Chapter I: Functions of Real Variables (4)

where [f](x) represents a monogenic branch of the function, in case it
be not everywhere single valued, and t is on the boundary of the
region. Describe now another region R0 lying entirely within R, and
let x be restricted to be within R0 or upon its boundary; then for any
point t on the boundary of R, the points z of the plane for which
zt^(-1) is real and positive and equal to or greater than 1, being
points for which |z| = |t| or |z| > |t|, are without the region R0,
and not infinitely near to its boundary points. Taking then an
arbitrary real positive [epsilon] we can determine a polynomial in
xt^(-1), say P(xt^(-1)), such that for all points x in R0 we have

|[1 - xt^(-1)]^(-1) - P[xt^(-1)]| < [epsilon];

the form of this polynomial may be taken the same for all points t on
the boundary of R, and hence, if E be a proper variable quantity of
modulus not greater than [epsilon],
_ _
| / dt | | / dt |
|2[pi]i[f](x) - | --[f](t)P(xt^(-1))| = | | --[f](t)E| <= [epsilon]LM,
| _/ t | | _/ t |

where L is the length of the path of integration, the boundary of R,
and M is a real positive quantity such that upon this boundary
|t^(-1)[f](t)| < M. If now

P(xt^(-1)) = c0 + c1xt^(-1) + ... + c_mx^mt^(-m),

and
_
1 /
------ | t^(-r-1)[f](t)dt = [mu]_r,
2[pi]i _/

this gives

|[f](x) - {c0[mu]0 + c1[mu]1x + ... + c_m[mu]_mx^m}| =< [epsilon]LM/2[pi],

where the quantities [mu]0, [mu]1, [mu]2, ... are the coefficients in
the expansion of [f](x) about the origin.

If then an arbitrary finite region be constructed of the kind
explained, excluding the barriers joining the singular points of
[f](x) to x = [oo], it is possible, corresponding to an arbitrary real
positive number [sigma], to determine a number m, and a polynomial
Q(x), of order m, such that for all interior points of this region

|[f](x) - Q(x)| < [sigma].

Hence as before, within this region [f](x) can be represented by a
series of polynomials, converging uniformly; when [f](x) is not a
single valued function the series represents one branch of the
function.

The same result can be obtained without the use of Cauchy's integral.
We explain briefly the character of the proof. If a monogenic function
of t, [phi](t) be capable of expression as a power series in t-x about
a point x, for |t - x| =< [rho], and for all points of this circle
|[phi](t)| < g, we know that |[phi]^(n)(x)| < g[rho]^(-n)(n!). Hence,
taking |z| < 1/3[rho], and, for any assigned positive integer [mu],
taking m so that for n > m we have ([mu] + n)^[mu] < (3/2)^n, we have

|[phi]^(([mu] + n))(x)z^n| [phi]^([mu] + n)(x)
|------------------------| < -------------------([mu] + n)^([mu])|z|^n
| n! | ([mu] + n)!

g /3 \ /[rho]\ g
< ---------------- ( -- )^n ( ----- )^n < --------------,
[rho]^([mu] + n) \2 / \ 3 / [rho]^[mu] 2^n

and therefore

_m
\ [phi]^([mu] + n)(x)
[phi]^([mu])(x + z) = /_ ------------------- z^n + [epsilon]_[mu],
n=0 n!

where

g _[oo] 1 g
|[epsilon]_([mu])| < ---------- \ --- < --------------
[rho]^[mu] /_ 2^n [rho]^[mu] 2^m
n=m+1

Now draw barriers as before, directed from the origin, joining the
singular point of [phi](z) to z = [oo], take a finite region excluding
all these barriers, let [rho] be a quantity less than the radii of
convergence of all the power series developments of [phi](z) about
interior points of this region, so chosen moreover that no circle of
radius [rho] with centre at an interior point of the region includes
any singular point of [phi](z), let g be such that |[phi](z)| < g for
all circles of radius [rho] whose centres are interior points of the
region, and, x being any interior point of the region, choose the
positive integer n so that 1/n |x| 1/3 - [rho]; then take the points
a1 = x/n, a2 = 2x/n, a3 = 3x/n, ... a_n = x; it is supposed that the
region is so taken that, whatever x may be, all these are interior
points of the region. Then by what has been said, replacing x, z
respectively by 0 and x/n, we have

_m1 [phi]^([mu] + [lambda]1)(0) /x \^[lambda]1
[phi]^([mu]) (a{1}) = \ --------------------------- ( -- ) + [alpha]_[mu]
/_ [lambda]{1}! \n /
[lambda]1=0

with

|[alpha]{[mu]}| < g/[rho]^[mu] 2^m1,

provided ([mu] + m1 + 1)^[mu] < (3/2)^(m1+1); in fact for [mu] =<
2n^(2n-2) it is sufficient to take m1 = n^2n; by another application
of the same inequality, replacing x, z respectively by a1 and x/n, we
have

_ m2 [phi]^([mu]+[lambda]2)(a1) /x \^[lambda]2
[phi]^([mu])(a2) = \ -------------------------- ( -- ) + [beta]'_[mu],
/_ [lambda]{2}! \n /
[lambda]2=0

where

|[beta]'[mu]| < g/[rho]^[mu] 2^m2

provided ([mu] + m2 + 1)^[mu] < {3/2}^(m2 + 1); we take m2 = n^(2n -
2), supposing [mu] < 2^(2n - 4). So long as [lambda]2 =< = m} =< n^(2n
- 2) and [mu] < 2n^(2n - 4) we have [mu] + [lambda]{2} < 2n^(2n - 2),
and we can use the previous inequality to substitute here for
[phi]^([mu] + [lambda]2) (a1). When this is done we find

_ m2 _ m1 [phi]^([mu] + [lambda]1 + [lambda]2)(0)
[phi]^([mu])(a2) = \ \ ---------------------------------------
/_ /_ [lambda]1! [lambda]2!
[lambda]2=0 [lambda]1=0

/x \ ^[lambda]1 + [lambda]2
( -- ) + [beta]_[mu],
\n /

where |[beta]_[mu]| < 2g/[rho]^[mu] 2^(m2), the numbers m1, m2 being
respectively n^2n and n^(2n - 2).

Applying then the original inequality to [phi]^([mu]) (a3) =
[phi]^([mu]) (a2 + x/n), and then using the series just obtained, we
find a series for [phi]^([mu]) (a3). This process being continued, we
finally obtain

_ m1 _ m2 _ m_n
[phi](x) = \ \ ... \ [phi]^(h)(0) /x \^h
/_ /_ /_ ----------- ( -- ) + [epsilon],
[lambda]1=0 [lambda]2=0 [lambda]_n=0 K \n /

where h = [lambda]1 + [lambda]2 + ... + [lambda]_n , K = [lambda]1!
[lambda]2! ... [lambda]_n!, m1 = n^(2n), m1 = n^(2n - 2), ... , m1 =
n^2, |[epsilon]| < 2g/2^(m_n).

By this formula [phi](x) is represented, with any required degree of
accuracy, by a polynomial, within the region in question; and thence
can be expressed as before by a series of polynomials converging
uniformly (and absolutely) within this region.

S 13. _Application of Cauchy's Theorem to the Determination of Definite Integrals._--Some reference must be made to a method whereby real definite integrals may frequently be evaluated by use of the theorem of the vanishing of the integral of a function of a complex variable round a contour within which the function is single valued and non singular.

We are to evaluate an integral [int][a to b] [f](x)dx; we form a
closed contour of which the portion of the real axis from x = a to x =
b forms a part, and consider the integral [int][f](z)dz round this
contour, supposing that the value of this integral can be determined
along the curve forming the completion of the contour. The contour
being supposed such that, within it, [f](z) is a single valued and
finite function of the complex variable z save at a finite number of
isolated interior points, the contour integral is equal to the sum of
the values of [int][f](z)dz taken round these points. Two instances
will suffice to explain the method. (1) The integral [int][0 to [oo]]
(tan x)/x dx is convergent if it be understood to mean the limit when
[epsilon], [zeta], [sigma], ... all vanish of the sum of the integrals

_1/2[pi]-[epsilon] _(3/2)[pi]-[zeta] _(5/2)[pi]-[sigma]
/ tan x / tan x / tan x
| ----- dx, | ----- dx, | ----- dx, ...
_/ 0 x _/1/2[pi]+[epsilon] x _/(3/2)[pi]+[zeta] x

Now draw a contour consisting in part of the whole of the positive and
negative real axis from x = -n[pi] to x = + n[pi], where n is a
positive integer, broken by semicircles of small radius whose centres
are the points x = [+-]1/2[pi], x = [+-]3/4[pi], ... , the contour
containing also the lines x = n[pi] and x = -n[pi] for values of y
between 0 and n[pi] tan [alpha], where [alpha] is a small fixed angle,
the contour being completed by the portion of a semicircle of radius
n[pi] sec [alpha] which lies in the upper half of the plane and is
terminated at the points x = [+-]n[pi], y = n[pi] tan [alpha]. Round
this contour the integral [int](tan z /z) dz has the value zero. The
contributions to this contour integral arising from the semicircles of
centres -1/2(2s - 1)[pi], + 1/2(2s - 1)[pi], supposed of the same
radius, are at once seen to have a sum which ultimately vanishes when
the radius of the semicircles diminishes to zero. The part of the
contour lying on the real axis gives what is meant by 2 [int][0 to
n[pi]](tan x / x) dx. The contribution to the contour integral from
the two straight portions at x = [+-]n[pi] is

_n[pi] tan [alpha]
/ / tan iy tan iy \
| idy ( ---------- - ----------- )
_/ 0 \n[pi] + iy -n[pi] + iy /

where i tan iy, = -[exp(y) - exp(-y)]/[exp(y) + exp(-y)], is a real
quantity which is numerically less than unity, so that the
contribution in question is numerically less than

_n[pi] tan [alpha]
/ 2n[pi]
| dy ---------------, that is than 2[alpha].
_/ 0 n^2[pi]^2 + y^2

Finally, for the remaining part of the contour, for which, with R =
n[pi] sec [alpha], we have z = R(cos [theta] + i sin [theta]) =
RE(i[theta]), we have

dz
-- = id[theta], i tan z =
z

exp(-R sin [theta]) E(iR cos [theta]) - exp(R sin [theta]) E(-iR cos [theta])
-----------------------------------------------------------------------------;
exp(-R sin [theta]) E(iR cos [theta]) + exp(R sin [theta]) E(-iR cos [theta])

when n and therefore R is very large, the limit of this contribution
to the contour integral is thus
_
/ [pi]-[alpha]
- | d[theta] = -([pi] - 2[alpha]).
_/ [alpha]

Making n very large the result obtained for the whole contour is
_
/ [oo] tan x
2 | ----- dx - ([pi] - 2[alpha]) - 2[alpha][epsilon] = 0;
_/ 0 x

where [epsilon] is numerically less than unity. Now supposing [alpha]
to diminish to zero we finally obtain
_
/ [oo] tan x [pi]
| ----- dx = ----
_/ 0 x 2

(2) For another case, to illustrate a different point, we may take the
integral
_
/ z^(a-1)
| ------- dz,
_/ 1 + z

wherein a is real quantity such that 0 < a < 1, and the contour
consists of a small circle, z = rE(i[theta]), terminated at the points
x = r cos [alpha], y = [+-] r sin [alpha], where [alpha] is small, of
the two lines y = [+-] r sin [alpha] for r cos [alpha] =< x =< R cos
[beta], where R sin [beta] = r sin [alpha], and finally of a large
circle z = RE(i[phi]), terminated at the points x = R cos [beta], y =
[+-] R sin [beta]. We suppose [alpha] and [beta] both zero, and that
the phase of z is zero for r cos a =< x =< R cos [beta], y = r sin
[alpha] = R sin [beta]. Then on r cos [alpha] =< x =< R cos [beta], y
= -r sin [alpha], the phase of z will be 2[pi], and z^([alpha] - 1)
will be equal to x^([alpha] - 1) exp (2[pi]i(a - 1)), where x is real
and positive. The two straight portions of the contour will thus
together give a contribution

_
/ R cos [beta] x^(a - 1)
[1 - exp (2[pi]i[alpha])] | --------- dx.
_/ r cos [alpha] 1 + x

It can easily be shown that if the limit of z[f](z) for z = 0 is zero,
the integral [int] [f](z)dz taken round an arc, of given angle, of a
small circle enclosing the origin is ultimately zero when the radius
of the circle diminishes to zero, and if the limit of z[f](z) for z =
[oo] is zero, the same integral taken round an arc, of given angle, of
a large circle whose centre is the origin is ultimately zero when the
radius of the circle increases indefinitely; in our case with [f](z) =
z^([alpha] - 1)/(1 + z), we have z[f](z) = z^a/(1 + z), which, for 0 <
a < 1, diminishes to zero both for z = 0 and for z = [oo]. Thus,
finally the limit of the contour integral when r = 0, R = [oo] is
_
/ [oo] x^([alpha] - 1)
[1 - exp (2[pi]i[alpha])] | --------------- dx.
_/ 0 1 + x

Within the contour [f](z) is single valued, and has a pole at z = 1;
at this point the phase of z is [pi] and z^(a - 1) is exp [i[pi](a -
1)] or - exp (i[pi]a); this is then the residue of [f](z) at z = -1;
we thus have
_
/ [oo] x^(a - 1)
[1 - exp (2[pi]ia)] | --------- dx = -2[pi]i exp (i[pi]a),
_/ 0 1 + x

that is
_
/ [oo] x^(a - 1)
| --------- dx = [pi] cosec (a[pi]).
_/ 0 1 + x

S 14. _Doubly Periodic Functions._--An excellent illustration of the preceding principles is furnished by the theory of single valued functions having in the finite part of the plane no singularities but poles, which have two periods.

Before passing to this it may be convenient to make here a few remarks
as to the periodicity of (single valued) monogenic functions. To say
that [f](z) is periodic is to say that there exists a constant [omega]
such that for every point z of the interior of the region of existence
of [f](z) we have [f](z + [omega]) = [f](z). This involves,
considering all existing periods [omega] = [rho] + i[sigma], that
there exists a lower limit of [rho]^2 + [sigma]^2 other than zero; for
otherwise all the differential coefficients of [f](z) would be zero,
and [f](z) a constant; we can then suppose that not both [rho] and
[sigma] are numerically less than [epsilon], where [epsilon] >
[sigma]. Hence, if g be any real quantity, since the range (-g, ... g)
contains only a finite number of intervals of length [epsilon], and
there cannot be two periods [omega] = [rho] + i[sigma] such that
[mu][epsilon] =< [rho] < ([mu] + 1)[epsilon], [nu][epsilon] =< [sigma]
< ([nu] + 1)[epsilon], where [mu], [nu] are integers, it follows that
there is only a finite number of periods for which both [rho] and
[sigma] are in the interval (-g ... g). Considering then all the
periods of the function which are real multiples of one period
[omega], and in particular those periods [lambda][omega] wherein 0 <
[lambda] =< 1, there is a lower limit for [lambda], greater than zero,
and therefore, since there is only a finite number of such periods for
which the real and imaginary parts both lie between -g and g, a least
value of [lambda], say [lambda]0. If [Omega] = [lambda]0[omega] and
[lambda] = M[lambda]0 + [lambda]', where M is an integer and 0 [< = ]
[lambda]' < [lambda]0, any period [lambda][omega] is of the form
M[Omega] + [lambda]'[omega]; since, however, [Omega], M[Omega] and
[lambda][omega] are periods, so also is [lambda]'[omega], and hence,
by the construction of [lambda]0, we have [lambda]' = 0; thus all
periods which are real multiples of [omega] are expressible in the
form M[Omega] where M is an integer, and [Omega] a period.

If beside [omega] the functions have a period [omega]' which is not a
real multiple of [omega], consider all existing periods of the form
[mu][omega] + [nu][omega]' wherein [mu], [nu] are real, and of these
those for which 0 [< = ] [mu] =< 1, 0 < [nu] =< 1; as before there is
a least value for [nu], actually occurring in one or more periods, say
in the period [Omega]' = [mu]0[omega] + [nu]0[omega]'; now take, if
[mu][omega] + [nu][omega]' be a period, [nu] = N'[nu]0 + [nu]', where
N' is an integer, and 0 =< [nu]' < [nu]0; thence [mu][omega] +
[nu][omega]' = [mu][omega] + N'([Omega]' - [mu]0[omega]) +
[nu]'[omega]'; take then [mu] - N[mu]0 = N[lambda]0 + [lambda]', where
N is an integer and [lambda]0 is as above, and 0 =< [lambda]' <
[lambda]0; we thus have a period N[Omega] + N'[Omega]' +
[lambda]'[omega] + [nu]'[omega]', and hence a period [lambda]'[omega]
+ [nu]'[omega]', wherein [lambda]' < [lambda]0, [nu]' < [nu]0; hence
[nu]' = 0 and [lambda]' = 0. All periods of the form [mu][omega] +
[nu][omega]' are thus expressible in the form N[Omega] + N'[Omega]',
where [Omega], [Omega]' are periods and N, N' are integers. But in
fact any complex quantity, P + iQ, and in particular any other
possible period of the function, is expressible, with [mu], [nu] real,
in the form [mu][omega] + [nu][omega]'; for if [omega] = [rho] +
i[sigma], [omega]' = [rho]' + i[sigma]', this requires only P =
[mu][rho] + [nu][rho]', Q = [mu][sigma] + [nu][sigma]', equations
which, since [omega]'/[omega] is not real, always give finite values
for [mu] and [nu].

It thus appears that if a single valued monogenic function of z be
periodic, either all its periods are real multiples of one of them,
and then all are of the form M[Omega], where [Omega] is a period and M
is an integer, or else, if the function have two periods whose ratio
is not real, then all its periods are expressible in the form N[Omega]
+ N'[Omega]', where [Omega], [Omega]' are periods, and N, N' are
integers. In the former case, putting [zeta] = 2[pi]iz/[Omega], and
the function [f](z) = [phi]([zeta]), the function [phi]([zeta]) has,
like exp ([zeta]), the period 2[pi]i, and if we take t = exp([zeta])
or [zeta] = [lambda](t) the function is a single valued function of t.
If then in particular [f](z) is an integral function, regarded as a
function of t, it has singularities only for t = 0 and t = [oo], and
may be expanded in the form [Sigma](-[oo] to [oo]) a_nt^n.

Taking the case when the single valued monogenic function has two
periods [omega], [omega]' whose ratio is not real, we can form a
network of parallelograms covering the plane of z whose angular points
are the points c + m[omega] + m'[omega]', wherein c is some constant
and m, m' are all possible positive and negative integers; choosing
arbitrarily one of these parallelograms, and calling it the primary
parallelogram, all the values of which the function is at all capable
occur for points of this primary parallelogram, any point, z', of the
plane being, as it is called, _congruent_ to a definite point, z, of
the primary parallelogram, z' - z being of the form m[omega] +
m'[omega]', where m, m' are integers. Such a function cannot be an
integral function, since then, if, in the primary parallelogram
|[f](z)| < M, it would also be the case, on a circle of centre the
origin and radius R, that |[f](z)| < M, and therefore, if
[Sigma]a_nz^n be the expansion of the function, which is valid for an
integral function for all finite values of z, we should have |a_n| <
MR^(-n), which can be made arbitrarily small by taking R large enough.
The function must then have singularities for finite values of z.

We consider only functions for which these are poles. Of these there
cannot be an infinite number in the primary parallelogram, since then
those of these poles which are sufficiently near to one of the
necessarily existing limiting points of the poles would be arbitrarily
near to one another, contrary to the character of a pole. Supposing
the constant c used in naming the corners of the parallelograms so
chosen that no pole falls on the perimeter of a parallelogram, it is
clear that the integral 1/(2[pi]i) [int][f](z)dz round the perimeter
of the primary parallelogram vanishes; for the elements of the
integral corresponding to two such opposite perimeter points as z, z +
[omega] (or as z, z + [omega]') are mutually destructive. This
integral is, however, equal to the sum of the residues of [f](z) at
the poles interior to the parallelogram. Which sum is therefore zero.
There cannot therefore be such a function having only one pole of the
first order in any parallelogram; we shall see that there can be such
a function with two poles only in any parallelogram, each of the first
order, with residues whose sum is zero, and that there can be such a
function with one pole of the second order, having an expansion near
this pole of the form (z - a)^(-2) + (power series in z - a).

Considering next the function [phi](z) = [[f](z)]^(-1) d[f](z)/dz, it
is easily seen that an ordinary point of [f](z) is an ordinary point
of [phi](z), that a zero of order m for [f](z) in the neighbourhood of
which [f](z) has a form, (z - a)^m multiplied by a power series, is a
pole of [phi](z) of residue m, and that a pole of [f](z) of order n is
a pole of [phi](z) of residue -n; manifestly [phi](z) has the two
periods of [f](z). We thus infer, since the sum of the residues of
[phi](z) is zero, that for the function [f](z), the sum of the orders
of its vanishing at points belonging to one parallelogram, [Sigma]m,
is equal to the sum of the orders of its poles, [Sigma]n; which is
briefly expressed by saying that the number of its zeros is equal to
the number of its poles. Applying this theorem to the function
[f](z) - A, where A is an arbitrary constant, we have the result, that
the function [f](z) assumes the value A in one of the parallelograms
as many times as it becomes infinite. Thus, by what is proved above,
every conceivable complex value does arise as a value for the doubly
periodic function [f](z) in any one of its parallelograms, and in fact
at least twice. The number of times it arises is called the _order_ of
the function; the result suggests a property of rational functions.

Consider further the integral [int] z [f]'(z)/[f](z) dz, where [f]'(z)
= d[f](z)/dz taken round the perimeter of the primary parallelogram;
the contribution to this arising from two opposite perimeter points
such as z and z + [omega] is of the form -[omega] [int] z
[f]'(z)/[f](z) dz, which, as z increases from z0 to z0 + [omega]',
gives, if [lambda] denote the generalized logarithm, -
[omega]{[lambda][[f](z0 + [omega]')] - [lambda][[f](z0)]}, that is,
since [f](z0 + [omega]') = [f](z0), gives 2[pi]iN[omega], where N is
an integer; similarly the result of the integration along the other
two opposite sides is of the form 2[pi]iN'[omega]', where N' is an
integer. The integral, however, is equal to 2[pi]i times the sum of
the residues of z[f]'(z)/[f](z) at the poles interior to the
parallelogram. For a zero, of order m, of [f](z) at z = a, the
contribution to this sum is 2[pi]ima, for a pole of order n at z = b
the contribution is -2[pi]inb; we thus infer that [Sigma]ma -
[Sigma]nb = N[omega] + N'[omega]'; this we express in words by saying
that the sum of the values of z where [f](z) = 0 within any
parallelogram is equal to the sum of the values of z where [f](z) =
[oo] save for integral multiples of the periods. By considering
similarly the function [f](z) - A where A is an arbitrary constant, we
prove that each of these sums is equal to the sum of the values of z
where the function takes the value A in the parallelogram.

We pass now to the construction of a function having two arbitrary periods [omega], [omega]' of unreal ratio, which has a single pole of the second order in any one of its parallelograms.

For this consider first the network of parallelograms whose corners
are the points [Omega] = m[omega] + m'[omega]', where m, m' take all
positive and negative integer values; putting a small circle about
each corner of this network, let P be a point outside all these
circles; this will be interior to a parallelogram whose corners in
order may be denoted by z0, z0 + [omega], z0 + [omega] + [omega]', z0
+ [omega]'; we shall denote z0, z0 + [omega] by A0, B0; this
parallelogram [Pi]0 is surrounded by eight other parallelograms,
forming with [Pi]0 a larger parallelogram [Pi]1, of which one side,
for instance, contains the points z0 - [omega] - [omega]', z0 -
[omega]', z0 - [omega]' + [omega], z0 - [omega]' + 2[omega], which we
shall denote by A1, B1, C1, D1. This parallelogram [Pi]1 is surrounded
by sixteen of the original parallelograms, forming with [Pi]1 a still
larger parallelogram [Pi]2 of which one side, for instance, contains
the points z0 - 2[omega] - 2[omega]', z0 - [omega] - 2[omega]', z0 -
2[omega]', z0 + [omega] - 2[omega]', z0 + 2[omega] - 2[omega]', z0 +
3[omega] - 2[omega]', which we shall denote by A2, B2, C2, D2, E2, F2.
And so on. Now consider the sum of the inverse cubes of the distances
of the point P from the corners of all the original parallelograms.
The sum will contain the terms

1 / 1 1 1 \ / 1 1 1 \
S0 = ----- + ( ----- + ----- + ----- ) + ( ----- + ----- + ... + ----- ) + ...
PA0^3 \PA1^3 PB1^3 PC1^3/ \PA2^3 PB2^3 PE2^3/

and three other sets of terms, each infinite in number, formed in a
similar way. If the perpendiculars from P to the sides A0B0, A1B1C1,
A2B2C2D2E2, and so on, be p, p + q, p + 2q and so on, the sum S0 is at
most equal to

1 3 5 2n + 1
--- + --------- + ---------- + ... + ---------- + ...
p^3 (p + q)^3 (p + 2q)^3 (p + nq)^3

of which the general term is ultimately, when n is large, in a ratio
of equality with 2q^(-3)n^(-2), so that the series S0 is convergent,
as we know the sum [Sigma]n^(-2) to be; this assumes that p[/ = ]0; if
P be on A0B0 the proof for the convergence of S0 - 1/PA0^3, is the
same. Taking the three other sums analogous to S0 we thus reach the
result that the series

[phi](z) = -2[Sigma](z - [Omega])^(-3),

where [Omega] is m[omega] + m'[omega]', and m, m' are to take all
positive and negative integer values, and z is any point outside small
circles described with the points [Omega] as centres, is _absolutely
convergent_. Its sum is therefore independent of the order of its
terms. By the nature of the proof, which holds for all positions of z
outside the small circles spoken of, the series is also clearly
_uniformly convergent_ outside these circles. Each term of the series
being a monogenic function of z, the series may therefore be
differentiated and integrated outside these circles, and represents a
monogenic function. It is clearly periodic with the periods [omega],
[omega]'; for [phi](z + [omega]) is the same sum as [phi](z) with the
terms in a slightly different order. Thus [phi](z + [omega]) =
[phi](z) and [phi](z + [omega]') = [phi](z).

Consider now the function
_ _ _
1 / z | 2 |
[f](z) = --- + | | [phi](z) + -- | dz,
z^2 _/ 0 |_ z^3 _|

where, for the subject of integration, the area of uniform convergence
clearly includes the point z = 0; this gives

d[f](z)
------- = [phi](z)
dz

and
_ _
1 | 1 1 |
[f](z) = --- + [Sigma]' | -------------- - -------- |,
z^2 |_ (z - [Omega])^2 [Omega]^2 _|

wherein [Sigma]' is a sum excluding the term for which m = 0 and m' =
0. Hence [f](z + [omega]) - [f](z) and [f](z + [omega]') - [f](z) are
both independent of z. Noticing, however, that, by its form, [f](z) is
an even function of z, and putting z = -1/2[omega], z = -1/2[omega]'
respectively, we infer that also [f](z) has the two periods [omega]
and [omega]'. In the primary parallelogram [Pi]0, however, [f](z) is
only infinite at z = 0 in the neighbourhood of which its expansion is
of the form z^(-2) + (power series in z). Thus [f](z) is such a doubly
periodic function as was to be constructed, having in any
parallelogram of periods only one pole, of the second order.

It can be shown that any single valued meromorphic function of z with [omega] and [omega]' as periods can be expressed rationally in terms of [f](z) and [phi](z), and that [[phi](z)]^2 is of the form 4[[f](z)]^3 + A[f](z) + B, where A, B are constants.

To prove the last of these results, we write, for |z| < |[Omega]|,

1 1 2z 3z^2
--------------- - --------- = --------- + --------- + ...,
(z - [Omega])^2 [Omega]^2 [Omega]^3 [Omega]^4

and hence, if [Sigma]'[Omega]^(-2n) = [sigma]_n, since
[Sigma]'[Omega]^(-(2n - 1)) = 0, we have, for sufficiently small z
greater than zero,

[f](z) = z^(-2) + 3[sigma]2.z^2 + 5[sigma]3.z^4 + ...

and

[phi](z) = -2z^(-3) + 6[sigma]2.z + 20[sigma]3.z^3 + ...;

using these series we find that the function

F(z) = [[phi](z)]^2 - 4[[f](z)]^3 + 60[sigma]2[f](z) + 140[sigma]3

contains no negative powers of z, being equal to a power series in z^2
beginning with a term in z^2. The function F(z) is, however, doubly
periodic, with periods [omega], [omega]', and can only be infinite
when either [f](z) or [phi](z) is infinite; this follows from its form
in [f](z) and [phi](z); thus in one parallelogram of periods it can be
infinite only when z = 0; we have proved, however, that it is not
infinite, but, on the contrary, vanishes, when z = 0. Being,
therefore, never infinite for finite values of z it is a constant, and
therefore necessarily always zero. Putting therefore [f](z) = [zeta]
and [phi](z) = d[zeta]/dz we see that

dz
------- = (4[zeta]^3 - 60[sigma]2[zeta] - 140[sigma]3)^(-1/2)
d[zeta]

Historically it was in the discussion of integrals such as
_
/
| d[zeta](4[zeta]^3 - 60[sigma]2.[zeta] - 140[sigma]3)^(-1/2),
_/

regarded as a branch of Integral Calculus, that the doubly periodic
functions arose. As in the familiar case
_
/ [zeta]
z = | (1 - [zeta]^2)^(-1/2) d[zeta],
_/ 0

where [zeta] = sin z, it has proved finally to be simpler to regard
[zeta] as a function of z. We shall come to the other point of view
below, under S 20, _Elliptic Integrals_.

To prove that any doubly periodic function F(z) with periods [omega], [omega]', having poles at the points z = a1, ... z = a_m of a parallelogram, these being, for simplicity of explanation, supposed to be all of the first order, is rationally expressible in terms of [phi](z) and [f](z), and we proceed as follows:--

Consider the expression

([zeta], 1)_m + [eta]([zeta], 1)_(m - 2)
[Phi](z) = ------------------------------------------
([zeta]- A1)([zeta] - A2)...([zeta] - A_m)

where A_s = [f](a_s), [zeta] is an abbreviation for [f](z) and [eta]
for [phi](z), and ([zeta], 1)_m, ([zeta], 1)_(m - 2), denote integral
polynomials in [zeta], of respective orders m and m - 2, so that there
are 2m unspecified, homogeneously entering, constants in the
numerator. It is supposed that no one of the points a1, ... a_m is one
of the points m[omega] + m'[omega]' where f(z) = [oo]. The function
[Phi](z) is a monogenic function of z with the periods [omega],
[omega]', becoming infinite (and having singularities) only when (1)
[zeta] = [oo] or (2) one of the factors [zeta] - A_s is zero. In a
period parallelogram including z = 0 the first arises only for z = 0;
since for [zeta] = [oo], [eta] is in a finite ratio to [zeta]^(3/2);
the function [Phi](z) for [zeta] = [oo] is not infinite provided the
coefficient of [zeta]^m in ([zeta], 1)_m is not zero; thus [Phi](z) is
regular about z = 0. When [zeta] - A_s = 0, that is [f](z) = f(a_s),
we have z = [+-]a_s + m[omega] + m'[omega]', and no other values of z, m
and m' being integers; suppose the unspecified coefficients in the
numerator so taken that the numerator vanished to the first order in
each of the m points -a1, -a2, ... -a_m; that is, if [phi](a_s) = B_s,
and therefore [phi](-a_s) = -B_s, so that we have the m relations

(A_s, 1)_m - B_s(A_s, 1)_(m - 2) = 0;

then the function [Phi](z) will only have the m poles a1, ... a_m.
Denoting further the m zeros of F(z) by a1', ... a_m', putting
[f](a_s') = A_s', [phi](a_s') = B_s', suppose the coefficients of the
numerator of [Phi](z) to satisfy the further m-1 conditions

(A_s', 1)_m + B_s'(A_s',1)_(m - 2) = 0

for s = 1, 2, ... (m - 1). The ratios of the 2m coefficients in the
numerator of [Phi](z) can always be chosen so that the m + (m - 1)
linear conditions are all satisfied. Consider then the ratio

F(z)/[Phi](z);

it is a doubly periodic function with no singularity other than the
one pole a_m'. It is therefore a constant, the numerator of [Phi](z)
vanishing spontaneously in a_m'. We have

F(z) = A[Phi](z),

where A is a constant; by which F(z) is expressed rationally in terms
of [f](z) and [phi](z), as was desired.

When z = 0 is a pole of F(z), say of order r, the other poles, each of
the first order, being a1, ... a_m, similar reasoning can be applied
to a function

([zeta], 1)_h + [eta]([zeta], 1)_k
----------------------------------,
([zeta] - A1)...([zeta] - A_m)

where h, k are such that the greater of 2h - 2m, 2k + 3 - 2m is equal
to r; the case where some of the poles a1, ... a_m are multiple is to
be met by introducing corresponding multiple factors in the
denominator and taking a corresponding numerator. We give a solution
of the general problem below, of a different form.

One important application of the result is the theorem that the
functions [f](z + t), [phi](z + t), which are such doubly periodic
function of z as have been discussed, can each be expressed, so far as
they depend on z, rationally in terms of [f](z) and [phi](z), and
therefore, so far as they depend on z and t, rationally in terms of
[f](z), [f](t), [phi](z) and [phi](t). It can in fact be shown, by
reasoning analogous to that given above, that
_ _
| [phi](z) - [phi](t) |^2
[f](z + t) + [f](z) + [f](t) = 1/4 | ------------------- |.
|_ [f](z) - [f](t) _|

This shows that if F(z) be any single valued monogenic function which
is doubly periodic and of meromorphic character, then F(z + t) is an
algebraic function of F(z) and F(t). Conversely any single valued
monogenic function of meromorphic character, F(z), which is such that
F(z + t) is an algebraic function of F(z) and F(t), can be shown to be
a doubly periodic function, or a function obtained from such by
degeneration (in virtue of special relations connecting the
fundamental constants).

The functions [f](z), [phi](z) above are usually denoted by RN(z),
RN'(z); further the fundamental differential equation is usually
written

(RN'z)^2 = 4(RNz)^3 - g2RNz - g3,

and the roots of the cubic on the right are denoted by e1, e2, e3; for
the odd function, RN'z, we have, for the congruent arguments
-1/2[omega]and 1/2[omega], RN'(1/2[omega]) = -RN'(-1/2[omega]) =
-RN'(1/2[omega]), and hence RN'(1/2[omega]) = 0; hence we can take e1
= RN(1/2[omega]), e2 = RN(1/2[omega] + 1/2[omega]'), e3 =
RN(1/2[omega]). It can then be proved that [RN(z) - e1][RN(z +
1/2[omega]) - e1] = (e1 - e2)(e1 - e3), with similar equations for the
other half periods. Consider more particularly the function RN(z) -
e1; like RN(z) it has a pole of the second order at z = 0, its
expansion in its neighbourhood being of the form z^(-2)(1 - e1z^2 +
Az^4 + ...); having no other pole, it has therefore either two zeros,
or a double zero in a period parallelogram ([omega], [omega]'). In
fact near its zero 1/2[omega] its expansion is (x - 1/2[omega])
RN'(1/2[omega]) + 1/2(z - 1/2[omega])^2 RN"(1/2[omega]) + ...; we have
seen that RN'(1/2[omega]) = 0; thus it has a zero of the second order
wherever it vanishes. Thus it appears that the square root [RN(z) -
e1]^1/2, if we attach a definite sign to it for some particular value
of z, is a single valued function of z; for it can at most have two
values, and the only small circuits in the plane which could lead to
an interchange of these values are those about either a pole or a
zero, neither of which, as we have seen, has this effect; the function
is therefore single valued for any circuit. Denoting the function, for
a moment, by [f]1(z), we have [f]1(z + [omega]) = [+-][f]1(z), [f]1(z
+ [omega]') = [+-][f]1(z); it can be seen by considerations of
continuity that the right sign in either of these equations does not
vary with z; not both these signs can be positive, since the function
has only one pole, of the first order, in a parallelogram ([omega],
[omega]'); from the expansion of [f]1(z) about z = 0, namely z^(-1) (1
- 1/2e1z^2 + ...), it follows that [f]1(z) is an odd function, and
hence [f]1(-1/2[omega]') = -[f]1(1/2[omega]'), which is not zero since
[[f]1(1/2[omega]')]^2 = e3 - e1, so that we have [f]1(z + [omega]') =
-[f]1(z); an equation f1(z + [omega]) = -[f]1(z) would then give
[f]1(z + [omega] + [omega]') = [f]1(z), and hence [f]1(1/2[omega] +
1/2[omega]') = [f]1(-1/2[omega] - 1/2[omega]'), of which the latter is
-[f]1(1/2[omega] + 1/2[omega]'); this would give [f]1(1/2[omega] +
1/2[omega]') = 0, while [[f]1 (1/2[omega] + 1/2[omega]')]^2 = e2 - e1.
We thus infer that [f]1(z + [omega]) = [f]1(z), [f]1(z + [omega]') =
-[f]1(z), [f]1(z + [omega] + [omega]') = -[f]1(z). The function
[f]1(z) is thus doubly periodic with the periods [omega] and
2[omega]'; in a parallelogram of which two sides are [omega] and
2[omega]' it has poles at z = 0, z = [omega]' each of the first order,
and zeros of the first order at z = 1/2[omega], z = 1/2[omega] +
[omega]'; it is thus a doubly periodic function of the second order
with two different poles of the first order in its parallelogram
([omega], 2[omega]'). We may similarly consider the functions [f]2(z)
= [RN(z) - e2]^1/2, [f]3(z) = [RN(z) - e3]^1/2; they give

[f]2(z + [omega] + [omega]') = [f]2(z), [f]2(z + [omega]) = -[f]2(z), [f]2(z + [omega]') = -[f]2(z),

[f]3(z + [omega]') = [f]3z, [f]3(z + [omega]) = -[f]3(z), [f]3(z + [omega] + [omega]') = -[f]3(z).

Taking u = z(e1 - e3)^1/2, with a definite determination of the
constant (e1 - e3)^1/2, it is usual, taking the preliminary signs so
that for z = 0 each of z[f]1(z), z[f]2(z), z[f]3(z) is equal to + 1,
to put

(e1 - e3)^1/2 [f]1(z) f2(z)
sn(u) = -------------, cn(u) = -------, dn(u) = -----,
[f]3(z) [f]3(z) f3(z)

k^2 = (e2 - e3)/(e1 - e3), K = 1/2[omega](e1 - e3)^1/2, iK' = 1/2[omega]'(e1 - e3)^1/2;

thus sn(u) is an odd doubly periodic function of the second order with
the periods 4K, 2iK, having poles of the first order at u = iK', u =
2K + iK', and zeros of the first order at u = 0, u = 2K; similarly
cn(u), dn(u) are even doubly periodic functions whose periods can be
written down, and sn^2(u) + cn^2(u) = 1, k^2sn^2(u) + dn^2(u) = 1; if
x = sn(u) we at once find, from the relations given here, that

du
-- = [(1 - x^2) (1 - k^2x^2)]^(-1/2);
dx

if we put x = sin[phi] we have

du
------ = [1 - k^2sin^2 [phi]]^(-1/2),
d[phi]

and if we call [phi] the amplitude of u, we may write [phi] = am(u), x
= sin.am(u), which explains the origin of the notation sn(u).
Similarly cn(u) is an abbreviation of cos.am(u), and dn(u) of
[Delta]am(u), where [Delta]([phi]) meant (1 - k^2sin^2 [phi])^1/2. The
addition equation for each of the functions [f]1(z), [f]2(z), [f]3(z)
is very simple, being

/(Pd) (Pd) \ [f](z) + [f](t) [f](z)[f]'(t) - [f](t)[f]'(z)
[f](z + t) = 1/2( ----- + ----- ) log --------------- = -----------------------------,
\(Pd)z (Pd)i/ [f](z) - [f](t) [f]^2(z) - [f]^2(t)

where f1'(z) means d[f]1(z)/dz, which is equal to -[f]2(z).[f]3(z),
and [f]^2(z) means [[f](z)]^2. This may be verified directly by
showing, if R denote the right side of the equation, that (Pd)R/(Pd)z
= (Pd)R/(Pd)t; this will require the use of the differential equation

[[f]1'^(z)]^2 = [[f]1^2(z) + e1 - e2] [[f]1^2(z) + e1 - e3],

and in fact we find

/ (Pd)^2 (Pd)^2\
( ------- - ------ ) log [[f](z) + [f](t)] = [f]^2(z) - [f]^2(t) =
\(Pd)z^2 dt^2 /

/ (Pd)^2 (Pd)^2\
( ------- - ------ ) log [[f](z) - [f](t)];
\(Pd)z^2 dt^2 /

hence it will follow that R is a function of z + t, and R is at once
seen to reduce to [f](z) when t = 0. From this the addition equation
for each of the functions sn(u), cn(u), dn(u) can be deduced at once;
if s1, c1, d1, s2, c2, d2 denote respectively sn(u1), cn(u1), dn(u1),
sn(u2), cn(u2), dn(u2), they can be put into the forms

sn(u1 + u2) = (s1c2d2 + s2c1d1)/D,

cn(u1 + u2) = (c1c2 - s1s2d1d2)/D,

dn(u1 + u2) = (d1d2 - k^2s1s2c1c2)/D,

where

D = 1 - k^2s1^2s2^2.

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