-- inverse Laplace transform for s / (s^2 + a s + b) -- from Wolfram T = (a exp(t (-1/2 sqrt(a^2 - 4 b) - a/2)) - a exp(t (1/2 sqrt(a^2 - 4 b) - a/2)) + sqrt(a^2 - 4 b) exp(t (-1/2 sqrt(a^2 - 4 b) - a/2)) + sqrt(a^2 - 4 b) exp(t (1/2 sqrt(a^2 - 4 b) - a/2)))/(2 sqrt(a^2 - 4 b)) T = eval(T, a^2 - 4 b, K^2) T1 = -a exp(1/2 K t) / (2 K) T2 = a exp(-1/2 K t) / (2 K) T3 = 1/2 exp(1/2 K t) T4 = 1/2 exp(-1/2 K t) check(T == (T1 + T2 + T3 + T4) exp(-1/2 a t)) -- factor out i k = sqrt(4 b - a^2) K = i k check(T1 + T2 == -a sin(1/2 k t) / k) check(T3 + T4 == cos(1/2 k t)) f = (cos(1/2 k t) - a sin(1/2 k t) / k) exp(-1/2 a t) check(T == f) F = s / (s^2 + a s + b) F "Inverse transform is" f Run