and in the portion of the finite part of the plane of
u which is outside these circles, the lower limit of the radii of
convergence of the expansions of [phi](u) is greater than zero; the
same will therefore be the case for the lower limit of the radii |[xi]
- x0| necessary for the continuations spoken of above provided that
the values of ([xi], [eta]) considered do not lead to infinitely
increasing values of v; there does not exist, however, any definite
point ([xi]0, [eta]0) in the neighbourhood of which the integral [int]
[([xi], [eta]) to (x0, y0)] d[xi]/[eta] increases indefinitely, it is
only by a path of infinite length that the integral can so increase.
We infer therefore that if ([xi],[eta]) be any point, where [eta]2 =
4[xi]^3 - g2[xi] - g3, and v be defined by
_
/ ([oo]) dx
v = | -- ,
_/ ([xi], [eta]) y

then [xi] = [phi](v) and [eta] = -[phi]'(v). Thus this equation
determines ([xi], [eta]) without ambiguity. In particular the additive
indeterminatenesses of the integral obtained by closed circuits of the
point of integration are periods of the function [phi](u); by
considerations advanced above it appears that these periods are sums
of integral multiples of two which may be taken to be
_ _
/ [oo] dx / [oo] dx
[omega] = 2| --, [omega]' = 2 | --;
_/ e1 y _/ e3 y