The introduction of the function [f]1(z) is equivalent to the
introduction of the function RN(z; [omega], 2[omega]') constructed
from the periods [omega], 2[omega]' as was RN(z) from [omega] and
[omega]'; denoting this function by RN1(z) and its differential
coefficient by RN'1(z), we have in fact
RN'1(z)
[f]1(z) = 1/2 ----------------------
RN1([omega]') - RN1(z)
as we see at once by considering the zeros and poles and the limit of
z[f]1(z) when z = 0. In terms of the function RN1(z) the original
function RN(z) is expressed by
RN(z) = RN1(z) + RN1(z + [omega]') - RN1([omega]'),
as a consideration of the poles and expansion near z = 0 will show.
A function having [omega], [omega]' for periods, with poles at two
arbitrary points a, b and zeros at a', b', where a' + b' = a + b save
for an expression m[omega] + m'[omega]', in which m, m' are integers,
is a constant multiple of
{RN[z - 1/2(a' + b')] - RN[a' - 1/2(a' + b')]} / {RN[z - 1/2(a + b)] - RN[a - 1/2(a + b)]};
if the expansion of this function near z = a be
_
[lambda](z - a)^(-1) + [mu] + \ [mu]_n(z - a)^n,
/_
n = 1
the expansion near z = b is
_
-[lambda](z - b)^(-1) + [mu] + \ (-1)^n [mu]_n (z - b)^n,
/_
n = 1
as we see by remarking that if z'- b = -(z - a) the function has the
same value at z and z'; hence the differential equation satisfied by
the function is easily calculated in terms of the coefficients in the
expansions.
From the function RN(z) we can obtain another function, termed the
Zeta-function; it is usually denoted by [zeta](z), and defined by
_ _ _
1 / [pi] | 1 | _ / 1 1 z \
[zeta](z) -- = | | --- - RN(z) |dz = \ ' ( ----------- + ------- + --------- ),
z _/ 0 |_ z^2 _| /_ \z - [Omega] [Omega] [Omega]^2/
for which as before we have equations
[zeta](z + [omega]) = [zeta](z) + 2[pi]i[eta],
[zeta](z + [omega]') = [zeta](z) + 2[pi]i[eta]',
where 2[eta], 2[eta]' are certain constants, which in this case do not
both vanish, since else [zeta](z) would be a doubly periodic function
with only one pole of the first order. By considering the integral
_
/
| [zeta](z)dz
_/
round the perimeter of a parallelogram of sides [omega], [omega]'
containing z = 0 in its interior, we find [eta][omega]' -
[eta]'[omega] = 1, so that neither of [eta], [eta]' is zero. We have
[zeta]'(z) = -RN(z). From [zeta](z) by means of the equation
_ _ _
[sigma](z) { / z | 1 | }
---------- = exp { | | [zeta](x) - -- |dz } =
z { _/ 0 |_ z _| }
_ _
| / 2 \ / z z^2 \ |
[Pi]' | ( 1 - ------- ) exp ( ------- + ---------- ) |,
|_ \ [Omega]/ \[Omega] 2[Omega]^2/ _|
we determine an integral function [sigma](z), termed the
Sigma-function, having a zero of the first order at each of the points
z = [Omega]; it can be seen to satisfy the equations
[sigma](z + [omega])
-------------------- = -exp [2[pi] i[eta](z + 1/2[omega])],
[sigma](z)
[sigma](z + [omega]')
--------------------- = -exp [2[pi] i[eta]'(z + 1/2[omega]')].
[sigma](z)
By means of these equations, if a1 + a2 + ... + a_m = a'1 + a'2 + ...
+ a'_m, it is readily shown that
[sigma](z - a'1)[sigma](z - a'2) ... [sigma](z - a'_m)
------------------------------------------------------
[sigma](z - a1[sigma](z - a2) ... [sigma](z - a_m)
is a doubly periodic function having a1, ... a_m as its simple poles,
and a'1, ... a'_m as its simple zeros. Thus the function [sigma](z)
has the important property of enabling us to write any meromorphic
doubly periodic function as a product of factors each having one zero
in the parallelogram of periods; these form a generalization of the
simple factors, z - a, which have the same utility for rational
functions of z. We have [zeta](z) = [sigma]'(z)/[sigma](z).
The functions [zeta](z), RN(z) may be used to write any meromorphic
doubly periodic function F(z) as a sum of terms having each only one
pole; for if in the expansion of F(z) near a pole z = a the terms with
negative powers of z-a be
A1(z - a)^(-1) + A2(z - a){-2} + ... + A_(m + 1)(z - a)^(-(m + 1)),
then the difference
A_(m + 1)
F(z) - A1[zeta](z - a) - A2[Fraktur](z - a)- ... + ---------(-1)^m RN^(m - 1)(z - a)
m!
will not be infinite at z = a. Adding to this a sum of further terms
of the same form, one for each of the poles in a parallelogram of
periods, we obtain, since the sum of the residues A is zero, a doubly
periodic function without poles, that is, a constant; this gives the
expression of F(z) referred to. The indefinite integral [int]F(z)dz
can then be expressed in terms of z, functions RN(z - a) and their
differential coefficients, functions [zeta](z - a) and functions
log[sigma](z - a).
S 15. _Potential Functions. Conformal Representation in
General._--Consider a circle of radius a lying within the region of
existence of a single valued monogenic function, u + iv, of the complex
variable z, = x + iy, the origin z = 0 being the centre of this circle.
If z = rE(i[phi]) = r(cos [phi] + i sin [phi]) be an internal point of
this circle we have
_
1 / (U + iV)
u + iv = ------ | -------- dt,
2[pi]i _/ t - z
where U + iV is the value of the function at a point of the
circumference and t = aE(i[theta]); this is the same as
_
1 / (U + iV) [1 - (r/a)E(i[theta] - i[phi])]
u + iv = ----- | ------------------------------------------ d[theta].
2[pi] _/ 1 + (r/a)^2 - 2(r/a) cos ([theta] - [phi])
If in the above formula we replace z by the external point
(a^2/r)E(i[phi]) the corresponding contour integral will vanish, so that
also
_
1 / (U + iV) [(r/a)^2 - (r/a)E(i[theta] - i[phi])]
0 = ----- | ---------------------------------------------- d[theta];
2[pi] _/ 1 + (r/a)^2 - 2(r/a) cos ([theta] - [phi])
hence by subtraction we have
_
1 / U(a^2 - r^2)
u = ----- | ------------------------------------- d[theta],
2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi])
and a corresponding formula for v in terms of V. If O be the centre of
the circle, Q be the interior point z, P the point aE(i[theta]) of the
circumference, and [omega] the angle which QP makes with OQ produced,
this integral is at once found to be the same as
_ _
1 / 1 /
u = ---- | Ud[omega] - ----- | Ud[theta]
[pi] _/ 2[pi] _/
of which the second part does not depend upon the position of z, and the
equivalence of the integrals holds for every arc of integration.
Conversely, let U be any continuous real function on the
circumference, U0 being the value of it at a point P0 of the
circumference, and describe a small circle with centre at P0 cutting
the given circle in A and B, so that for all points P of the arc AP0B
we have |U - U0| < [epsilon], where [epsilon] is a given small real
quantity. Describe a further circle, centre P0 within the former,
cutting the given circle in A' and B', and let Q be restricted to lie
in the small space bounded by the arc A'P0B' and this second circle;
then for all positions of P upon the greater arc AB of the original
circle QP^2 is greater than a definite finite quantity which is not
zero, say QP^2 > D^2. Consider now the integral
_
1 / (a^2 - r^2)
u' = ----- | U ------------------------------------ d[theta], =
2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi]
_ _
1 / 1 /
---- | Ud[omega] - ----- | Ud[theta],
[pi] _/ 2[pi] _/
which we evaluate as the sum of two, respectively along the small arc
AP0B and the greater arc AB. It is easy to verify that, for the whole
circumference,
_
1 / (a^2 - r^2)
U0 = ----- | U0 ------------------------------------ d[theta] =
2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi]
_ _
1 / 1 /
---- | U0d[omega] - ----- | U0d[theta].
[pi] _/ 2[pi] _/
Hence we can write
_ _
1 / 1 /
u' - U0 = ----- | (U - U0) d[omega] - ----- | (U - U0) d[theta] +
2[pi] _/AP0B 2[pi] _/AP0B
_
1 / (a^2 - r^2)
----- | (U - U0) ----------- d[theta].
2[pi] _/AB QP^2
If the finite angle between QA and QB be called [Phi] and the finite
angle AOB be called [Theta], the sum of the first two components is
numerically less than
[epsilon]
--------- ([Phi] + [Theta]).
2[pi]
If the greatest value of |(U - U0)| on the greater arc AB be called H,
the last component is numerically less than
H
--- (a^2 - r^2),
D^2
of which, when the circle, of centre P0, passing through A'B' is
sufficiently small, the factor a^2 - r^2 is arbitrarily small. Thus it
appears that u' is a function of the position of Q whose limit, when
Q, interior to the original circle, approaches indefinitely near to
P0, is U0. From the form
_ _
1 / 1 /
u' = ---- | Ud[omega] - ----- | Ud[theta],
[pi] _/ 2[pi] _/
since the inclination of QP to a fixed direction is, when Q varies, P
remaining fixed, a solution of the differential equation
(Pd)^2[psi] (Pd)^2
----------- + ------- = 0,
(Pd)x^2 (Pd)y^2
where z, = x + iy, is the point Q, we infer that u' is a
differentiable function satisfying this equation; indeed, when r < a,
we can write
_
1 / (a^2 - r^2)
----- | U ------------------------------------- d[theta]
2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi])
_ _ _
1 / | r r^2 |
= ----- | U | 1 + 2 -- cos ([theta] - [phi]) + 2 --- cos 2([theta] - [phi]) + ...| d[theta]
2[pi] _/ |_ a a^2 _|
= a0 + a1x + b1y + a2(x^2 - y^2) + 2b2xy + ...,
where
_ _ _
1 / 1 / U cos[theta] 1 / U sin[theta]
a0 = ----- | Ud[theta], a1 = ---- | ------------ d[theta], b1 = ---- | ------------ d[theta],
2[pi] _/ [pi] _/ a [pi] _/ a
_ _
1 / U cos 2[theta] 1 / U sin 2[theta]
a2 = ---- | -------------- d[theta], b2 = ---- | -------------- d[theta].
[pi] _/ a^2 [pi] _/ a^2
In this series the terms of order n are sums, with real coefficients,
of the various integral polynomials of dimension n which satisfy the
equation (Pd)^2[psi]/(Pd)x^2 + (Pd)^2[psi]/(Pd)y^2; the series is thus
the real part of a power series in z, and is capable of
differentiation and integration within its region of convergence.
Conversely we may suppose a function, P, defined for the interior of a
finite region R of the plane of the real variables x, y, capable of
expression about any interior point x0, y0 of this region by a power
series in x - x0, y - y0, with real coefficients, these various series
being obtainable from one of them by continuation. For any region R0
interior to the region specified, the radii of convergence of these
power series will then have a lower limit greater than zero, and hence
a finite number of these power series suffice to specify the function
for all points interior to R0. Each of these series, and therefore the
function, will be differentiable; suppose that at all points of R0 the
function satisfies the equation
(Pd)^2P (Pd)P^2
------- + ------- = 0,
(Pd)x^2 (Pd)y^2
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
these quantities cannot therefore have a real ratio, for else, being
periods of a monogenic function, they would, as we have previously
seen, be each integral multiples of another period; there would then
be a closed path for (x, y), starting from an arbitrary point (x0,
y0), other than one enclosing two of the points (e1, 0), (e2, 0), (e3,
0), ([oo], [oo]), which leads back to the initial point (x0, y0),
which is impossible. On the whole, therefore, it appears that the
function [phi](u) agrees with the function RN(u) previously discussed,
and the discussion of the elliptic integrals can be continued in the
manner given under S 14, _Doubly Periodic Functions_.
S 21. _Modular Functions._--One result of the previous theory is the
remarkable fact that if
_ _
/ [oo] dx / [oo] dx
[omega] = 2| --, [omega]' = 2 | --,
_/ e1 y _/ e3 y
where y^2 = 4(x - e1) (x - e2) (x - e3), then we have
e1 = (1/2[omega])^(-2) + [Sigma]' {[(m + 1/2)[omega] + m'[omega]']^(-2) -
[m[omega] + m'[omega]']^(-2)},
and a similar equation for e3, where the summation refers to all integer
values of m and m' other than the one pair m = 0, m' = 0. This, with
similar results, has led to the consideration of functions of the
complex ratio [omega]'/[omega].
It is easy to see that the series for RN(u), u^(-2) + [Sigma] [(u +
m[omega] + m'[omega]')^2-(m[omega] + m'[omega]')^2], is unaffected by
replacing [omega], [omega]' by two quantities [Omega], [Omega]' equal
respectively to p[omega] + q[omega]', p'[omega]' + q'[omega]', where
p, q, p', q' are any integers for which pq' - p'q = [+-]1; further it
can be proved that all substitutions with integer coefficients [Omega]
= p[omega] + q[omega]', [Omega]' = p'[omega] + q'[omega]', wherein pq'
- p'q = 1, can be built up by repetitions of the two particular
substitutions ([Omega] = -[omega]', [Omega]' = [omega]), ([Omega] =
[omega], [Omega]' = [omega] + [omega]'). Consider the function of the
ratio [omega]'/[omega] expressed by
h = -RN (1/2[omega]') / RN(1/2[omega]);
it is at once seen from the properties of the function RN(u) that by
the two particular substitutions referred to we obtain the
corresponding substitutions for h expressed by
h' = 1/h, h' = 1 - h;
thus, by all the integer substitutions [Omega] = p[omega] + q[omega]',
[Omega]' = p'[omega] + q'[omega]', in which pq' - p'q = 1, the
function h can only take one of the six values h, 1/h, 1 - h, 1/(1 -
h), h/(h - 1), (h - 1)/h, which are the roots of an equation in
[theta],
(1 - [theta] + [theta]^2)^3 (1 - h + h^2)^3
--------------------------- = ---------------;
[theta]^2(1 - [theta])^2 h^2(1 - h)^2
the function of [tau], = [omega]'/[omega], expressed by the right
side, is thus unaltered by every one of the substitutions [tau]' = (p'
+ q'[tau] / p + q[tau]), wherein p, q, p', q' are integers having pq'
- p'q = 1. If the imaginary part [sigma], of [tau], which we may write
[tau] = [rho] + i[sigma], is positive, the imaginary part of [tau]',
which is equal to [sigma](pq' - p'q)/[(p + q[rho])^2 + q^2[sigma]^2],
is also positive; suppose [sigma] to be positive; it can be shown that
the upper half of the infinite plane of the complex variable [tau] can
be divided into regions, all bounded by arcs of circles (or straight
lines), no two of these regions overlapping, such that any
substitution of the kind under consideration, [tau]' = (p' +
q'[tau])/(p + q[tau]) leads from an arbitrary point [tau], of one of
these regions, to a point [tau]' of another; taking [tau] = [rho] +
i[sigma], one of these regions may be taken to be that for which -1/2
< [rho] < 1/2, [rho]^2 + [sigma]^2 > 1, together with the points for
which [rho] is negative on the curves limiting this region; then every
other region is obtained from this so-called fundamental region by one
and only one of the substitutions [tau] = (p' + q'[tau])/(p + q[tau]),
and hence by a definite combination of the substitutions [tau]' =
-1/[tau], [tau]' = 1 + [tau]. Upon the infinite half plane of [tau],
the function considered above,
4 [RN^2(1/2[omega]) + RN(z(1/2[omega]) RN(1/2[omega]') + RN^2 (1/2[omega]')]^3
z([tau]) = -- ----------------------------------------------------------------------------
27 RN^2(1/2[omega]) RN^2(1/2[omega]') [RN(1/2[omega]) + RN(1/2[omega]')]^2
is a single valued monogenic function, whose only essential
singularities are the points [tau]' = (p' + q'[tau])/(p + q[tau]) for
which [tau] = [oo], namely those for which [tau]' is any real rational
value; the real axis is thus a line over which the function z([tau])
cannot be continued, having an essential singularity in every arc of
it, however short; in the fundamental region, z([tau]) has thus only
the single essential singularity, r = [rho] + i[sigma], where [sigma]
= [oo]; in this fundamental region z([tau]) takes any assigned complex
value just once, the relation z([tau]') = z([tau]) requiring, as can
be shown, that [tau]' is of the form (p' + q'[tau])/(p + q[tau]), in
which p, q, p', q' are integers with pq' - p'q = 1; the function
z([tau]) has thus a similar behaviour in every other of the regions.
The division of the plane into regions is analogous to the division of
the plane, in the case of doubly periodic functions, into
parallelograms; in that case we considered only functions without
essential singularities, and in each of the regions the function
assumed every complex value twice, at least. Putting, as another
function of [tau], J([tau]) = z([tau])[z([tau]) - 1], it can be shown
that J([tau]) = 0 for [tau] = exp (2/3[pi]i), that J([tau]) = 1 for
[tau] = i, these being values of [tau] on the boundary of the
fundamental region; like z([tau]) it has an essential singularity for
[tau] = [rho] + i[sigma], [sigma] = + [oo]. In the theory of linear
differential equations it is important to consider the inverse
function [tau](J); this is infinitely many valued, having a cycle of
three values for circulation of J about J = 0 (the circuit of this
point leading to a linear substitution for [tau] of period 3, such as
[tau]' = -(1 + [tau])^(-1)), having a cycle of two values about J = 1
(the circuit leading to a linear substitution for [tau] of period 2,
such as [tau]' = -[tau]^(-1)), and having a cycle of infinitely many
values about J = [oo] (the circuit leading to a linear substitution
for [tau] which is not periodic, such as [tau]' = 1 + [tau]). These
are the only singularities for the function [tau](J). Each of the
functions
_ _
| RN(1/2[omega]) + 2RN(1/2[omega]') |^(1/8)
[J([tau])]^(1/3), [J([tau])-1]^1/2, | - --------------------------------- | ,
|_ RN(1/2[omega]) - RN(1/2)[omega]') _|
beside many others (see below), is a single valued function of [tau],
and is expressible without ambiguity in terms of the single valued
function of [tau],
/i[pi][tau]\ [oo]
[eta]([tau]) = exp( ---------- ) [Pi] [1-exp (2i[pi]n[tau])],
\ 12 / n=1
/i[pi][tau]\ _[oo]
= exp(----------- ) \ (-1)^m exp [(3m^2 + m) i[pi][tau]].
\ 12 / /_
m = -[oo]
It should be remarked, however, that [eta]([tau]) is not unaltered by
all the substitutions we have considered; in fact
[eta](-[tau]^(-1)) = (-i[tau])1/2[eta]([tau]), [eta](1 + [tau]) =
exp (1/12 i[pi]) [eta]([tau]).
The aggregate of the substitutions [tau]' = (p' + q'[tau])/(p +
q[tau]), wherein p, q, p', q' are integers with pq' - p'q = 1,
represents a _Group_; the function J([tau]), unaltered by all these
substitutions, is called a _Modular Function_. More generally any
function unaltered by all the substitutions of a group of linear
substitutions of its variable is called an _Automorphic Function_. A
rational function, of its variable h, of this character, is the
function (1 - h + h^2)^3 h^(-2)(1 - h)^(-2) presenting itself
incidentally above; and there are other rational functions with a
similar property, the group of substitutions belonging to any one of
these being, what is a very curious fact, associable with that of the
rotations of one of the regular solids, about an axis through its
centre, which bring the solid into coincidence with itself. Other
automorphic functions are the double periodic functions already
discussed; these, as we have seen, enable us to solve the algebraic
equation y^2 = 4x^3 - g2x - g3 (and in fact many other algebraic
equations, see below, under S 23, _Geometrical Applications of
Elliptic Functions_) in terms of single valued functions x = RN(u), y
= -RN'(u). A similar utility, of a more extended kind, belongs to
automorphic functions in general; but it can be shown that such
functions necessarily have an infinite number of essential
singularities except for the simplest cases.
The modular function J([tau]) considered above, unaltered by the group
of linear substitutions [tau]' = (p' + q'[tau]) / (p + q[tau]), where
p, q, p', q' are integers with pq' - p'q = 1, may be taken as the
independent variable x of a differential equation of the third order,
of the form
s''' 3 /s''\^2 1 - a^2 1 - [beta]^2 [alpha]^2 + [beta]^2 - [gamma]^2 - 1
---- - -- ( --- ) = ---------- + ------------ + ------------------------------------,
s' 2 \ s'/ 2(x - 1)^2 2x^2 2x(x - 1)
where s' = ds/dx, &c., of which the dependent variable s is equal to
[tau]. A differential equation of this form is satisfied by the
quotient of two independent integrals of the linear differential
equation of the second order satisfied by the hypergeometric
functions. If the solution of the differential equation for s be
written s([alpha], [beta], [gamma], x), we have in fact [tau] = s(1/2,
1/3, 0, J). If we introduce also the function of [tau] given by
2RN (1/2[omega]') + f V(1/2[omega])
[lambda] = -----------------------------------,
RN(1/2[omega]') - RN(1/2[omega])
we similarly have [tau] = s(0, 0, 0, [lambda]); this function [lambda]
is a single valued function of [tau], which is also a modular
function, being unaltered by a group of integral substitutions also of
the form [tau]' = (p' + q'[tau])/(p + q[tau]), with pq' - p'q = 1, but
with the restriction that p' and q are even integers, and therefore p
and q' are odd integers. This group is thus a subgroup of the general
modular group, and is in fact of the kind called a self-conjugate
subgroup. As in the general case this subgroup is associated with a
subdivision of the plane into regions of which any one is obtained
from a particular region, called the fundamental region, by a
particular one of the substitutions of the subgroup. This fundamental
region, putting [tau] = [rho] + i[sigma], may be taken to be that
given by -1 < [rho] < 1, ([rho] + 1/2)^2 + [sigma]^2 > 1/4, ([rho] -
1/2)^2 + [sigma]^2 > 1/4, and is built up of six of the regions which
arose for the general modular group associated with J([tau]). Within
this fundamental region, [lambda] takes every complex value just once,
except the values [lambda] = 0, 1, [oo], which arise only at the
angular points [tau] = 0, [tau] = [oo], [tau] = -1 and the equivalent
point [tau] = 1; these angular points are essential singularities for
the function [lambda]([tau]). For [lambda]([tau]) as for J([tau]), the
region of existence is the upper half plane of [tau], there being an
essential singularity in every length of the real axis, however short.
If, beside the plane of [tau], we take a plane to represent the values
of [lambda], the function [tau] = s(0, 0, 0, [lambda]) being
considered thereon, the values of [tau] belonging to the interior of
the fundamental region of the [tau]-plane considered above, will
require the consideration of the whole of the [lambda]-plane taken
once with the exception of the portions of the real axis lying between
-[oo] and 0 and between 1 and + [oo], the two sides of the first
portion corresponding to the circumferences of the [tau]-plane
expressed by ([rho] + 1/2)^2 + [sigma]^2 = 1/4, ([rho] - 1/2)^2 +
[sigma]^2 = 1/4, while the two sides of the latter portion, for which
[lambda] is real and > 1, correspond to the lines of the [tau]-plane
expressed by [rho] = [+-]1. The line for which [lambda] is real,
positive and less than unity corresponds to the imaginary axis of the
[tau]-plane, lying in the interior of the fundamental region. All the
values of [tau] = s(0, 0, 0, [lambda]) may then be derived from those
belonging to the fundamental region of the [tau]-plane by making
[lambda] describe a proper succession of circuits about the points
[lambda] = 0, [lambda] = 1; any such circuit subjects [tau] to a
linear substitution of the subgroup of [tau] considered, and
corresponds to a change of [tau] from a point of the fundamental
region to a corresponding point of one of the other regions.
S 22. _A Property of Integral Functions deduced from the Theory of
Modular Functions._--Consider now the function exp(z), for finite values
of z; for such values of z, exp(z) never vanishes, and it is impossible
to assign a closed circuit for z in the finite part of the plane of z
which will make the function [lambda] = exp(z) pass through a closed
succession of values in the plane of [lambda] having [lambda] = 0 in its
interior; the function s[0, 0, 0, exp(z)], however z vary in the finite
part of the plane, will therefore never be subjected to those linear
substitutions imposed upon s(0, 0, 0, [lambda]) by a circuit of [lambda]
about [lambda] = 0; more generally, if [phi](z) be an integral function
of z, never becoming either zero or unity for finite values of z, the
function [lambda] = [phi](z), however z vary in the finite part of the
plane, will never make, in the plane of [lambda], a circuit about either
[lambda] = 0 or [lambda] = 1, and s(0, 0, 0, [lambda]), that is s[0, 0,
0, [phi](z)], will be single valued for all finite values of z; it will
moreover remain finite, and be monogenic. In other words, s[0, 0, 0,
[phi](z)] is also an integral function--whose imaginary part, moreover,
by the property of s(0, 0, 0, [lambda]), remains positive for all finite
values of z. In that case, however, exp{is[0, 0, 0, [phi](z)]} would
also be an integral function of z with modulus less than unity for all
finite values of z. If, however, we describe a circle of radius R in the
z plane, and consider the greatest value of the modulus of an integral
function upon this circle, this certainly increases indefinitely as R
increases. We can infer therefore that _an integral function [phi](z)
which does not vanish for any finite value of z, takes the value unity
and hence_ (by considering the function A^(-1)[phi](z)) _takes every
other value for some definite value of z_; or, an integral function for
which both the equations [phi](z) = A, [phi](z) = B are unsatisfied by
definite values of z, does not exist, A and B being arbitrary constants.
A similar theorem can be proved in regard to the values assumed by the
function [phi](z) for points z of modulus greater than R, however
great R may be, also with the help of modular functions. In general
terms it may be stated that it is a very exceptional thing for an
integral function not to assume every complex value an infinite number
of times.
Another application of modular functions is to prove that the function
s([alpha], [beta], [gamma], [lambda]) is a single valued function of
[tau] = s(0, 0, 0, [lambda]); for, putting [tau]' = ([tau] - i)/([tau]
+ i), the values of [tau]' which correspond to the singular points
[lambda] = 0, 1, [oo] of s([alpha], [beta], [gamma], [lambda]), though
infinite in number, all lie on the circumference of the circle
|[tau]'| = 1, within which therefore s([alpha], [beta], [gamma], x) is
expressible in a form [Sigma] [n = 0 to [oo]] a_n[tau]'^n. More
generally any monogenic function of [lambda] which is single valued
save for circuits of the points [lambda] = 0, 1, [oo], is a single
valued function of [tau] = s(0, 0, 0, [lambda]). Identifying [lambda]
with the square of the modulus in Legendre's form of the elliptical
integral, we have [tau] = iK'/K, where
_ _
/1 dt /1 dt
K = | ----------------------------------, K' = | ---------------------------------------;
_/0 [root][1 - t^2] [1 - [lambda]t^2] _/0 [root][1 - t^2] [1 - (1 - [lambda])t^2]
functions such as [lambda]^1/4, (1 - [lambda])^1/4, [[lambda](1 -
[lambda])]^1/4, which have only [lambda] = 0, 1, [oo] as singular
points, were expressed by Jacobi as power series in q =
e^(i[pi][tau]), and therefore, at least for a limited range of values
of [tau], as single valued functions of [tau]; it follows by the
theorem given that any product of a root of [lambda] and a root of 1 -
[lambda] is a single valued function of [tau]. More generally the
differential equation
d^2y dy
x(1 - x) ---- + [[gamma] - ([alpha] + [beta] + 1)x] -- -[alpha][beta][gamma] = 0
dx^2 dx
may be solved by expressing both the independent and dependent
variables as single valued functions of a single variable [tau], the
expression for the independent variable being x = [lambda]([tau]).
S 23. _Geometrical Applications of Elliptic Functions._--Consider any
irreducible algebraic equation rational in x, y, f(x, y) = 0, of such a
form that the equation represents a plane curve of order n with 1/2n(n
-3) double points; taking upon this curve n-3 arbitrary fixed points,
draw through these and the double points the most general curve of order
n -2; this will intersect [f] in n(n - 2) - n(n - 3) - (n - 3) = 3
other points, and will contain homogeneously at least 1/2(n - 1)n -
1/2n(n -3) - (n - 3) = 3 arbitrary constants, and so will be of the form
[lambda][phi] + [lambda]1[phi]1 + [lambda]2[phi]2 + ... = 0, wherein
[lambda]3, [lambda]4, ... are in general zero. Put now [xi] =
[phi]1/[phi], [eta] = [phi]2/[phi] and eliminate x, y between these
equations and [f](x, y) = 0, so obtaining a rational irreducible
equation F([xi], [eta]) = 0, representing a further plane curve. To any
point (x, y) of [f] will then correspond a definite point ([xi], [eta])
of F.
For a general position of (x, y) upon [f] the equations [phi]1(x',
y')/[phi](x', y') = [phi]1(x, y)/[phi](x, y), [phi]2(x', y')/[phi](x',
y') = [phi]2(x, y)/[phi](x, y), subject to [f](x', y') = 0, will have
the same number of solutions (x', y'); if their only solution is x' =
x, y' = y, then to any position ([xi],[eta]) of F will conversely
correspond only one position (x, y) of [f]. If these equations have
another solution beside (x, y), then any curve [lambda][phi] +
[lambda]1[phi]1 + [lambda]2[phi]2 = 0 which passes (through the double
points of [f] and) through the n - 2 points of [f] constituted by the
fixed n-3 points and a point (x0, y0), will necessarily pass through a
further point, say (x0', y0'), and will have only one further
intersection with [f]; such a curve, with the n - 2 assigned points,
beside the double points, of [f], will be of the form [mu][psi] +
[mu]1[psi]1 + ... = 0, where [mu]2, [mu]3, ... are generally zero;
considering the curves [psi] + t[psi]1 = 0, for variable t, one of
these passes through a further arbitrary point of [f], by choosing t
properly, and conversely an arbitrary value of t determines a single
further point of [f]; the co-ordinates of the points of [f] are thus
rational functions of a parameter t, which is itself expressible
rationally by the co-ordinates of the point; it can be shown
algebraically that such a curve has not 1/2(n - 3)n but 1/2(n - 3)n +
1 double points. We may therefore assume that to every point of F
corresponds only one point of [f], and there is a birational
transformation between these curves; the coefficients in this
transformation will involve rationally the co-ordinates of the n-3
fixed points taken upon [f], that is, at the least, by taking these to
be consecutive points, will involve the co-ordinates of one point of
[f], and will not be rational in the coefficients of [f] unless we can
specify a point of [f] whose co-ordinates are rational in these. The
curve F is intersected by a straight line a[xi] + b[eta] + c = 0 in as
many points as the number of unspecified intersections of [f] with
a[phi] + b[phi]1 + c[phi]2 = 0, that is, 3; or F will be a cubic
curve, without double points.
Such a cubic curve has at least one point of inflection Y, and if a
variable line YPQ be drawn through Y to cut the curve again in P and
Q, the locus of a point R such that YR is the harmonic mean of YP and
YQ, is easily proved to be a straight line. Take now a triangle of
reference for homogeneous co-ordinates XYZ, of which this straight
line is Y = 0, and the inflexional tangent at Y is Z = 0; the equation
of the cubic curve will then be of the form
ZY^2 = aX^3 + bX^2Z + cXZ^2 + dZ^3;
by putting X equal to [lambda]X + [mu]Z, that is, choosing a suitable
line through Y to be X = 0, and choosing [lambda] properly, this is
reduced to the form
ZY^2 = 4X^3 - g2XZ^2 -g3Z^3,
of which a representation is given, valid for every point, in terms of
the elliptic functions RN(u), RN'(u), by taking X = ZRN(u), Y =
ZRN'(u). The value of u belonging to any point is definite save for
sums of integral multiples of the periods of the elliptic functions,
being given by
_
/ (x) ZdX - XdZ
u = | ---------,
_/ ([oo]) ZY
where ([oo]) denotes the point of inflection.
It thus appears that the co-ordinates of any point of a plane curve,
[f], of order n with 1/2(n - 3)n double points are expressible as
elliptic functions, there being, save for periods, a definite value of
the argument u belonging to every point of the curve. It can then be
shown that if a variable curve, [phi], of order m be drawn, passing
through the double points of the curve, the values of the argument u
at the remaining intersections of [phi] with [f], have a sum which is
unaffected by variation of the coefficients of [phi], save for
additive aggregates of the periods. In virtue of the birational
transformation this theorem can be deduced from the theorem that if
any straight line cut the cubic y^2 = 4x^3 - g2x - g3, in points (u1),
(u2), (u3), the sum u1 + u2 + u3 is zero, or a period; or the general
theorem is a corollary from Abel's theorem proved under S 17,
_Integrals of Algebraic Functions_. To prove the result directly for
the cubic we remark that the variation of one of the intersections (x,
y) of the cubic with the straight line y = mx + n, due to a variation
[delta]m, [delta]n in m and n, is obtained by differentiation of the
equation for the three abscissae, namely the equation
F(x) = 4x^3 - g2x - g3 - (mx + n)^2 = 0,
and is thus given by
dx x[delta]m + [delta]n
-- = 2 --------------------,
y F'(x)
and the sum of three such fractions as that on the right for the three
roots of F(x) = 0 is zero; hence u1 + u2 + u3 is independent of the
straight line considered; if in particular this become the inflexional
tangent each of u1, u2, u3 vanishes. It may be remarked in passing
that x1 + x2 + x3 = 1/4m^2, and hence is 1/4{(y1 - y2)/(x1 - x2)}^2; so
that we have another proof of the addition equation for the function
RN(u). From this theorem for the cubic curve many of its geometrical
properties, as for example those of its inflections, the properties of
inscribed polygons, of the three kinds of corresponding points, and
the theory of residuation, are at once obvious. And similar results
hold for the curve of order n with 1/2(n - 3)n double points.
S 24. _Integrals of Algebraic Functions in Connexion with the Theory of
Plane Curves._--The developments which have been explained in connexion
with elliptic functions may enable the reader to appreciate the vastly
more extensive theory similarly arising for any algebraical
irrationality, [f](x, y) = o.
The algebraical integrals [int] R(x, y)dx associated with this may as
before be divided into those of the _first kind_, which have no
infinities, those of the _second kind_, possessing only algebraical
infinities, and those of the _third kind_, for which logarithmic
infinities enter. Here there is a certain number, p, greater than
unity, of linearly independent integrals of the first kind; and this
number p is unaltered by any birational transformation of the
fundamental equation [f](x, y) = 0; a rational function can be
constructed with poles of the first order at p + 1 arbitrary positions
(x, y), satisfying [f](x, y) = 0, but not with a fewer number unless
their positions are chosen properly, a property we found for the case
p = 1; and p is the number of linearly independent curves of order n-3
passing through the double points of the curve of order n expressed by
[f](x, y) = 0. Again any integral of the second kind can be expressed
as a sum of p integrals of this kind, with poles of the first order at
arbitrary positions, together with rational functions and integrals of
the first kind; and an integral of the second kind can be found with
one pole of the first order of arbitrary position, and an integral of
the third kind with two logarithmic infinities, also of arbitrary
position; the corresponding properties for p = 1 are proved above.
There is, however, a difference of essential kind in regard to the
inversion of integrals of the first kind; if u = [int] R(x, y)dx be
such an integral, it can be shown, in common with all algebraic
integrals associated with [f](x, y) = 0, to have 2p linearly
independent additive constants of indeterminateness; the upper limit
of the integral cannot therefore, as we have shown, be a single valued
function of the value of the integral. The corresponding theorem, if
[int] R_i(x, y)dx denote one of the integrals of the first kind, is
that the p equations
_ _
/ /
| R_i (x1, y1)dx1 + ... + | Ri (x_p, y_p) dx_p = u_i,
_/ _/
determine the rational symmetric functions of the p positions (x1,
y1), ... (x_p, y_p) as single valued functions of the p variables, u1,
... u_p. It is thus necessary to enter into the theory of functions of
several independent variables; and the equation [f](x, y) = 0 is thus
not, in this way, capable of solution by single valued functions of
one variable. That solution in fact is to be sought with the help of
automorphic functions, which, however, as has been remarked, have, for
p > 1, an infinite number of essential singularities.
S 25. _Monogenic Functions of Several Independent Variables._--A
monogenic function of several independent complex variables u_i, ... u_p
is to be regarded as given by an aggregate of power series all
obtainable by continuation from any one of them in a manner analogous to
that before explained in the case of one independent variable. The
singular points, defined as the limiting points of the range over which
such continuation is possible, may either be _poles_, or _polar points
of indetermination_, or _essential singularities_.
A pole is a point (u1^(0), ... u_p^(0)) in the neighbourhood of which
the function is expressible as a quotient of converging power series
in u1 - u1^(0) ... u_p - u_p^(0); of these the denominator series D
must vanish at (u1^(0), ... u_p^(0)), since else the fraction is
expressible as a power series and the point is not a singular point,
but the numerator series N must not also vanish at (u1^(0), ...
u_p^(0)), or if it does, it must be possible to write D = M0, N = MN0,
where M is a converging power series vanishing at (u1^(0), ...
u_p^(0)), and N0 is a converging power series, in (u1 - u1^(0) ... u_p
- u_p^(0)), not so vanishing. A polar point of indetermination is a
point about which the function can be expressed as a quotient of two
converging power series, both of which vanish at the point. As in such
a simple case as (Ax + By) / (ax + by), about x = 0, y = 0, it can be
proved that then the function can be made to approach to any
arbitrarily assigned value by making the variables u1, ... u_p
approach to u1^(0), ... u_p^(0) by a proper path. It is the necessary
existence of such polar points of indetermination, which in case p > 2
are not merely isolated points, which renders the theory essentially
more difficult than that of functions of one variable. An essential
singularity is any which does not come under one of the two former
descriptions and includes very various possibilities. A point at
infinity in this theory is one for which any one of the variables u1,
... u_p is indefinitely great; such points are brought under the
preceding definitions by means of the convention that for u_i^(0) =
[oo], the difference u_i - u_i^(0) is to be understood to stand for
u_i^(-1) . This being so, a single valued function of u1, ... u_p
without essential singularities for infinite or finite values of the
variables can be shown, by induction, to be, as in the case of p = 1,
necessarily a rational function of the variables. A function having no
singularities for finite values of all the variables is as before
called an integral function; it is expressible by a power series
converging for all finite values of the variables; a single valued
function having for finite values of the variables no singularities
other than poles or polar points of indetermination is called a
meromorphic function; as for p = 1 such a function can be expressed as
a quotient of two integral functions having no common zero point other
than the points of indetermination of the function; but the proof of
this theorem is difficult.
The single valued functions which occur, as explained above, in the
inversion of algebraic integrals of the first kind, for p > 1, are
meromorphic. They must also be periodic, unaffected that is when the
variables u1, ... u_p are _simultaneously_ increased each by a proper
constant, these being the additive constants of indeterminateness for
the p integrals [int] R_i(x, y)dx arising when (x, y) makes a closed
circuit, the same for each integral. The theory of such single valued
meromorphic periodic functions is simpler than that of meromorphic
functions of several variables in general, as it is sufficient to
consider only finite values of the variables; it is the natural
extension of the theory of doubly periodic functions previously
discussed. It can be shown to reduce, though the proof of this
requires considerable developments of which we cannot speak, to the
theory of a single integral function of u1, ... u_p, called the _Theta
Function_. This is expressible as a series of positive and negative
integral powers of quantities exp (c1u1), exp (c2u2), ... exp (c_p
u_p), wherein c1, ... c_p are proper constants; for p = 1 this theta
function is essentially the same as that above given under a different
form (see S 14, _Doubly Periodic Functions_), the function [sigma](u).
In the case of p = 1, all meromorphic functions periodic with the same
two periods have been shown to be rational functions of two of them
connected by a single algebraic equation; in the same way all
meromorphic functions of p variables, periodic with the same sets of
simultaneous periods, 2p sets in all, can be shown to be expressible
rationally in terms of p + 1 such periodic functions connected by a
single algebraic equation. Let x1, ... x_p, y denote p + 1 such
functions; then each of the partial derivatives dx_i/(Pd)u_i will
equally be a meromorphic function of the same periods, and so
expressible rationally in terms of x1, ... x_p, y; thus there will
exist p equations of the form
dx_i = R1 du1 + ... + R_p du_p,
and hence p equations of the form
du_i = H_(i, 1)dx1 + ... + H_(i, p)dx_p,
wherein H_(i, j) are rational functions of x1, ... x_p, y, these being
connected by a fundamental algebraic (rational) equation, say [f](x1,
... x_p, y) = 0. This then is the generalized form of the
corresponding equation for p = 1.
S 26. _Multiply-Periodic Functions and the Theory of Surfaces._--The
theory of algebraic integrals [int] R(x, y)dx, wherein x, y are
connected by a rational equation [f](x, y) = 0, has developed
concurrently with the theory of algebraic curves; in particular the
existence of the number p invariant by all birational transformations is
one result of an extensive theory in which curves capable of birational
correspondence are regarded as equivalent; this point of view has made
possible a general theory of what might otherwise have remained a
collection of isolated theorems.
In recent years developments have been made which point to a similar
unity of conception as possible for surfaces, or indeed for algebraic
constructs of any number of dimensions. These developments have been
in two directions, at first followed independently, but now happily
brought into the most intimate connexion. On the analytical side, E.
Picard has considered the possibility of classifying integrals of the
form [int](Rds + Sdy), belonging to a surface [f](x, y, z) = 0,
wherein R and S are rational functions of x, y, z, according as they
are (1) everywhere finite, (2) have poles, which then lie along curves
upon the surface, or (3) have logarithmic infinities, also then lying
along curves, and has brought the theory to a high degree of
perfection. On the geometrical side A. Clebsch and M. Noether, and
more recently the Italian school, have considered the geometrical
characteristics of a surface which are unaltered by birational
transformation. It was first remarked that for surfaces of order n
there are associated surfaces of order n-4, having properties in
relation thereto analogous to those of curves of order n-3 for a plane
curve of order n; if such a surface [f](x, y, z) = 0 have a double
curve with triple points triple also for the surface, and [phi](x, y,
z) = 0 be a surface of order n - 4 passing through the double curve,
the double integral
_ _
/ / [phi] dx dy
| | -----------
_/ _/ (Pd)f/(Pd)z
is everywhere finite; and, the most general everywhere finite integral
of this form remains invariant in a birational transformation of the
surface [f], the theorem being capable of generalization to algebraic
constructs of any number of dimensions. The number of linearly
independent surfaces of order n - 4, possessing the requisite
particularity in regard to the singular lines and points of the
surface, is thus a number invariant by birational transformation, and
the equality of these numbers for two surfaces is a necessary
condition of their being capable of such transformation. The number of
surfaces of order m having the assigned particularity in regard to the
singular points and lines of the fundamental surface can be given by a
formula for a surface of given singularity; but the value of this
formula for m = n - 4 is not in all cases equal to the actual number
of surfaces of order n - 4 with the assigned particularity, and for a
cone (or ruled surface) is in fact negative, being the negative of the
deficiency of the plane section of the cone. Nevertheless this number
for m = n - 4 is also found to be invariant for birational
transformation. This number, now denoted by p_a, is then a second
invariant of birational transformation. The former number, of actual
surfaces of order n - 4 with the assigned particularity in regard to
the singularities of the surface, is now denoted by p_g. The
difference p_g - p_a, which is never negative, is a most important
characteristic of a surface. When it is zero, as in the case of the
general surface of order n, and in a vast number of other ordinary
cases, the surface is called regular.
On a plane algebraical curve we may consider linear series of sets of
points, obtained by the intersection with it of curves [lambda][phi] +
[lambda]1[phi]1 + ... = 0, wherein [lambda], [lambda]1, ... are
variable coefficients; such a series consists of the sets of points
where a rational function of given poles, belonging to the construct
[f](x, y) = 0, has constant values. And we may consider series of sets
of points determined by variable curves whose coefficients are
algebraical functions, not necessarily rational functions, of
parameters. Similarly on a surface we may consider linear systems of
curves, obtained by the intersection with the given surface of
variable surfaces [lambda][phi] + [lambda]1[phi]1 + ... = 0, and may
consider algebraic systems, of which the individual curve is given by
variable surfaces whose coefficients are algebraical, not necessarily
rational, functions of parameters. Of a linear series upon a plane
curve there are two numbers manifestly invariant in birational
transformation, the _order_, which is the number of points forming a
set of the series, and the _dimension_, which is the number of
parameters [lambda]1/[lambda], [lambda]2/[lambda], ... entering
linearly in the equation of the series. The series is _complete_ when
it is not contained in a series of the same order but of higher
dimension. So for a linear system of curves upon a surface, we have
three invariants for birational transformation; the _order_, being in
the number of variable intersections of two curves of the system, the
_dimension_, being the number of linear parameters [lambda]1/[lambda],
[lambda]2/[lambda], ... in the equation for the system, and the
_deficiency_ of the individual curves of the system. Upon any curve of
the linear system the other curves of the system define a linear
series, called the _characteristic_ series; but even when the linear
system is complete, that is, not contained in another linear system of
the same order and higher dimension, it does not follow that the
characteristic series is complete; it may be contained in a series
whose dimension is greater by p_g - p_a than its own dimension. When
this is so it can be shown that the linear system of curves is
contained in an algebraic system whose dimension is greater by p_g -
p_a than the dimension of the linear system. The extra p = p_g - p_a
variable parameters so entering may be regarded as the independent
co-ordinates of an algebraic construct [f](y, x1, ... x_p) = 0; this
construct has the property that its co-ordinates are single valued
meromorphic functions of p variables, which are periodic, possessing
2p systems of periods; the p variables are expressible in the forms
_
/
u_i = | R1(x, y) dx1 + ... + R_p(x, y) dx_p,
_/
wherein R_i(x, y) denotes a rational function of x1, ... x_p and y.
The original surface has correspondingly p integrals of the form
[int](R dx + S dy), wherein R, S are rational in x, y, z, which are
everywhere finite; and it can be shown that it has no other such
integrals. From this point of view, then, the number p, = p_g - p_a
is, for a surface, analogous to the deficiency of a plane curve;
another analogy arises in the comparison of the theorems: for a plane
curve of zero deficiency there exists no algebraic series of sets of
points which does not consist of sets belonging to a linear series;
for a surface for which p_g - p_a = 0 there exists no algebraic system
of curves not contained in a linear system.
But whereas for a plane curve of deficiency zero, the co-ordinates of
the points of the curve are rational functions of a single parameter,
it is not necessarily the case that for a surface having p_g - p_a = 0
the co-ordinates of the points are rational functions of two
parameters; it is necessary that p_g - p_a = 0, but this is not
sufficient. For surfaces, beside the p_g linearly independent surfaces
of order n - 4 having a definite particularity at the singularities of
the surface, it is useful to consider surfaces of order k(n - 4), also
having each a definite particularity at the singularities, the number
of these, not containing the original surface as component, which are
linearly independent, is denoted by P_k. It can then be stated that a
sufficient condition for a surface to be rational consists of the two
conditions p_a = 0, P2 = 0. More generally it becomes a problem to
classify surfaces according to the values of the various numbers which
are invariant under birational transformation, and to determine for
each the simplest form of surface to which it is birationally
equivalent. Thus, for example, the hyperelliptic surface discussed by
Humbert, of which the co-ordinates are meromorphic functions of two
variables of the simplest kind, with four sets of periods, is
characterized by p_g = 1, p_a = -1; or again, any surface possessing a
linear system of curves of which the order exceeds twice the
deficiency of the individual curves diminished by two, is reducible by
birational transformation to a ruled surface or is a rational surface.
But beyond the general statement that much progress has already been
made in this direction, of great interest to the student of the theory
of functions, nothing further can be added here.
BIBLIOGRAPHY.--The learner will find a lucid introduction to the
theory in E. Goursat, _Cours d'analyse mathematique_, t. ii. (Paris,
1905), or, with much greater detail, in A.R. Forsyth, _Theory of
Functions of a Complex Variable_ (2nd ed., Cambridge, 1900); for
logical rigour in the more difficult theorems, he should consult W.F.
Osgood, _Lehrbuch der Functionentheorie_, Bd. i. (Leipzig, 1906-1907);
for greater precision in regard to the necessary quasi-geometrical
axioms, beside the indications attempted here, he should consult W.H.
Young, _The Theory of Sets of Points_ (Cambridge, 1906), chs.
viii.-xiii., and C. Jordan, _Cours d'analyse_, t. i. (Paris, 1893),
chs. i., ii.; a comprehensive account of the _Theory of Functions of
Real Variables_ is by E.W. Hobson (Cambridge, 1907). Of the theory
regarded as based after Weierstrass upon the theory of power series,
there is J. Harkness and F. Morley, _Introduction to the Theory of
Analytic Functions_ (London, 1898), an elementary treatise; for the
theory of the convergence of series there is also T.J. I'A. Bromwich,
_An Introduction to the Theory of Infinite Series_ (London, 1908); but
the student should consult the collected works of Weierstrass (Berlin,
1894 ff.), and the writings of Mittag-Leffler in the early volumes of
the _Acta mathematica_; earlier expositions of the theory of functions
on the basis of power series are in C. Meray, _Lecons nouvelles sur
l'analyse infinitesimale_ (Paris, 1894), and in Lagrange's books on
the Theory of Functions. An account of the theory of potential in its
applications to the present theory is found in most treatises; in
particular consult E. Picard, _Traite d'analyse_, t. ii. (Paris,
1893). For elliptic functions there is an introductory book, P. Appell
and E. Lacour, _Principes de la theorie des fonctions elliptiques et
applications_ (Paris, 1897), beside the treatises of G.H. Halphen,
_Traite des fonctions elliptiques et de leurs applications_ (three
parts, Paris, 1886 ff.), and J. Tannery et J. Molk, _Elements de la
theorie des fonctions elliptiques_ (Paris, 1893 ff.); a book, A.G.
Greenhill, _The Applications of Elliptic Functions_ (London, 1892),
shows how the functions enter in problems of many kinds. For modular
functions there is an extensive treatise, F. Klein and R. Fricke,
_Theorie der elliptischen Modulfunctionen_ (Leipzig, 1890); see also
the most interesting smaller volume, F. Klein, _Uber das Ikosaeder_
(Leipzig, 1884) (also obtainable in English). For the theory of
Riemann's surface, and algebraic integrals, an interesting
introduction is P. Appeil and E. Goursat, _Theorie des fonctions
algebriques et de leurs integrales_; for Abelian functions see also H.
Stahl, _Theorie der Abel'schen Functionen_ (Leipzig, 1896), and H.F.
Baker, _An Introduction to the Theory of Multiply Periodic Functions_
(Cambridge, 1907), and H.F. Baker, _Abel's Theorem and the Allied
Theory, including the Theory of the Theta Functions_ (Cambridge,
1897); for theta functions of one variable a standard work is C.G.
Jacobi, _Fundamenta nova, &c._ (Konigsberg, 1828); for the general
theory of theta functions, consult W. Wirtinger, _Untersuchungen uber
Theta-Functionen_ (Leipzig, 1895). For a history of the theory of
algebraic functions consult A. Brill and M. Noether, _Die Entwicklung
der Theorie der algebraischen Functionen in alterer und neuerer Zeit,
Bericht der deutschen Mathematiker-Vereinigung_ (1894); and for a
special theory of algebraic functions, K. Hensel and G. Landsberg,
_Theorie der algebraischen Function u.s.w._ (Leipzig, 1902). The
student will, of course, consult also Riemann's and Weierstrass's
_Ges. Werke_. For the applications to geometry in general an important
contribution, of permanent value, is E. Picard and G. Simart, _Theorie
des fonctions algebriques de deux variables independantes_ (Paris,
1897-1906). This work contains, as Note v. t. ii. p. 485, a valuable
summary by MM. Castelnuovo and Enriques, _Sur quelques resultats
nouveaux dans la theorie des surfaces algebriques_, containing many
references to the numerous memoirs to be found, for the most part, in
the transactions of scientific societies and the mathematical journals
of Italy.
Beside the books above enumerated there exists an unlimited number of
individual memoirs, often of permanent importance and only
imperfectly, or too elaborately, reproduced in the pages of the
volumes in which the student will find references to them. The German
_Encyclopaedia of Mathematics_, and the Royal Society's _Reference
Catalogue of Current Scientific Literature, Pure Mathematics_,
published yearly, should also be consulted. (H. F. Ba.)