Laplace transforms of functions teᵗ

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.)

(i) L{t eᵗ}

L{t eᵗ} = −d/ds [1/(s − 1)] = 1/(s − 1)²      (s > 1)

(ii) L{sinh at · sin at}

Write sinh at = (e^(at) − e^(−at))/2 and use the first shifting theorem:

L{sinh at sin at} = (1/2) [ a/((s − a)² + a²) − a/((s + a)² + a²) ]
                  = (a/2) · [(s + a)² + a² − (s − a)² − a²] / [(s² + 2a²)² − 4a²s²]
                  = (a/2) · 4as/(s⁴ + 4a⁴)
                  = 2a² s/(s⁴ + 4a⁴)

Answers: (i) 1/(s − 1)²; (ii) 2a²s/(s⁴ + 4a⁴).