Part 31

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