Relationships between denominator constants from fractional decompositionPerforming fractional decomposition on the fraction a ( x − c ) + b ( x − y 1 ) ( x − y 2 ) = A x − y 1 + B x − y 2 (1)where x is a variable and the rest are constants, you find A = a ( y 1 − c ) + b y 1 − y 2 B = a ( y 2 − c ) + b y 2 − y 1 If you try to check these values by inserting A and B into equation (1), you arrive at this equation a ( x − c ) + b ( x − y 1 ) ( x − y 2 ) = ( y 1 − y 2 ) 2 [ a ( x − c ) + b ] ( x − y 1 ) ( x − y 2 ) which implies that ( y 1 − y 2 ) 2 = 1, despite these being just arbitrary constants. Is this true? Where does this come from? I came upon this from checking the values obtained from fractional decomposition for a dynamic model on a first order system.

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Relationships between denominator constants from fractional decompositionPerforming fractional decomposition on the fraction            a      (      x      −      c      )      +      b              (      x      −              y        1            )      (      x      −              y        2            )        =      A          x      −              y        1              +      B          x      −              y        2            (1)where x is a variable and the rest are constants, you find  A  =            a      (              y        1            −      c      )      +      b                      y        1            −              y        2              B  =            a      (              y        2            −      c      )      +      b                      y        2            −              y        1            If you try to check these values by inserting A and B into equation (1), you arrive at this equation            a      (      x      −      c      )      +      b              (      x      −              y        1            )      (      x      −              y        2            )        =            (              y        1            −              y        2                    )        2            [      a      (      x      −      c      )      +      b      ]              (      x      −              y        1            )      (      x      −              y        2            )      which implies that   (      y    1    −      y    2        )    2    =  1, despite these being just arbitrary constants. Is this true? Where does this come from? I came upon this from checking the values obtained from fractional decomposition for a dynamic model on a first order system.

Vicente Macias · Accepted Answer

Your A and B are correct. Then you had a mistake somewhere. Inserting A and B on the RHS gives the following steps:            (      a      (              y        1            −      c      )      +      b      )              (              y        1            −              y        2            )      (      x      −              y        1            )        −            (      a      (              y        2            −      c      )      +      b      )              (              y        1            −              y        2            )      (      x      −              y        2            )          =            (      a      (              y        1            −      c      )      +      b      )      (      x      −              y        2            )              (              y        1            −              y        2            )      (      x      −              y        1            )      (      x      −              y        2            )        −            (      a      (              y        2            −      c      )      +      b      )      (      x      −              y        1            )              (              y        1            −              y        2            )      (      x      −              y        1            )      (      x      −              y        2            )          =            a      (      x      −      c      )      (              y        1            −              y        2            )      +      b      (              y        1            −              y        2            )              (              y        1            −              y        2            )      (      x      −              y        1            )      (      x      −              y        2            )          =            a      (      x      −      c      )      +      b              (      x      −              y        1            )      (      x      −              y        2            )

Relationships between denominator constants from fractional decomposition Performing fractional decomposition on the fraction (a(x-c)+b)/((x-y_1)(x-y_2))=(A)/(x-y_1) + (B)/(x-y_2) where x is a variable and the rest are constants, you find A = (a(y_1-c)+b)/(y_1-y_2) B =(a(y_2-c)+b)/(y_2-y_1)

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