Transcription of Partial Fraction Decomposition for Inverse Laplace …
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Partial Fraction Decomposition for Inverse Laplace …
Partial Fraction Decomposition for Inverse Laplace Trans-formUsually Partial fractions method starts withpolynomial long divisionin order torepresent a Fraction as a sum of a polynomial and an another Fraction , where thedegree of the polynomial in the numerator of the new fractionis less than the degreeof the polynomial in its denominator:s3+ 1s2+ 1=s+ s+ 1s2+ , however,neverhave to do this polynomial long division, when Partial FractionDecomposition is applied to problems from Chapter important fact in Chapter 6 is that we use only the followingthreetypesof a(s a)2+b2, (s a)2+b2, (s a)n,because we know the corresponding Inverse Laplace 1 s a(s a)2+b2 =eatcos(bt), 1 b(s a)2+b2 =eatsin(bt),(1) 1 1s a =eat,L 1 1(s a)2 =teat,L 1 1(s a)3 =t22eat,(2)L 1 1(s a)4 =t36eat,L 1 1(s a)5 =t424eat, ..L 1 1(s a)n+1 =tnn!
On the right-hand side the coefficients in front of s2, s, and 1 are 1, 0, and 1, respec- tively. On the left-hand side the coefficients in front of s2, s, and 1 are A + B + C, C −B, and −A, respectively.
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