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Partial fraction expansion confusion


Derivation of the general forms of partial fractionsWhy do you need two fractions for partial fraction decomposition with repeated factors?Integration - Partial Fraction DecompositionPartial Fraction Expansion of Transfer FunctionHow to solve Partial Fraction- Improper FractionsPartial Fraction Solution?Extra Square in Partial FractionLaurent Expansion partial fractionsComplicated partial fraction expansionIntegration of Partial Fraction ExpansionSimple partial fraction expansionConfusion with how partial fractions work













1












$begingroup$


Can someone please explain why: $$frac{1}{s^2(s+2)}=frac{A}{s}+frac{B}{s^2}+frac{C}{(s+2)}$$



And not:$$frac{1}{s^2(s+2)}=frac{A}{s^2}+frac{B}{(s+2)}$$



I'm a bit confused where the extra s term comes from in the first equation.










share|cite|improve this question









$endgroup$












  • $begingroup$
    There are many answers available on MSE, i.e. here and here
    $endgroup$
    – callculus
    1 hour ago
















1












$begingroup$


Can someone please explain why: $$frac{1}{s^2(s+2)}=frac{A}{s}+frac{B}{s^2}+frac{C}{(s+2)}$$



And not:$$frac{1}{s^2(s+2)}=frac{A}{s^2}+frac{B}{(s+2)}$$



I'm a bit confused where the extra s term comes from in the first equation.










share|cite|improve this question









$endgroup$












  • $begingroup$
    There are many answers available on MSE, i.e. here and here
    $endgroup$
    – callculus
    1 hour ago














1












1








1





$begingroup$


Can someone please explain why: $$frac{1}{s^2(s+2)}=frac{A}{s}+frac{B}{s^2}+frac{C}{(s+2)}$$



And not:$$frac{1}{s^2(s+2)}=frac{A}{s^2}+frac{B}{(s+2)}$$



I'm a bit confused where the extra s term comes from in the first equation.










share|cite|improve this question









$endgroup$




Can someone please explain why: $$frac{1}{s^2(s+2)}=frac{A}{s}+frac{B}{s^2}+frac{C}{(s+2)}$$



And not:$$frac{1}{s^2(s+2)}=frac{A}{s^2}+frac{B}{(s+2)}$$



I'm a bit confused where the extra s term comes from in the first equation.







partial-fractions






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 1 hour ago









stuartstuart

1968




1968












  • $begingroup$
    There are many answers available on MSE, i.e. here and here
    $endgroup$
    – callculus
    1 hour ago


















  • $begingroup$
    There are many answers available on MSE, i.e. here and here
    $endgroup$
    – callculus
    1 hour ago
















$begingroup$
There are many answers available on MSE, i.e. here and here
$endgroup$
– callculus
1 hour ago




$begingroup$
There are many answers available on MSE, i.e. here and here
$endgroup$
– callculus
1 hour ago










4 Answers
4






active

oldest

votes


















2












$begingroup$

If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $frac{A}{as+b}+frac{B}{(as+b)^2}+cdots+frac{Z}{(as+b)^n}$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $frac{A}{s}+frac{B}{s^2}$.






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    That is because for
    $$frac{as^2+bs+c}{s^2(s+2)}=frac{A}{s^2}+frac{B}{(s+2)},$$
    the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



    Or more simply, consider the example
    $$
    frac{s+1}{s^2}=frac{1}{s^2}+frac{1}{s}
    $$






    share|cite|improve this answer









    $endgroup$





















      2












      $begingroup$

      The general result is the following.




      Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
      $$frac{p(s)}{q(s)}=frac{r_1(s)}{q_1(s)}+frac{r_2(s)}{q_2(s)} .$$




      In your case the denominator factorises as $s^2$ times $s+2$ so you have
      $$frac1{s^2(s+2)}=frac{As+B}{s^2}+frac{C}{s+2} .$$
      It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



      Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.






      share|cite|improve this answer









      $endgroup$





















        0












        $begingroup$

        One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



        By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



        $$frac{1}{4(s+2)}$$



        For large $s$ we can expand this in powers of $1/s$:



        $$frac{1}{4(s+2)} = frac{1}{4 s}frac{1}{1+frac{2}{s}} = frac{1}{4s} - frac{1}{2 s^2} + mathcal{O}left(frac{1}{s^3}right)$$



        The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



        $$frac{1}{2 s^2}-frac{1}{4s} $$



        The complete partial fraction expansion is thus given by:



        $$frac{1}{2 s^2}-frac{1}{4s} + frac{1}{4(s+2)} $$






        share|cite|improve this answer









        $endgroup$














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          4 Answers
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          active

          oldest

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          4 Answers
          4






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $frac{A}{as+b}+frac{B}{(as+b)^2}+cdots+frac{Z}{(as+b)^n}$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $frac{A}{s}+frac{B}{s^2}$.






          share|cite|improve this answer









          $endgroup$


















            2












            $begingroup$

            If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $frac{A}{as+b}+frac{B}{(as+b)^2}+cdots+frac{Z}{(as+b)^n}$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $frac{A}{s}+frac{B}{s^2}$.






            share|cite|improve this answer









            $endgroup$
















              2












              2








              2





              $begingroup$

              If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $frac{A}{as+b}+frac{B}{(as+b)^2}+cdots+frac{Z}{(as+b)^n}$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $frac{A}{s}+frac{B}{s^2}$.






              share|cite|improve this answer









              $endgroup$



              If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $frac{A}{as+b}+frac{B}{(as+b)^2}+cdots+frac{Z}{(as+b)^n}$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $frac{A}{s}+frac{B}{s^2}$.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered 1 hour ago









              Julian MejiaJulian Mejia

              39328




              39328























                  2












                  $begingroup$

                  That is because for
                  $$frac{as^2+bs+c}{s^2(s+2)}=frac{A}{s^2}+frac{B}{(s+2)},$$
                  the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



                  Or more simply, consider the example
                  $$
                  frac{s+1}{s^2}=frac{1}{s^2}+frac{1}{s}
                  $$






                  share|cite|improve this answer









                  $endgroup$


















                    2












                    $begingroup$

                    That is because for
                    $$frac{as^2+bs+c}{s^2(s+2)}=frac{A}{s^2}+frac{B}{(s+2)},$$
                    the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



                    Or more simply, consider the example
                    $$
                    frac{s+1}{s^2}=frac{1}{s^2}+frac{1}{s}
                    $$






                    share|cite|improve this answer









                    $endgroup$
















                      2












                      2








                      2





                      $begingroup$

                      That is because for
                      $$frac{as^2+bs+c}{s^2(s+2)}=frac{A}{s^2}+frac{B}{(s+2)},$$
                      the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



                      Or more simply, consider the example
                      $$
                      frac{s+1}{s^2}=frac{1}{s^2}+frac{1}{s}
                      $$






                      share|cite|improve this answer









                      $endgroup$



                      That is because for
                      $$frac{as^2+bs+c}{s^2(s+2)}=frac{A}{s^2}+frac{B}{(s+2)},$$
                      the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



                      Or more simply, consider the example
                      $$
                      frac{s+1}{s^2}=frac{1}{s^2}+frac{1}{s}
                      $$







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered 1 hour ago









                      Holding ArthurHolding Arthur

                      1,350417




                      1,350417























                          2












                          $begingroup$

                          The general result is the following.




                          Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
                          $$frac{p(s)}{q(s)}=frac{r_1(s)}{q_1(s)}+frac{r_2(s)}{q_2(s)} .$$




                          In your case the denominator factorises as $s^2$ times $s+2$ so you have
                          $$frac1{s^2(s+2)}=frac{As+B}{s^2}+frac{C}{s+2} .$$
                          It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



                          Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.






                          share|cite|improve this answer









                          $endgroup$


















                            2












                            $begingroup$

                            The general result is the following.




                            Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
                            $$frac{p(s)}{q(s)}=frac{r_1(s)}{q_1(s)}+frac{r_2(s)}{q_2(s)} .$$




                            In your case the denominator factorises as $s^2$ times $s+2$ so you have
                            $$frac1{s^2(s+2)}=frac{As+B}{s^2}+frac{C}{s+2} .$$
                            It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



                            Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.






                            share|cite|improve this answer









                            $endgroup$
















                              2












                              2








                              2





                              $begingroup$

                              The general result is the following.




                              Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
                              $$frac{p(s)}{q(s)}=frac{r_1(s)}{q_1(s)}+frac{r_2(s)}{q_2(s)} .$$




                              In your case the denominator factorises as $s^2$ times $s+2$ so you have
                              $$frac1{s^2(s+2)}=frac{As+B}{s^2}+frac{C}{s+2} .$$
                              It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



                              Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.






                              share|cite|improve this answer









                              $endgroup$



                              The general result is the following.




                              Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
                              $$frac{p(s)}{q(s)}=frac{r_1(s)}{q_1(s)}+frac{r_2(s)}{q_2(s)} .$$




                              In your case the denominator factorises as $s^2$ times $s+2$ so you have
                              $$frac1{s^2(s+2)}=frac{As+B}{s^2}+frac{C}{s+2} .$$
                              It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



                              Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered 1 hour ago









                              DavidDavid

                              69.7k668131




                              69.7k668131























                                  0












                                  $begingroup$

                                  One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



                                  By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



                                  $$frac{1}{4(s+2)}$$



                                  For large $s$ we can expand this in powers of $1/s$:



                                  $$frac{1}{4(s+2)} = frac{1}{4 s}frac{1}{1+frac{2}{s}} = frac{1}{4s} - frac{1}{2 s^2} + mathcal{O}left(frac{1}{s^3}right)$$



                                  The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



                                  $$frac{1}{2 s^2}-frac{1}{4s} $$



                                  The complete partial fraction expansion is thus given by:



                                  $$frac{1}{2 s^2}-frac{1}{4s} + frac{1}{4(s+2)} $$






                                  share|cite|improve this answer









                                  $endgroup$


















                                    0












                                    $begingroup$

                                    One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



                                    By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



                                    $$frac{1}{4(s+2)}$$



                                    For large $s$ we can expand this in powers of $1/s$:



                                    $$frac{1}{4(s+2)} = frac{1}{4 s}frac{1}{1+frac{2}{s}} = frac{1}{4s} - frac{1}{2 s^2} + mathcal{O}left(frac{1}{s^3}right)$$



                                    The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



                                    $$frac{1}{2 s^2}-frac{1}{4s} $$



                                    The complete partial fraction expansion is thus given by:



                                    $$frac{1}{2 s^2}-frac{1}{4s} + frac{1}{4(s+2)} $$






                                    share|cite|improve this answer









                                    $endgroup$
















                                      0












                                      0








                                      0





                                      $begingroup$

                                      One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



                                      By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



                                      $$frac{1}{4(s+2)}$$



                                      For large $s$ we can expand this in powers of $1/s$:



                                      $$frac{1}{4(s+2)} = frac{1}{4 s}frac{1}{1+frac{2}{s}} = frac{1}{4s} - frac{1}{2 s^2} + mathcal{O}left(frac{1}{s^3}right)$$



                                      The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



                                      $$frac{1}{2 s^2}-frac{1}{4s} $$



                                      The complete partial fraction expansion is thus given by:



                                      $$frac{1}{2 s^2}-frac{1}{4s} + frac{1}{4(s+2)} $$






                                      share|cite|improve this answer









                                      $endgroup$



                                      One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



                                      By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



                                      $$frac{1}{4(s+2)}$$



                                      For large $s$ we can expand this in powers of $1/s$:



                                      $$frac{1}{4(s+2)} = frac{1}{4 s}frac{1}{1+frac{2}{s}} = frac{1}{4s} - frac{1}{2 s^2} + mathcal{O}left(frac{1}{s^3}right)$$



                                      The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



                                      $$frac{1}{2 s^2}-frac{1}{4s} $$



                                      The complete partial fraction expansion is thus given by:



                                      $$frac{1}{2 s^2}-frac{1}{4s} + frac{1}{4(s+2)} $$







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered 27 mins ago









                                      Count IblisCount Iblis

                                      8,50221534




                                      8,50221534






























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