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
| J3 = ½ƒ′(θ) ∫ | dz |
| (z − θ) s |
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 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 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−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, J1 denote 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
| s + s0 | + z √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 √a0 or 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
| s + s0 | + z √a0 − J1 + J2 + (a0z02 + 2a1z0) ∫ | dz | = 0; |
| z − z0 | s |
and there exists the analogous equation
| s | + z √a0 − J1 + J3 + (a0θ2 + 2a1θ) ∫ | dz | . |
| z − θ | s |
Consider now the integral
| P = ∫ ( | s + s0 | + z √a0 ) | dz | ; |
| z − z0 | 2s |