Partial Fraction Decomposition with complex polesI have a function which I'd like to perform partial fraction decomposition on, to allow easier inverse laplace transform. F ( s ) = 1 s ( s 2 + 140 s + 10 4 ) I begin with finding the poles s = 0 , s = − 70 ± j ⋅ 10 51 To which I then try putting F ( s ) in this form: F ( s ) = A s + B s − 70 + j ⋅ 10 51 + B ∗ s − 70 − j ⋅ 10 51 Because one unknown ( B ∗ ) is just the complex conjugate of B, I only need to find out what A, and B is. A = F ( 0 ) = 10 − 4 B = F ( − 70 + j ⋅ 10 51 ) ≈ 6.31514 ⋅ 10 9 + j ⋅ 6.44274 ⋅ 10 9 Where the last step was done in Mathematica.Answer, according to several other people, is supposed to be B = − 70 + j ⋅ 71.41, which looks a lot nicer, but I'm not sure HOW they got to that answer.

Question

Partial Fraction Decomposition with complex polesI have a function which I&#039;d like to perform partial fraction decomposition on, to allow easier inverse laplace transform.  F  (  s  )  =      1          s      (              s        2            +      140      s      +              10        4            )      I begin with finding the poles  s  =  0  ,  s  =  −  70  ±  j  ⋅  10      51  To which I then try putting   F  (  s  ) in this form:  F  (  s  )  =      A    s    +      B          s      −      70      +      j      ⋅      10              51              +            B      ∗              s      −      70      −      j      ⋅      10              51            Because one unknown (      B    ∗  ) is just the complex conjugate of   B, I only need to find out what   A, and   B is.  A  =  F  (  0  )  =      10          −      4        B  =  F  (  −  70  +  j  ⋅  10      51    )  ≈  6.31514  ⋅      10    9    +  j  ⋅  6.44274  ⋅      10    9  Where the last step was done in Mathematica.Answer, according to several other people, is supposed to be   B  =  −  70  +  j  ⋅  71.41, which looks a lot nicer, but I&#039;m not sure HOW they got to that answer.

Perman7z · Accepted Answer

Note that the partial fraction expansion of   F  (  s  ) is:  F  (  s  )  =      A    s    +      B          s      −              s        0              +            B      ∗              s      −              s        0        ∗            where       s    0    =  −  70  +  i  10      51   and       s    0    ∗    =      s    0    =  −  70  −  i  10      51   are the complex-conjugate roots of       s    2    +  140  s  +      10    4  Note that we have                    A                            =                  lim                      s            →            0                          s        F        (        s        )                                                        =                  10                      −            4                                            and                    B                            =                  lim                      s            →                          s              0                                      (        s        −                  s          0                )        F        (        s        )                                                        =                  1                                    s              0                        (                          s              0                        −                          s              0              ∗                        )                                                                          =                              −            70            −            i            10                          51                                            i            20                          51                                      10              4                                                                                      =                              −            51            +            i            7                          51                                1020000