THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

644 Appendix C. Using Laplace Transforms to Solve Differential Equations(a)(b)Figure C.3. Graphical residue approach for complex roots.We observe that the coefficient of the second term, which represents the residueat s =−α – iβ is the complex conjugate of the coefficient of the first term that isthe residue at s =–α + iβ. Equation (C.14) can be expressed asX(s) =Me iφs + α − iβ +Me−iφs + α + iβ+ additonal termswhereM =z 1 z 2P 1 P 2 P 3 P 4, φ = θ 2 + θ 6 − (θ 1 + θ 3 + θ 4 + θ 5 )