∫ψ(z) dz = z ψ(z) + 1⁄3 z3 Π(z) + 1⁄3 ∫z3 φ(z) dz,
∫zψ(z) dz = ½ z2ψ(z) + 1⁄8 z4 Π(z) + 1⁄8 ∫z4 φ(z) dz.
In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write
∫∞0 ψ(z) dz = 1⁄3 ∫∞2 z3 φ(z) dz, ∫∞0 zψ(z) dz = 1⁄8 ∫∞0 z4 φ(z) dz; (36)
so that
| K0 = | 2π | ∫∞0z3 φ(z) dz, T0 = | π | ∫∞0z4 φ(z) dz. (37) |
| 3 | 8 |
A few examples of these formulae will promote an intelligent comprehension of the subject. One of the simplest suppositions open to us is that
φ(ƒ) = eβƒ. (38)
From this we obtain
Π(z) = β-1 e-βz, ψ(z) = β-3 (βz + 1)e-βz, (39)