Real and Simple or Distinct Poles

If all the poles of F1(s) are of first order, Equation may be written in terms of a partial fraction expansion as

where p₁, p₂,..., p_n are distinct, and K₁, K₂,..., K_n are nonzero finite constants. For any k, the evaluation of the residue Kk of F₁(s) corresponding to the pole s = pk is done by multiplying both sides by (s - pk) and letting s → pk,

Equation is valid for k = 1, 2, . . . , n. Once the K's are determined in Equation, the inverse Laplace transform of each of the terms can be written easily in order to obtain the complete time solution.