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 definite order. There exists a generalization of another theorem also proved above for doubly periodic functions, namely, that the sum of the values of the argument in one parallelogram of periods for which a doubly periodic function takes a given value is independent of that value; this generalization, known as Abel’s Theorem, is given § 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 ƒ(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
| ∫ | dz | , ∫ | zdz | , ∫ | dz |
| s | s | (z − c)s |
can be expressed by rational and logarithms of rational functions of s and z; but it can be shown that every integral ∫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 as Abel’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−1 in the expansions of the function
| R(s, z) | dz | λ ( | H(s, z) − B | ), |
| dt | H(s, z) − A |
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
| [ | 1 | R(s, z) | dz | ]t−1 = 0, |
| H(s, z) − μ | dt |
wherein μ is a constant. (For illustrations see below, under § 20, Elliptic Integrals.)
§ 18. Indeterminateness of Algebraic Integrals.—The theorem that the integral ∫ xa ƒ(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 ∫ z1 z−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 = ∫ z1 z−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)12 = −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
| ∫ −10 | dz | + ∫ 0−1 | dz | + ∫ 10 | dz | + ∫ 01 | dz | , |
| (1 − z2)12 | −(1 − z2)12 | −(1 − z2)12 | (1 − z2)12 |
where, in each case, (1 − z2)12 is real and positive; that is, it gives
| −4 ∫ 10 | dz |
| (1 − z2)12 |
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
| u = ∫ (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 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/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 algebraic
