6.3. Limits only on bounded intervals🔗

For each fixed t,

P_h(t)/h² → t² + 1/4.

This follows from

P_h(t) = 4(sinh(ht/2)² + sin(h/4)²).

Convergence is uniform on each bounded t interval. Alternatively, for h ≤ 1 and 0 ≤ t ≤ M, use

cosh(u)−1 ≤ (u²/2) cosh(u),   1−cos(v) ≤ v²/2

with u=ht and v=h/2 to bound P_h(t)/h² by M² cosh(M)+1/4. Continuity of g gives domination on every compact interval. Hence those integrals converge to the integrals with weight (t²+1/4)^k cosh(at). This step does not assert integrability on the full boundary half-line.