| ∫ | 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)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
| ∫ −10 | dz | + ∫ 0−1 | dz | + ∫ 10 | dz | + ∫ 01 | dz | , |
| (1 − z²)1/2 | −(1 − z²)1/2 | −(1 − z²)1/2 | (1 − z²)1/2 |
where, in each case, (1 − z²)1/2 is real and positive; that is, it gives