Extended definition of the initial value theorem applied to the bilateral Laplace transform

If the bilateral Laplace transform of f(t) is in the form of a rational function FII(S) = anS^n-1+ an-1S^n-2+ ... + a1/Sⁿ+ b1S^n-1+ ... + bn, then it is shown that f^(k-1)(0+) - f^(k-1)(0-) = (-1)^{k - 1}Δk, where f^(k)(t) is the kth derivative of f(t), Δ1= an, and Δk= Σi=1^k-1(-1)ⁱ⁺¹biΔk-i+ (-1)^k-1an-k+1, k = 2, 3, ..., n.