Partial fraction failure for $frac{1}{sin(x-a)sin(x-b)}$












1












$begingroup$


Here in this answer I have tried to perform partial fraction for $frac{1}{sin(x-a)sin(x-b)}$ as follows:
$$frac{1}{sin(x-a)sin(x-b)}=frac{p}{sin(x-a)}+frac{q}{sin(x-b)} quad(1)$$
Multiplying $(1)$ by $sin(x-a)$ and plugging $x=a$ yields:
$$left(frac{1}{sin(x-b)}=p+qunderset{rightarrow 0}{frac{sin(x-a)}{sin(x-b)}}right)bigg|_{x=a}Rightarrow p=frac{1}{sin(a-b)}$$
Also to obtain q, multiply $(1)$ by $sin(x-b)$ and plugg $x=b$ gives:
$$left(frac{1}{sin(x-a)}=punderset{rightarrow 0}{frac{sin(x-b)}{sin(x-a)}}+qright)bigg|_{x=b}Rightarrow q=frac{1}{sin(b-a)}=-p$$
So that means we can rewrite the original expression as: $$frac{1}{sin(x-a)sin(x-b)}=frac{1}{sinleft(a-bright)}left(frac{1}{sinleft(x-aright)}-frac{1}{sinleft(x-bright)}right)$$
However as pointed out in the comments this doesn't verify. Can anyone point out where I went sideways with this?



Also the same approach can be found here. Is there a rigurous way to prove that we can't perform partial fraction in this case?



Further question, what a about a general case such as proving/disproving: $${prod_{k=1}^nfrac{1}{sin(x-x_k)}}neqsum_{k=1}^n frac{a_k}{sin(x-x_k)}$$










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$endgroup$












  • $begingroup$
    Write $$sin(a-b)=sin{x-b-(x-a)}=?$$
    $endgroup$
    – lab bhattacharjee
    Dec 20 '18 at 12:46






  • 1




    $begingroup$
    What's the goal here? To find constants $p,q$ that make (1) hold? But perhaps there are no such constants.
    $endgroup$
    – lulu
    Dec 20 '18 at 12:48










  • $begingroup$
    Still doesn't add up... // Sure, so that we can evaluate the integral afterwards, but you're probably right, I haven't thought that we can't find such $p$ and $q$.
    $endgroup$
    – Zacky
    Dec 20 '18 at 12:49


















1












$begingroup$


Here in this answer I have tried to perform partial fraction for $frac{1}{sin(x-a)sin(x-b)}$ as follows:
$$frac{1}{sin(x-a)sin(x-b)}=frac{p}{sin(x-a)}+frac{q}{sin(x-b)} quad(1)$$
Multiplying $(1)$ by $sin(x-a)$ and plugging $x=a$ yields:
$$left(frac{1}{sin(x-b)}=p+qunderset{rightarrow 0}{frac{sin(x-a)}{sin(x-b)}}right)bigg|_{x=a}Rightarrow p=frac{1}{sin(a-b)}$$
Also to obtain q, multiply $(1)$ by $sin(x-b)$ and plugg $x=b$ gives:
$$left(frac{1}{sin(x-a)}=punderset{rightarrow 0}{frac{sin(x-b)}{sin(x-a)}}+qright)bigg|_{x=b}Rightarrow q=frac{1}{sin(b-a)}=-p$$
So that means we can rewrite the original expression as: $$frac{1}{sin(x-a)sin(x-b)}=frac{1}{sinleft(a-bright)}left(frac{1}{sinleft(x-aright)}-frac{1}{sinleft(x-bright)}right)$$
However as pointed out in the comments this doesn't verify. Can anyone point out where I went sideways with this?



Also the same approach can be found here. Is there a rigurous way to prove that we can't perform partial fraction in this case?



Further question, what a about a general case such as proving/disproving: $${prod_{k=1}^nfrac{1}{sin(x-x_k)}}neqsum_{k=1}^n frac{a_k}{sin(x-x_k)}$$










share|cite|improve this question











$endgroup$












  • $begingroup$
    Write $$sin(a-b)=sin{x-b-(x-a)}=?$$
    $endgroup$
    – lab bhattacharjee
    Dec 20 '18 at 12:46






  • 1




    $begingroup$
    What's the goal here? To find constants $p,q$ that make (1) hold? But perhaps there are no such constants.
    $endgroup$
    – lulu
    Dec 20 '18 at 12:48










  • $begingroup$
    Still doesn't add up... // Sure, so that we can evaluate the integral afterwards, but you're probably right, I haven't thought that we can't find such $p$ and $q$.
    $endgroup$
    – Zacky
    Dec 20 '18 at 12:49
















1












1








1





$begingroup$


Here in this answer I have tried to perform partial fraction for $frac{1}{sin(x-a)sin(x-b)}$ as follows:
$$frac{1}{sin(x-a)sin(x-b)}=frac{p}{sin(x-a)}+frac{q}{sin(x-b)} quad(1)$$
Multiplying $(1)$ by $sin(x-a)$ and plugging $x=a$ yields:
$$left(frac{1}{sin(x-b)}=p+qunderset{rightarrow 0}{frac{sin(x-a)}{sin(x-b)}}right)bigg|_{x=a}Rightarrow p=frac{1}{sin(a-b)}$$
Also to obtain q, multiply $(1)$ by $sin(x-b)$ and plugg $x=b$ gives:
$$left(frac{1}{sin(x-a)}=punderset{rightarrow 0}{frac{sin(x-b)}{sin(x-a)}}+qright)bigg|_{x=b}Rightarrow q=frac{1}{sin(b-a)}=-p$$
So that means we can rewrite the original expression as: $$frac{1}{sin(x-a)sin(x-b)}=frac{1}{sinleft(a-bright)}left(frac{1}{sinleft(x-aright)}-frac{1}{sinleft(x-bright)}right)$$
However as pointed out in the comments this doesn't verify. Can anyone point out where I went sideways with this?



Also the same approach can be found here. Is there a rigurous way to prove that we can't perform partial fraction in this case?



Further question, what a about a general case such as proving/disproving: $${prod_{k=1}^nfrac{1}{sin(x-x_k)}}neqsum_{k=1}^n frac{a_k}{sin(x-x_k)}$$










share|cite|improve this question











$endgroup$




Here in this answer I have tried to perform partial fraction for $frac{1}{sin(x-a)sin(x-b)}$ as follows:
$$frac{1}{sin(x-a)sin(x-b)}=frac{p}{sin(x-a)}+frac{q}{sin(x-b)} quad(1)$$
Multiplying $(1)$ by $sin(x-a)$ and plugging $x=a$ yields:
$$left(frac{1}{sin(x-b)}=p+qunderset{rightarrow 0}{frac{sin(x-a)}{sin(x-b)}}right)bigg|_{x=a}Rightarrow p=frac{1}{sin(a-b)}$$
Also to obtain q, multiply $(1)$ by $sin(x-b)$ and plugg $x=b$ gives:
$$left(frac{1}{sin(x-a)}=punderset{rightarrow 0}{frac{sin(x-b)}{sin(x-a)}}+qright)bigg|_{x=b}Rightarrow q=frac{1}{sin(b-a)}=-p$$
So that means we can rewrite the original expression as: $$frac{1}{sin(x-a)sin(x-b)}=frac{1}{sinleft(a-bright)}left(frac{1}{sinleft(x-aright)}-frac{1}{sinleft(x-bright)}right)$$
However as pointed out in the comments this doesn't verify. Can anyone point out where I went sideways with this?



Also the same approach can be found here. Is there a rigurous way to prove that we can't perform partial fraction in this case?



Further question, what a about a general case such as proving/disproving: $${prod_{k=1}^nfrac{1}{sin(x-x_k)}}neqsum_{k=1}^n frac{a_k}{sin(x-x_k)}$$







trigonometry proof-verification partial-fractions






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edited Dec 20 '18 at 13:29







Zacky

















asked Dec 20 '18 at 12:44









ZackyZacky

7,79511062




7,79511062












  • $begingroup$
    Write $$sin(a-b)=sin{x-b-(x-a)}=?$$
    $endgroup$
    – lab bhattacharjee
    Dec 20 '18 at 12:46






  • 1




    $begingroup$
    What's the goal here? To find constants $p,q$ that make (1) hold? But perhaps there are no such constants.
    $endgroup$
    – lulu
    Dec 20 '18 at 12:48










  • $begingroup$
    Still doesn't add up... // Sure, so that we can evaluate the integral afterwards, but you're probably right, I haven't thought that we can't find such $p$ and $q$.
    $endgroup$
    – Zacky
    Dec 20 '18 at 12:49




















  • $begingroup$
    Write $$sin(a-b)=sin{x-b-(x-a)}=?$$
    $endgroup$
    – lab bhattacharjee
    Dec 20 '18 at 12:46






  • 1




    $begingroup$
    What's the goal here? To find constants $p,q$ that make (1) hold? But perhaps there are no such constants.
    $endgroup$
    – lulu
    Dec 20 '18 at 12:48










  • $begingroup$
    Still doesn't add up... // Sure, so that we can evaluate the integral afterwards, but you're probably right, I haven't thought that we can't find such $p$ and $q$.
    $endgroup$
    – Zacky
    Dec 20 '18 at 12:49


















$begingroup$
Write $$sin(a-b)=sin{x-b-(x-a)}=?$$
$endgroup$
– lab bhattacharjee
Dec 20 '18 at 12:46




$begingroup$
Write $$sin(a-b)=sin{x-b-(x-a)}=?$$
$endgroup$
– lab bhattacharjee
Dec 20 '18 at 12:46




1




1




$begingroup$
What's the goal here? To find constants $p,q$ that make (1) hold? But perhaps there are no such constants.
$endgroup$
– lulu
Dec 20 '18 at 12:48




$begingroup$
What's the goal here? To find constants $p,q$ that make (1) hold? But perhaps there are no such constants.
$endgroup$
– lulu
Dec 20 '18 at 12:48












$begingroup$
Still doesn't add up... // Sure, so that we can evaluate the integral afterwards, but you're probably right, I haven't thought that we can't find such $p$ and $q$.
$endgroup$
– Zacky
Dec 20 '18 at 12:49






$begingroup$
Still doesn't add up... // Sure, so that we can evaluate the integral afterwards, but you're probably right, I haven't thought that we can't find such $p$ and $q$.
$endgroup$
– Zacky
Dec 20 '18 at 12:49












3 Answers
3






active

oldest

votes


















3












$begingroup$

What is really happening is that $xmapstosin(x-x_k)$ is a rational function of $exp(ix)$, so $displaystyleprod_{k=1}^nfrac{1}{sin(x-x_k)}$ has a partial fraction decomposition of the form
begin{align*}
prod_{k=1}^nfrac{1}{sin(x-x_k)}
&=prod_{k=1}^nfrac{2iexp(-ix_k)exp(ix)}{[exp(ix)]^2-[exp(ix_k)]^2}
\
&=sum_{k=1}^nleft(frac{alpha^-_k}{exp(ix)+exp(ix_k)}+frac{alpha^+_k}{exp(ix)-exp(ix_k)}right)
end{align*}

with $alpha_k^pminmathbb{C}$. To be able to say
$$
prod_{k=1}^nfrac{1}{sin(x-x_k)}=sum_{k=1}^n frac{a_k}{sin(x-x_k)}
$$

is thus the condition $alpha_k^+=alpha_k^-$ which we can check holds if and only if $n$ is odd.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    The problem is there are no constants $p,,q$ satisfying the intended identity. The reason $frac{1}{(x-a)(x-b)}=frac{p}{x-a}+frac{q}{x-b}$ works is because the denominators are degree-$1$ polynomials, allowing the numerators to be degree-$0$. More generally distinct functions $f,,g$ give $frac{1}{fg}=frac{p}{f}+frac{q}{g}$ with functions $p=frac{1}{g-f},,q=-p$, among other options. But usually there isn't a solution with constant $p,,q$.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Thanks! Is there a direct way to prove that there isn't a solution?
      $endgroup$
      – Zacky
      Dec 20 '18 at 12:59










    • $begingroup$
      @Zacky You've effectively already found it. Your calculation uniquely determined what $p,,q$ could be if they were constant, but we can verify it doesn't work.
      $endgroup$
      – J.G.
      Dec 20 '18 at 13:02










    • $begingroup$
      Can you please take a look here too: math.stackexchange.com/questions/1882536/… that is invalid too right?
      $endgroup$
      – Zacky
      Dec 20 '18 at 13:03






    • 1




      $begingroup$
      @Zacky I can't find a numerical counterexample to that, even with a computer search. I think the result is correct, even though the method of finding it needs more justification. Why does it only work with three sines, not two? Probably because the coefficients' ratios depend on $a,,b,,c$ in the three-sines case.
      $endgroup$
      – J.G.
      Dec 20 '18 at 13:19



















    1












    $begingroup$

    We actually have
    $$frac1{sin(x-a)sin(x-b)} = frac1{sin(x-b)-sin(x-a)}left(frac1{sin(x-a)}-frac1{sin(x-b)}right)$$
    That factor of $sin(x-b)-sin(x-a)$ is equal to $sin(a-b)$ at $a$ and at $b$, but not everywhere; it isn't constant.



    Partial fractions are a theorem of linear algebra. If the factors we're dealing with aren't linear or at least polynomial, the theorem isn't valid. There is no partial fraction expansion of the type you're looking for here.






    share|cite|improve this answer









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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      3












      $begingroup$

      What is really happening is that $xmapstosin(x-x_k)$ is a rational function of $exp(ix)$, so $displaystyleprod_{k=1}^nfrac{1}{sin(x-x_k)}$ has a partial fraction decomposition of the form
      begin{align*}
      prod_{k=1}^nfrac{1}{sin(x-x_k)}
      &=prod_{k=1}^nfrac{2iexp(-ix_k)exp(ix)}{[exp(ix)]^2-[exp(ix_k)]^2}
      \
      &=sum_{k=1}^nleft(frac{alpha^-_k}{exp(ix)+exp(ix_k)}+frac{alpha^+_k}{exp(ix)-exp(ix_k)}right)
      end{align*}

      with $alpha_k^pminmathbb{C}$. To be able to say
      $$
      prod_{k=1}^nfrac{1}{sin(x-x_k)}=sum_{k=1}^n frac{a_k}{sin(x-x_k)}
      $$

      is thus the condition $alpha_k^+=alpha_k^-$ which we can check holds if and only if $n$ is odd.






      share|cite|improve this answer









      $endgroup$


















        3












        $begingroup$

        What is really happening is that $xmapstosin(x-x_k)$ is a rational function of $exp(ix)$, so $displaystyleprod_{k=1}^nfrac{1}{sin(x-x_k)}$ has a partial fraction decomposition of the form
        begin{align*}
        prod_{k=1}^nfrac{1}{sin(x-x_k)}
        &=prod_{k=1}^nfrac{2iexp(-ix_k)exp(ix)}{[exp(ix)]^2-[exp(ix_k)]^2}
        \
        &=sum_{k=1}^nleft(frac{alpha^-_k}{exp(ix)+exp(ix_k)}+frac{alpha^+_k}{exp(ix)-exp(ix_k)}right)
        end{align*}

        with $alpha_k^pminmathbb{C}$. To be able to say
        $$
        prod_{k=1}^nfrac{1}{sin(x-x_k)}=sum_{k=1}^n frac{a_k}{sin(x-x_k)}
        $$

        is thus the condition $alpha_k^+=alpha_k^-$ which we can check holds if and only if $n$ is odd.






        share|cite|improve this answer









        $endgroup$
















          3












          3








          3





          $begingroup$

          What is really happening is that $xmapstosin(x-x_k)$ is a rational function of $exp(ix)$, so $displaystyleprod_{k=1}^nfrac{1}{sin(x-x_k)}$ has a partial fraction decomposition of the form
          begin{align*}
          prod_{k=1}^nfrac{1}{sin(x-x_k)}
          &=prod_{k=1}^nfrac{2iexp(-ix_k)exp(ix)}{[exp(ix)]^2-[exp(ix_k)]^2}
          \
          &=sum_{k=1}^nleft(frac{alpha^-_k}{exp(ix)+exp(ix_k)}+frac{alpha^+_k}{exp(ix)-exp(ix_k)}right)
          end{align*}

          with $alpha_k^pminmathbb{C}$. To be able to say
          $$
          prod_{k=1}^nfrac{1}{sin(x-x_k)}=sum_{k=1}^n frac{a_k}{sin(x-x_k)}
          $$

          is thus the condition $alpha_k^+=alpha_k^-$ which we can check holds if and only if $n$ is odd.






          share|cite|improve this answer









          $endgroup$



          What is really happening is that $xmapstosin(x-x_k)$ is a rational function of $exp(ix)$, so $displaystyleprod_{k=1}^nfrac{1}{sin(x-x_k)}$ has a partial fraction decomposition of the form
          begin{align*}
          prod_{k=1}^nfrac{1}{sin(x-x_k)}
          &=prod_{k=1}^nfrac{2iexp(-ix_k)exp(ix)}{[exp(ix)]^2-[exp(ix_k)]^2}
          \
          &=sum_{k=1}^nleft(frac{alpha^-_k}{exp(ix)+exp(ix_k)}+frac{alpha^+_k}{exp(ix)-exp(ix_k)}right)
          end{align*}

          with $alpha_k^pminmathbb{C}$. To be able to say
          $$
          prod_{k=1}^nfrac{1}{sin(x-x_k)}=sum_{k=1}^n frac{a_k}{sin(x-x_k)}
          $$

          is thus the condition $alpha_k^+=alpha_k^-$ which we can check holds if and only if $n$ is odd.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 20 '18 at 15:19









          user10354138user10354138

          7,5472925




          7,5472925























              1












              $begingroup$

              The problem is there are no constants $p,,q$ satisfying the intended identity. The reason $frac{1}{(x-a)(x-b)}=frac{p}{x-a}+frac{q}{x-b}$ works is because the denominators are degree-$1$ polynomials, allowing the numerators to be degree-$0$. More generally distinct functions $f,,g$ give $frac{1}{fg}=frac{p}{f}+frac{q}{g}$ with functions $p=frac{1}{g-f},,q=-p$, among other options. But usually there isn't a solution with constant $p,,q$.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Thanks! Is there a direct way to prove that there isn't a solution?
                $endgroup$
                – Zacky
                Dec 20 '18 at 12:59










              • $begingroup$
                @Zacky You've effectively already found it. Your calculation uniquely determined what $p,,q$ could be if they were constant, but we can verify it doesn't work.
                $endgroup$
                – J.G.
                Dec 20 '18 at 13:02










              • $begingroup$
                Can you please take a look here too: math.stackexchange.com/questions/1882536/… that is invalid too right?
                $endgroup$
                – Zacky
                Dec 20 '18 at 13:03






              • 1




                $begingroup$
                @Zacky I can't find a numerical counterexample to that, even with a computer search. I think the result is correct, even though the method of finding it needs more justification. Why does it only work with three sines, not two? Probably because the coefficients' ratios depend on $a,,b,,c$ in the three-sines case.
                $endgroup$
                – J.G.
                Dec 20 '18 at 13:19
















              1












              $begingroup$

              The problem is there are no constants $p,,q$ satisfying the intended identity. The reason $frac{1}{(x-a)(x-b)}=frac{p}{x-a}+frac{q}{x-b}$ works is because the denominators are degree-$1$ polynomials, allowing the numerators to be degree-$0$. More generally distinct functions $f,,g$ give $frac{1}{fg}=frac{p}{f}+frac{q}{g}$ with functions $p=frac{1}{g-f},,q=-p$, among other options. But usually there isn't a solution with constant $p,,q$.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Thanks! Is there a direct way to prove that there isn't a solution?
                $endgroup$
                – Zacky
                Dec 20 '18 at 12:59










              • $begingroup$
                @Zacky You've effectively already found it. Your calculation uniquely determined what $p,,q$ could be if they were constant, but we can verify it doesn't work.
                $endgroup$
                – J.G.
                Dec 20 '18 at 13:02










              • $begingroup$
                Can you please take a look here too: math.stackexchange.com/questions/1882536/… that is invalid too right?
                $endgroup$
                – Zacky
                Dec 20 '18 at 13:03






              • 1




                $begingroup$
                @Zacky I can't find a numerical counterexample to that, even with a computer search. I think the result is correct, even though the method of finding it needs more justification. Why does it only work with three sines, not two? Probably because the coefficients' ratios depend on $a,,b,,c$ in the three-sines case.
                $endgroup$
                – J.G.
                Dec 20 '18 at 13:19














              1












              1








              1





              $begingroup$

              The problem is there are no constants $p,,q$ satisfying the intended identity. The reason $frac{1}{(x-a)(x-b)}=frac{p}{x-a}+frac{q}{x-b}$ works is because the denominators are degree-$1$ polynomials, allowing the numerators to be degree-$0$. More generally distinct functions $f,,g$ give $frac{1}{fg}=frac{p}{f}+frac{q}{g}$ with functions $p=frac{1}{g-f},,q=-p$, among other options. But usually there isn't a solution with constant $p,,q$.






              share|cite|improve this answer









              $endgroup$



              The problem is there are no constants $p,,q$ satisfying the intended identity. The reason $frac{1}{(x-a)(x-b)}=frac{p}{x-a}+frac{q}{x-b}$ works is because the denominators are degree-$1$ polynomials, allowing the numerators to be degree-$0$. More generally distinct functions $f,,g$ give $frac{1}{fg}=frac{p}{f}+frac{q}{g}$ with functions $p=frac{1}{g-f},,q=-p$, among other options. But usually there isn't a solution with constant $p,,q$.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Dec 20 '18 at 12:52









              J.G.J.G.

              33.1k23251




              33.1k23251












              • $begingroup$
                Thanks! Is there a direct way to prove that there isn't a solution?
                $endgroup$
                – Zacky
                Dec 20 '18 at 12:59










              • $begingroup$
                @Zacky You've effectively already found it. Your calculation uniquely determined what $p,,q$ could be if they were constant, but we can verify it doesn't work.
                $endgroup$
                – J.G.
                Dec 20 '18 at 13:02










              • $begingroup$
                Can you please take a look here too: math.stackexchange.com/questions/1882536/… that is invalid too right?
                $endgroup$
                – Zacky
                Dec 20 '18 at 13:03






              • 1




                $begingroup$
                @Zacky I can't find a numerical counterexample to that, even with a computer search. I think the result is correct, even though the method of finding it needs more justification. Why does it only work with three sines, not two? Probably because the coefficients' ratios depend on $a,,b,,c$ in the three-sines case.
                $endgroup$
                – J.G.
                Dec 20 '18 at 13:19


















              • $begingroup$
                Thanks! Is there a direct way to prove that there isn't a solution?
                $endgroup$
                – Zacky
                Dec 20 '18 at 12:59










              • $begingroup$
                @Zacky You've effectively already found it. Your calculation uniquely determined what $p,,q$ could be if they were constant, but we can verify it doesn't work.
                $endgroup$
                – J.G.
                Dec 20 '18 at 13:02










              • $begingroup$
                Can you please take a look here too: math.stackexchange.com/questions/1882536/… that is invalid too right?
                $endgroup$
                – Zacky
                Dec 20 '18 at 13:03






              • 1




                $begingroup$
                @Zacky I can't find a numerical counterexample to that, even with a computer search. I think the result is correct, even though the method of finding it needs more justification. Why does it only work with three sines, not two? Probably because the coefficients' ratios depend on $a,,b,,c$ in the three-sines case.
                $endgroup$
                – J.G.
                Dec 20 '18 at 13:19
















              $begingroup$
              Thanks! Is there a direct way to prove that there isn't a solution?
              $endgroup$
              – Zacky
              Dec 20 '18 at 12:59




              $begingroup$
              Thanks! Is there a direct way to prove that there isn't a solution?
              $endgroup$
              – Zacky
              Dec 20 '18 at 12:59












              $begingroup$
              @Zacky You've effectively already found it. Your calculation uniquely determined what $p,,q$ could be if they were constant, but we can verify it doesn't work.
              $endgroup$
              – J.G.
              Dec 20 '18 at 13:02




              $begingroup$
              @Zacky You've effectively already found it. Your calculation uniquely determined what $p,,q$ could be if they were constant, but we can verify it doesn't work.
              $endgroup$
              – J.G.
              Dec 20 '18 at 13:02












              $begingroup$
              Can you please take a look here too: math.stackexchange.com/questions/1882536/… that is invalid too right?
              $endgroup$
              – Zacky
              Dec 20 '18 at 13:03




              $begingroup$
              Can you please take a look here too: math.stackexchange.com/questions/1882536/… that is invalid too right?
              $endgroup$
              – Zacky
              Dec 20 '18 at 13:03




              1




              1




              $begingroup$
              @Zacky I can't find a numerical counterexample to that, even with a computer search. I think the result is correct, even though the method of finding it needs more justification. Why does it only work with three sines, not two? Probably because the coefficients' ratios depend on $a,,b,,c$ in the three-sines case.
              $endgroup$
              – J.G.
              Dec 20 '18 at 13:19




              $begingroup$
              @Zacky I can't find a numerical counterexample to that, even with a computer search. I think the result is correct, even though the method of finding it needs more justification. Why does it only work with three sines, not two? Probably because the coefficients' ratios depend on $a,,b,,c$ in the three-sines case.
              $endgroup$
              – J.G.
              Dec 20 '18 at 13:19











              1












              $begingroup$

              We actually have
              $$frac1{sin(x-a)sin(x-b)} = frac1{sin(x-b)-sin(x-a)}left(frac1{sin(x-a)}-frac1{sin(x-b)}right)$$
              That factor of $sin(x-b)-sin(x-a)$ is equal to $sin(a-b)$ at $a$ and at $b$, but not everywhere; it isn't constant.



              Partial fractions are a theorem of linear algebra. If the factors we're dealing with aren't linear or at least polynomial, the theorem isn't valid. There is no partial fraction expansion of the type you're looking for here.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                We actually have
                $$frac1{sin(x-a)sin(x-b)} = frac1{sin(x-b)-sin(x-a)}left(frac1{sin(x-a)}-frac1{sin(x-b)}right)$$
                That factor of $sin(x-b)-sin(x-a)$ is equal to $sin(a-b)$ at $a$ and at $b$, but not everywhere; it isn't constant.



                Partial fractions are a theorem of linear algebra. If the factors we're dealing with aren't linear or at least polynomial, the theorem isn't valid. There is no partial fraction expansion of the type you're looking for here.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  We actually have
                  $$frac1{sin(x-a)sin(x-b)} = frac1{sin(x-b)-sin(x-a)}left(frac1{sin(x-a)}-frac1{sin(x-b)}right)$$
                  That factor of $sin(x-b)-sin(x-a)$ is equal to $sin(a-b)$ at $a$ and at $b$, but not everywhere; it isn't constant.



                  Partial fractions are a theorem of linear algebra. If the factors we're dealing with aren't linear or at least polynomial, the theorem isn't valid. There is no partial fraction expansion of the type you're looking for here.






                  share|cite|improve this answer









                  $endgroup$



                  We actually have
                  $$frac1{sin(x-a)sin(x-b)} = frac1{sin(x-b)-sin(x-a)}left(frac1{sin(x-a)}-frac1{sin(x-b)}right)$$
                  That factor of $sin(x-b)-sin(x-a)$ is equal to $sin(a-b)$ at $a$ and at $b$, but not everywhere; it isn't constant.



                  Partial fractions are a theorem of linear algebra. If the factors we're dealing with aren't linear or at least polynomial, the theorem isn't valid. There is no partial fraction expansion of the type you're looking for here.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 20 '18 at 12:53









                  jmerryjmerry

                  17k11633




                  17k11633






























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