| ∫ β α | sin mz | dz = | 1 | ∫ γ α sin mz dz + | 1 | ∫ β γ sin mz dz |
| sin z | sin α | sin β |
where α < γ < β, hence
| | ∫ β α | sin mz | dz | < | 2 | ( | 1 | + | 1 | ) < | 4 | ; |
| sin z | m | sin α | sin β | m sin α |
a precisely similar proof shows that | ∫ β α (sin mz / z) dz | < 4 / mα, hence the integrals ∫ β α (sin mz / sin z) dz, ∫ β α (sin mz / z) dz, converge to the limit zero, as m is indefinitely increased.
3. If α > 0, | ∫ ∞ α (sin θ / θ) dθ | cannot exceed ½ π. For by the mean-value theorem | ∫ h α (sin θ / θ) dθ | < 2/α + 2/h,
hence | Lh = ∞ ∫ h α (sin θ / θ) dθ | ≦ 2/α
in particular if α ≧ π | ∫ ∞ α (sin θ / θ) dθ | ≦ 2/π < π/2.
Again d/dα ∫ ∞ α (sin θ / θ) dθ = − (sin α) / α, α > 0,
therefore ∫ ∞ α (sin θ / θ) dθ increases as α diminishes, when θ < α < π;
but lim α=0∫ ∞ α (sin θ / θ) dθ = π/2, hence | ∫ ∞ α (sin θ / θ) dθ | < π/2,