Part 29

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

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, _/