3.2 Partial Fraction Expansion

• Laplace transform in most case appear in a rational form of s:

• The coefficient ak and bk are real numbers for k = 1, 2, ..., n
• m < n : F(s) is a proper rational function
• m >= n : F(s) is an improper rational function
• In proper rational function, roots of N(s) (found by setting N(s) = 0) are called zeros of F(s) ; root of D(s) (found by setting D(s) = 0) are called poles of F(s)
• Distinct poles:
• If all the poles (p1, p2, p3, ..., pn) are distinct, F(s):

 

• Using partial fraction expansion method, F(s):

• Residue rk :

• Complex poles:

• Complex poles occur in complex conjugate pairs, the number of complex poles is even

• Multiple (Repeated) poles:

• Partial Fraction Expansion:

• Residue for repeated poles: