The commonest case of the occurrence of multiple valued functions is that in which the function s satisfies an algebraic equation ƒ(s, z) = p0sn+ p1sn−1+ ... + pn= 0, wherein p0, p1, ... pnare integral polynomials in z. Assuming ƒ(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 snvanishes, as also the values of z for which beside ƒ(s, z) = 0 we have also ∂f(s, z)/∂s = 0, and also in general the point z = ∞, 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 ∂f(s, z)/∂s = 0 is satisfied by one or more of the roots s of ƒ(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/rby ω(z − r)1/r, where ω, equal to exp (2πih/r), is one of the rth roots of unity. Putting then z − c = trwe may say that the r roots of a cycle are given by a single power series in t, an increase of 2π in the phase of t giving an increase of 2πr 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 2πr, is to be looked upon as defining a singleplaceamong the aggregate of values of z and s which satisfy ƒ(s, z) = 0; two such places may be at the samepoint(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 ∂f(s, z)/∂s = 0 there are less than n places. Similar remarks hold for the neighbourhood of z = ∞; there may be n places whose neighbourhood is given by n power series in z− 1or 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 ƒ(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−1when c is infinite, and similarly the expression s − d by s−1when d is infinite. The last case arises when we consider the finite values of z for which the polynomial coefficient of snvanishes. Of such a pair of expressions we may obtain a continuation by writing t = t0+ λ1τ + λ2τ² + ..., where τ is a new variable and λ1is not zero; in particular for an ordinary finite place this equation simply becomes t = t0+ τ. 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 ƒ(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 ƒ(s, z) = 0 defines amonogenic algebraic construct. With less accuracy we may say that an irreducible algebraic equation ƒ(s, z) = 0 determines a single monogenic function s of z.Any rational function of z and s, where ƒ(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 tmH(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, = −μ, the function is infinite to order μ at the place. More generally, if A be an arbitrary constant, and, near (c, d), R(s, z) −A is of the form tmH(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 μ 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 = ∞, 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 μ 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 ∞) the same number of times on the algebraic construct; this number is called theorderof 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 ƒ(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 ƒ(s, z) = 0; about any place (c, d) for which z = c + P(t), s = d + Q(t), expand the productR(s, z)dzdtin 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−1is zero. This we express by[R(s, z)dz]t−1= 0.dtThe 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 = ∞ 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,Σ (A1+A2+ ... +Am)+ Pzh+ Qzh−1+ ... + R,z − a(z − a)²(z − a)mthe sum ΣA1of 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 = ∞; an obvious result. In general, if for a finite place of the algebraic construct associated with ƒ(s, z) = 0, whose neighbourhood is given by z = c + tr, s = d + Q(t), there be a coefficient of t−1in R(s, z) dz/dt, this will be r times the coefficient of t−rin R(s, z) or R[d + Q(t), c + tr], namely will be the coefficient of t−rin the sum of the r series obtainable from R [d + Q(t), c + tr] by replacing t by ωt, where ω is an rth root of unity; thus the sum of the coefficients of t−1in 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)−1in R(s1, z) + R(s2z) + ... + R(sn, z), where s1, ... snare the n values of s for a value of z near to z = c; this latter sum Σ R(si, z) is, however, a rational function of z only. Similarly, near z = ∞, for a place given by z−1=tr, s = d + Q(t), or s−1= Q(t), the coefficient of t−1in R(s, z) dz/dt is equal to −r times the coefficient of trin R[d + Q(t), t−r], that is equal to the negative coefficient of z−lin the sum of the r series R[d + Q(ωt), t−r], so that, as before, the sum of the coefficients of t−1in R(s, z) dz/dt at the various places which arise for z = ∞ is equal to the negative coefficient of z− 1in the same rational function of z, Σ R(si, 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,1dR(s, z);R(s, z)dzat a zero of R(s, z) near which R(s, z) = tmH(t), we have1dR(s, z)dz=d{λ [R(s, z)] },R(s, z)dzdtdtwhere λ denotes the generalized logarithmic function, that is equal tomt−1+ power series in t;similarly at a place for which R(s, z) = t−μK(t); the theorem[1dR(s, z)dz]t−1= 0R(s, z)dzdtthus gives Σm = Σμ, 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 s² = 4z³ + 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 definiteorder. 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 § 17 below.
The commonest case of the occurrence of multiple valued functions is that in which the function s satisfies an algebraic equation ƒ(s, z) = p0sn+ p1sn−1+ ... + pn= 0, wherein p0, p1, ... pnare integral polynomials in z. Assuming ƒ(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 snvanishes, as also the values of z for which beside ƒ(s, z) = 0 we have also ∂f(s, z)/∂s = 0, and also in general the point z = ∞, 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 ∂f(s, z)/∂s = 0 is satisfied by one or more of the roots s of ƒ(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/rby ω(z − r)1/r, where ω, equal to exp (2πih/r), is one of the rth roots of unity. Putting then z − c = trwe may say that the r roots of a cycle are given by a single power series in t, an increase of 2π in the phase of t giving an increase of 2πr 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 2πr, is to be looked upon as defining a singleplaceamong the aggregate of values of z and s which satisfy ƒ(s, z) = 0; two such places may be at the samepoint(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 ∂f(s, z)/∂s = 0 there are less than n places. Similar remarks hold for the neighbourhood of z = ∞; there may be n places whose neighbourhood is given by n power series in z− 1or 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 ƒ(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−1when c is infinite, and similarly the expression s − d by s−1when d is infinite. The last case arises when we consider the finite values of z for which the polynomial coefficient of snvanishes. Of such a pair of expressions we may obtain a continuation by writing t = t0+ λ1τ + λ2τ² + ..., where τ is a new variable and λ1is not zero; in particular for an ordinary finite place this equation simply becomes t = t0+ τ. 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 ƒ(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 ƒ(s, z) = 0 defines amonogenic algebraic construct. With less accuracy we may say that an irreducible algebraic equation ƒ(s, z) = 0 determines a single monogenic function s of z.
Any rational function of z and s, where ƒ(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 tmH(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, = −μ, the function is infinite to order μ at the place. More generally, if A be an arbitrary constant, and, near (c, d), R(s, z) −A is of the form tmH(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 μ 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 = ∞, 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 μ 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 ∞) the same number of times on the algebraic construct; this number is called theorderof 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 Æ’(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 Æ’(s, z) = 0; about any place (c, d) for which z = c + P(t), s = d + Q(t), expand the product
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−1is zero. This we express by
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 = ∞ 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,
the sum ΣA1of 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 = ∞; an obvious result. In general, if for a finite place of the algebraic construct associated with ƒ(s, z) = 0, whose neighbourhood is given by z = c + tr, s = d + Q(t), there be a coefficient of t−1in R(s, z) dz/dt, this will be r times the coefficient of t−rin R(s, z) or R[d + Q(t), c + tr], namely will be the coefficient of t−rin the sum of the r series obtainable from R [d + Q(t), c + tr] by replacing t by ωt, where ω is an rth root of unity; thus the sum of the coefficients of t−1in 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)−1in R(s1, z) + R(s2z) + ... + R(sn, z), where s1, ... snare the n values of s for a value of z near to z = c; this latter sum Σ R(si, z) is, however, a rational function of z only. Similarly, near z = ∞, for a place given by z−1=tr, s = d + Q(t), or s−1= Q(t), the coefficient of t−1in R(s, z) dz/dt is equal to −r times the coefficient of trin R[d + Q(t), t−r], that is equal to the negative coefficient of z−lin the sum of the r series R[d + Q(ωt), t−r], so that, as before, the sum of the coefficients of t−1in R(s, z) dz/dt at the various places which arise for z = ∞ is equal to the negative coefficient of z− 1in the same rational function of z, Σ R(si, 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,
at a zero of R(s, z) near which R(s, z) = tmH(t), we have
where λ 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−μK(t); the theorem
thus gives Σm = Σμ, 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 s² = 4z³ + 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 definiteorder. 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 § 17 below.
§ 17.Integrals of Algebraic Functions.—In treatises on Integral Calculus it is proved that if R(z) denote any rational function, an indefinite integral ∫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 s² = az² + bz + c and R(s, z) be any rational function of s, z any integral ∫R(s, z)dz can be evaluated in terms of rational functions of s, z and logarithms of such functions; the simplest case is ∫s− 1dz or ∫(az² + bz + c)−1/2dz. More generally if f(s, z) = 0 be such a relation connecting s, z that when θ is an appropriate rational function of s and z both s and z are rationally expressible, in virtue of ƒ(s, z) = 0 in terms of θ, the integral ∫R(s, z)dz is reducible to a form ∫H(θ)dθ, where H(θ) is rational in θ, 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 ∫R(s, z)dz wherein s² is a cubic polynomial in z. The answer is in the negative. For instance, no one of the three integrals
can be expressed by rational and logarithms of rational functions of s and z; but it can be shown that every integral ∫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 § 20,Elliptic Integrals). A similar theorem is true when s² = quartic polynomial in z; in fact when s² = 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 ƒ(s, z) = 0 is of more general type. There exists then, however, a very general theorem, known asAbel’s Theorem, which may be enunciated as follows: Beside the rational function R(s, z) occurring in the integral ∫R(s, z)dz, consider another rational function H(s, z); let (a1), ... (am) denote the places of the construct associated with the fundamental equation ƒ(s, z) = 0, for which H(s, z) is equal to one value A, each taken with its proper multiplicity, and let (b1), ... (bm) denote the places for which H(s, z) = B, where B is another value; then the sum of the m integrals∫(bi)(ai)R(s, z)dz is equal to the sum of the coefficients of t−1in the expansions of the function
where λ 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
wherein μ is a constant. (For illustrations see below, under § 20,Elliptic Integrals.)
§ 18.Indeterminateness of Algebraic Integrals.—The theorem that the integral∫xaƒ(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 ƒ(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 ƒ(z) is not a single valued function, so that the final value of ƒ(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∫z1z−1dz is liable to an additive indeterminateness equal to the value obtained by a closed path about z = 0, which is equal to 2πi; if we put u =∫z1z−1dz and consider z as a function of u, then we must regard this function as unaffected by the addition of 2πi to its argument u; we know in fact that z = exp (u) and is a single valued function of u, with the period 2πi. Or again the integral∫z0(1 + z²)−1dz 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 =∫z0(1 + z²)−1dz, the function z of u is periodic with period π, this being the function tan (u). Next we take the integral u =∫(z)(0)(1 − z²)−1/2dz, 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 − z²)−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 − z²)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∫−10dz+∫0−1dz+∫10dz+∫01dz,(1 − z²)1/2−(1 − z²)1/2−(1 − z²)1/2(1 − z²)1/2where, in each case, (1 − z²)1/2is real and positive; that is, it gives−4∫10dz(1 − z²)1/2or 2π. Thus the additive indeterminateness of the integral is of the form 2kπ, where k is an integer, and the function z of u, which is sin (u), has 2π for period. Take now the caseu =∫(z)(z0)dz,√{ (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 z0describe a closed loop about the point z = a, consisting, suppose, of a straight path from z0to 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 z0along 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 z0with 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 z0enclosing both a and b but not either c or d, the value obtained will be A − B, and on returning to z0the 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/ζ be reduced to a circuit about ζ = 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 ∫ 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 algebraicequation ƒ(s, z) = 0. Such an integral ƒK(z, s)dz is called an Abelian Integral.
For instance, the integral∫z1z−1dz is liable to an additive indeterminateness equal to the value obtained by a closed path about z = 0, which is equal to 2πi; if we put u =∫z1z−1dz and consider z as a function of u, then we must regard this function as unaffected by the addition of 2πi to its argument u; we know in fact that z = exp (u) and is a single valued function of u, with the period 2πi. Or again the integral∫z0(1 + z²)−1dz 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 =∫z0(1 + z²)−1dz, the function z of u is periodic with period π, this being the function tan (u). Next we take the integral u =∫(z)(0)(1 − z²)−1/2dz, 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 − z²)−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 − z²)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
where, in each case, (1 − z²)1/2is real and positive; that is, it gives
or 2π. Thus the additive indeterminateness of the integral is of the form 2kπ, where k is an integer, and the function z of u, which is sin (u), has 2π for period. Take now the case
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 z0describe a closed loop about the point z = a, consisting, suppose, of a straight path from z0to 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 z0along 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 z0with 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 z0enclosing both a and b but not either c or d, the value obtained will be A − B, and on returning to z0the 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/ζ be reduced to a circuit about ζ = 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 ∫ 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 algebraicequation ƒ(s, z) = 0. Such an integral ƒK(z, s)dz is called an Abelian Integral.
§ 19.Reversion of an Algebraic Integral.—In a limited number of cases the equation u = ∫ [z0to 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 § 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
u =∫zz0[(z − a) (z − b) (z − c) (z − d) (z − e) (z − f) ]−1/2dz,
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.
§ 20.Elliptic Integrals.—An integral of the form ∫ 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 anelliptic 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 z0for 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 + t², we shall obtain s = tQ(t), where Q(t) is a power series in t; when z is very large and s² is a quartic polynomial in z, if we put z−1= t, we shall find s−1= t²Q(t); when z is very large and s² is a cubic polynomial in z, if we put z−1= t², we shall find s−l= t³Q(t). By means of substitutions of these forms the character of the integral ∫ R(z, s)dz may be investigated for any position of z; in any case it takes a form ∫ [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 ∫ s−1dz; it is easily seen that neither for any finite nor for infinite values of z can negative powers of t enter; the integral iseverywhere finite, and is said to be ofthe first kind; it can, moreover, be shown without difficulty that no integral ∫ R(z, s)dz, save a constant multiple of ∫ s−1dz, has this property. Consider next, s² being of the form a0z4+ 4a1z³ + ..., wherein a0may be zero, the integral ∫ (a0z² + 2a1z) s−1dz; 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 √a0a particular square root of a0when a0is not zero, the integral becomes infinite for z = ∞ for both signs of s, the value of A being + √a0or − √a0according as s is √a0·z² (1 + [2a1/a0] z−1+ ... ) or is the negative of this; hence the integral J1=∫( [a0z² + 2a1z]/s + √a0) dz becomes infinite when z is infinite, for the former sign of s, its infinite term being 2√a0·t−1or 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 = ∞ 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 = ∞, its infinite part being 2√a1·t−1or 2√a1·√z. Similarly if z0be any finite value of z which is not a root of the polynomial Æ’(z) to which s² is equal, and s0denotes a particular one of the determinations of s for z=z0, the integralJ2=∫ {s²0+ ½(z − z0) ƒ′(z0)+s0}dz,(z − z0)² s(z − z0)²wherein ƒ′(z) = dÆ’(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 = θ be a root of Æ’(z), in which case the corresponding value of s is zero, the integralJ3= ½ƒ′(θ)∫dz(z − θ) sbecomes infinite for z=0, its infinite part being, if z − θ = t², equal to −[ƒ′(θ)]½ t−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 ∫[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 thesecond 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 s² = Æ’(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 = ∞, (2) z = z0, (3) z = a, is given respectively by (s d/dz)m−1J1, (s d/dz)m−1J2, (s d/dz)m−1J3, 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, J3and their differential coefficients just written down, obtain, as the result, an integral without algebraic infinities. But, in fact, if J, J1denote any two of the three integrals J1, J2, J3, there exists an equation AJ + BJ′ + CÆ’s−1dz = rational function of s, z, where A, B, C are properly chosen constants. For the rational functions + s0+ z √a0z − z0is 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 √a0or 2 √a1√z when a0= 0. It does not become infinite for any other pair (z, s) satisfying the relation s2= Æ’(z); this is in accordance with the easily verified equations + s0+ z √a0− J1+ J2+ (a0z02+ 2a1z0)∫dz= 0;z − z0sand there exists the analogous equations+ z √a0− J1+ J3+ (a0θ2+ 2a1θ)∫dz.z − θsConsider now the integralP =∫ (s + s0+ z √a0)dz;z − z02sthis 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 = ∞, 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 (z1/2) when a0= 0. And, if Æ’(θ) = 0, the integralP1=∫ (s+ z √a0)dzz − θ2sis infinite at z = θ, s = 0 with an infinite part log t, that is log (z − θ)1/2, is not infinite for any other finite value of z, and is infinite like P for z = ∞. 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 J3and the rational functions of the form (s d/dz)m−1J1with constant multiples of integrals such as P or P1, with constant multiples of the integral u = ∫s−1dz, and with rational functions, can be reduced to an integral H becoming infinite only for z = ∞, 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 (z1/2). Such an integral H = ∫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 = ∞ 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 ass + s0+ z √a0ors+ z √a0z − z0z − θand their differential coefficients.The consideration of elliptic integrals is therefore reducible to that of the threeu =∫dz,   J =∫ (a0z2+ 2a1z+ z √a0)dz,   P =∫ (s + s0+ z √a0)dzssz − z02srespectively of the first, second and third kind. Now the equation s2= a0z4+ ... = a0(z − θ) (z − φ) (z − ψ) (z − χ), by puttingy = 2s (z − θ)−2[a0(θ − φ) (θ − ψ) (θ − χ) ]−1/2x =1+1(1+1+1)z − θ3θ − φθ − ψθ − χis at once reduced to the form y2= 4x3− 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 integralsu =∫dx,   J =∫xdx,   P =∫y + y0dx.yyx − x02yOf these consider the first, puttingu =∫(∞)(x)dx,ywhere 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= t2, y−1= 2t3(1 − ¼ g2t4− ¼ g3t6)−1/2, we haveu =∫t0(1 +1â„8g2t4+ ... ) dt = t +1â„40g2t5+ ...,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 formx = u−2+1â„20g2u2+ ...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 φ(u). In the range in question it is single valued and satisfies the differential equation[φ′(u)]2= 4[φ(u)]3− g2φ(u) − g3;in terms of it we can write x = φ(u), y = − φ′(u), and, φ′(u) being an odd function, the sign attached to y in the original integral for x = ∞ is immaterial. Now for any two values u, v in the range in question consider the functionF(u, v) = ¼[φ′(u) − φ′(v)]2− φ(u) − φ(v);φ(u) − φ(v)it is at once seen, from the differential equation, to be such that ∂F/∂u = ∂F/∂v; it is therefore a function of u + v; supposing |u + v| < R we infer therefore, by putting v = 0, thatφ(u + v) = ¼[φ′(u) − φ′(v)]2− φ(u) − φ(v).φ(u) − φ(v)By repetition of this equation we infer that if u1, ... unbe any arguments each of which is in absolute value less than R, whose sum is also in absolute value less than R, then φ(u1+ ... + un) is a rational function of the 2n functions φ(us), φ′(us); and hence, if |u| < R, thatφ(u) = H[φ(u),   φ′(u) ],nnwhere H is some rational function of the arguments φ(u/n), φ′(u/n). In fact, however, so long as |u/n| < R, each of the functions φ(u/n), φ′(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 [φ(u/n), φ′(u/n)] is single valued and without singularities other than poles so long as |u| < nR; it agrees with φ(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 [φ′(u)]2= 4[φ(u)]3− g2φ(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 φ(u + α), wherein α is arbitrary, being such. Taking now any range of values of u, from u = 0, and putting for any value of u, x = φ(u), y = −φ′(u), so that y2=4x3-g2x-g3, we clearly haveu =∫(∞)(x, y)dx;yconversely if x0= φ(u0), y0= −φ′(u0) and ξ, η be any values satisfying η2= 4ξ2− g2ξ − g3, which are sufficiently near respectively to x0, y0, while v is defined byv − u0= −∫(ξ, η)(x0, y0)dξ,ηthen ξ, η are respectively φ(v) and −φ′(v); for this equation leads to an expansion for ξ − x0in terms of v = u0and only one such expansion, and this is obtained by the same work as would be necessary to expand φ(v) when v is near to u0; the function φ(u) can therefore be continued by the help of this equation, from v = u0, provided the lower limit of |ξ − x0| necessary for the expansions is not zero in the neighbourhood of any value (x0, y0). In fact the function φ(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 φ(u) is greater than zero; the same will therefore be the case for the lower limit of the radii |ξ − x0| necessary for the continuations spoken of above provided that the values of (ξ, η) considered do not lead to infinitely increasing values of v; there does not exist, however, any definite point (ξ0, η0) in the neighbourhood of which the integral∫(ξ, η)(x0, y0)dξ/η increases indefinitely, it is only by a path of infinite length that the integral can so increase. We infer therefore that if (ξ, η) be any point, where η2= 4ξ3− g2ξ − g3, and v be defined byv =∫(∞)(ξ, η)dx,ythen ξ = φ(v) and η = −φ′(v). Thus this equation determines (ξ, η) without ambiguity. In particular the additive indeterminatenesses of the integral obtained by closed circuits of the point of integration are periods of the function φ(u); by considerations advanced above it appears that these periods are sums of integral multiples of two which may be taken to beω = 2∫∞e1dx,   ω′ = 2∫∞e3dx;yythese 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), (∞, ∞), which leads back to the initial point (x0, y0), which is impossible. On the whole, therefore, it appears that the function φ(u) agrees with the function â„œ(u) previously discussed, and the discussion of the elliptic integrals can be continued in the manner given under § 14,Doubly Periodic Functions.
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 z0for 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 + t², we shall obtain s = tQ(t), where Q(t) is a power series in t; when z is very large and s² is a quartic polynomial in z, if we put z−1= t, we shall find s−1= t²Q(t); when z is very large and s² is a cubic polynomial in z, if we put z−1= t², we shall find s−l= t³Q(t). By means of substitutions of these forms the character of the integral ∫ R(z, s)dz may be investigated for any position of z; in any case it takes a form ∫ [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 ∫ s−1dz; it is easily seen that neither for any finite nor for infinite values of z can negative powers of t enter; the integral iseverywhere finite, and is said to be ofthe first kind; it can, moreover, be shown without difficulty that no integral ∫ R(z, s)dz, save a constant multiple of ∫ s−1dz, has this property. Consider next, s² being of the form a0z4+ 4a1z³ + ..., wherein a0may be zero, the integral ∫ (a0z² + 2a1z) s−1dz; 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 √a0a particular square root of a0when a0is not zero, the integral becomes infinite for z = ∞ for both signs of s, the value of A being + √a0or − √a0according as s is √a0·z² (1 + [2a1/a0] z−1+ ... ) or is the negative of this; hence the integral J1=∫( [a0z² + 2a1z]/s + √a0) dz becomes infinite when z is infinite, for the former sign of s, its infinite term being 2√a0·t−1or 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 = ∞ 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 = ∞, its infinite part being 2√a1·t−1or 2√a1·√z. Similarly if z0be any finite value of z which is not a root of the polynomial ƒ(z) to which s² is equal, and s0denotes a particular one of the determinations of s for z=z0, the integral
wherein ƒ′(z) = dƒ(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 = θ be a root of ƒ(z), in which case the corresponding value of s is zero, the integral
becomes infinite for z=0, its infinite part being, if z − θ = t², equal to −[ƒ′(θ)]½ t−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 ∫[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 thesecond 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 s² = ƒ(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 = ∞, (2) z = z0, (3) z = a, is given respectively by (s d/dz)m−1J1, (s d/dz)m−1J2, (s d/dz)m−1J3, 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, J3and their differential coefficients just written down, obtain, as the result, an integral without algebraic infinities. But, in fact, if J, J1denote any two of the three integrals J1, J2, J3, there exists an equation AJ + BJ′ + Cƒs−1dz = rational function of s, z, where A, B, C are properly chosen constants. For the rational function
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 √a0or 2 √a1√z when a0= 0. It does not become infinite for any other pair (z, s) satisfying the relation s2= ƒ(z); this is in accordance with the easily verified equation
and there exists the analogous equation
Consider now the integral
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 = ∞, 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 (z1/2) when a0= 0. And, if ƒ(θ) = 0, the integral
is infinite at z = θ, s = 0 with an infinite part log t, that is log (z − θ)1/2, is not infinite for any other finite value of z, and is infinite like P for z = ∞. 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 J3and the rational functions of the form (s d/dz)m−1J1with constant multiples of integrals such as P or P1, with constant multiples of the integral u = ∫s−1dz, and with rational functions, can be reduced to an integral H becoming infinite only for z = ∞, 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 (z1/2). Such an integral H = ∫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 = ∞ 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
and their differential coefficients.
The consideration of elliptic integrals is therefore reducible to that of the three
respectively of the first, second and third kind. Now the equation s2= a0z4+ ... = a0(z − θ) (z − φ) (z − ψ) (z − χ), by putting
y = 2s (z − θ)−2[a0(θ − φ) (θ − ψ) (θ − χ) ]−1/2
is at once reduced to the form y2= 4x3− 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
Of these consider the first, putting
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= t2, y−1= 2t3(1 − ¼ g2t4− ¼ g3t6)−1/2, we have
u =∫t0(1 +1â„8g2t4+ ... ) dt = t +1â„40g2t5+ ...,
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+1â„20g2u2+ ...
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 φ(u). In the range in question it is single valued and satisfies the differential equation
[φ′(u)]2= 4[φ(u)]3− g2φ(u) − g3;
in terms of it we can write x = φ(u), y = − φ′(u), and, φ′(u) being an odd function, the sign attached to y in the original integral for x = ∞ is immaterial. Now for any two values u, v in the range in question consider the function
it is at once seen, from the differential equation, to be such that ∂F/∂u = ∂F/∂v; it is therefore a function of u + v; supposing |u + v| < R we infer therefore, by putting v = 0, that
By repetition of this equation we infer that if u1, ... unbe any arguments each of which is in absolute value less than R, whose sum is also in absolute value less than R, then φ(u1+ ... + un) is a rational function of the 2n functions φ(us), φ′(us); and hence, if |u| < R, that
where H is some rational function of the arguments φ(u/n), φ′(u/n). In fact, however, so long as |u/n| < R, each of the functions φ(u/n), φ′(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 [φ(u/n), φ′(u/n)] is single valued and without singularities other than poles so long as |u| < nR; it agrees with φ(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 [φ′(u)]2= 4[φ(u)]3− g2φ(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 φ(u + α), wherein α is arbitrary, being such. Taking now any range of values of u, from u = 0, and putting for any value of u, x = φ(u), y = −φ′(u), so that y2=4x3-g2x-g3, we clearly have
conversely if x0= φ(u0), y0= −φ′(u0) and ξ, η be any values satisfying η2= 4ξ2− g2ξ − g3, which are sufficiently near respectively to x0, y0, while v is defined by
then ξ, η are respectively φ(v) and −φ′(v); for this equation leads to an expansion for ξ − x0in terms of v = u0and only one such expansion, and this is obtained by the same work as would be necessary to expand φ(v) when v is near to u0; the function φ(u) can therefore be continued by the help of this equation, from v = u0, provided the lower limit of |ξ − x0| necessary for the expansions is not zero in the neighbourhood of any value (x0, y0). In fact the function φ(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 φ(u) is greater than zero; the same will therefore be the case for the lower limit of the radii |ξ − x0| necessary for the continuations spoken of above provided that the values of (ξ, η) considered do not lead to infinitely increasing values of v; there does not exist, however, any definite point (ξ0, η0) in the neighbourhood of which the integral∫(ξ, η)(x0, y0)dξ/η increases indefinitely, it is only by a path of infinite length that the integral can so increase. We infer therefore that if (ξ, η) be any point, where η2= 4ξ3− g2ξ − g3, and v be defined by
then ξ = φ(v) and η = −φ′(v). Thus this equation determines (ξ, η) without ambiguity. In particular the additive indeterminatenesses of the integral obtained by closed circuits of the point of integration are periods of the function φ(u); by considerations advanced above it appears that these periods are sums of integral multiples of two which may be taken to be
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), (∞, ∞), which leads back to the initial point (x0, y0), which is impossible. On the whole, therefore, it appears that the function φ(u) agrees with the function ℜ(u) previously discussed, and the discussion of the elliptic integrals can be continued in the manner given under § 14,Doubly Periodic Functions.
§ 21.Modular Functions.—One result of the previous theory is the remarkable fact that if
where y2= 4(x − e1) (x − e2) (x − e3), then we have
e1= (½ω)−2+ Σ′ {[(m + ½) ω + m′ω′]−2− [mω + m′ω′]−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 ω′/ω.