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Laplace transforms of functions tcoshat

Short answer

A sufficient condition: piecewise continuous and of exponential order. a) Differentiate s/(s²−a²) → (s²+a²)/(s²−a²)². b) Integrate 2/(s²+4) → tan⁻¹(2/s).

State the existence condition for the Laplace transform. Obtain the Laplace transform of a) t cosh at b) sin 2t / t.

Existence condition. The Laplace transform L{f(t)} = ∫₀^∞ e^(−st) f(t) dt exists for s > k if f(t) is piecewise continuous on every finite interval [0, T] and of exponential order, that is |f(t)| ≤ M e^(kt) for all t ≥ T. (These conditions are sufficient, not necessary.)

a) L{t cosh at}

Use L{t f(t)} = −dF/ds with F(s) = L{cosh at} = s/(s² − a²).

−d/ds [s/(s² − a²)] = −[(s² − a²) − 2s²]/(s² − a²)²
                    = (s² + a²)/(s² − a²)²        (s > |a|)

b) L{sin 2t / t}

Use L{f(t)/t} = ∫ₛ^∞ F(u) du, with F(s) = 2/(s² + 4). (sin 2t / t → 2 as t → 0.)

∫ₛ^∞ 2/(u² + 4) du = [tan⁻¹(u/2)]ₛ^∞ = π/2 − tan⁻¹(s/2) = tan⁻¹(2/s)

Answers: a) (s² + a²)/(s² − a²)²; b) tan⁻¹(2/s) = cot⁻¹(s/2).

✓ Answer checked by Claude Opus 5.5 (high effort)

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