Where does this “factorization” of a meromorphic function come from?












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I encounter a remark in reading a book which states:




If $f(x) in mathbb R[x]$ is a monic polynomial, i.e., $f(x) = x^n + a_{n-1} x^{n-1} + dots + a_0$, then
begin{align*}
frac{ f'(x) }{f(x) } = frac{n}{x} left( 1- g(x) right),
end{align*}

where $g(x) = frac{x^{n-2} + b_{n-3} x^{n-3} + dots + b_0}{x^{n-1} + c_{n-2} x^{n-2} + dots + c_0}$ for some real coefficients $b_i, c_j$'s.




Expanding the right-hand side and multiplying the numerator-denominator on both sides, then equating the coefficients of each term, it seems to me this is possible. But I am wondering whether there is a generalized theorem on meromorphic function which can tell me this directly.










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    0












    $begingroup$


    I encounter a remark in reading a book which states:




    If $f(x) in mathbb R[x]$ is a monic polynomial, i.e., $f(x) = x^n + a_{n-1} x^{n-1} + dots + a_0$, then
    begin{align*}
    frac{ f'(x) }{f(x) } = frac{n}{x} left( 1- g(x) right),
    end{align*}

    where $g(x) = frac{x^{n-2} + b_{n-3} x^{n-3} + dots + b_0}{x^{n-1} + c_{n-2} x^{n-2} + dots + c_0}$ for some real coefficients $b_i, c_j$'s.




    Expanding the right-hand side and multiplying the numerator-denominator on both sides, then equating the coefficients of each term, it seems to me this is possible. But I am wondering whether there is a generalized theorem on meromorphic function which can tell me this directly.










    share|cite|improve this question









    $endgroup$















      0












      0








      0


      0



      $begingroup$


      I encounter a remark in reading a book which states:




      If $f(x) in mathbb R[x]$ is a monic polynomial, i.e., $f(x) = x^n + a_{n-1} x^{n-1} + dots + a_0$, then
      begin{align*}
      frac{ f'(x) }{f(x) } = frac{n}{x} left( 1- g(x) right),
      end{align*}

      where $g(x) = frac{x^{n-2} + b_{n-3} x^{n-3} + dots + b_0}{x^{n-1} + c_{n-2} x^{n-2} + dots + c_0}$ for some real coefficients $b_i, c_j$'s.




      Expanding the right-hand side and multiplying the numerator-denominator on both sides, then equating the coefficients of each term, it seems to me this is possible. But I am wondering whether there is a generalized theorem on meromorphic function which can tell me this directly.










      share|cite|improve this question









      $endgroup$




      I encounter a remark in reading a book which states:




      If $f(x) in mathbb R[x]$ is a monic polynomial, i.e., $f(x) = x^n + a_{n-1} x^{n-1} + dots + a_0$, then
      begin{align*}
      frac{ f'(x) }{f(x) } = frac{n}{x} left( 1- g(x) right),
      end{align*}

      where $g(x) = frac{x^{n-2} + b_{n-3} x^{n-3} + dots + b_0}{x^{n-1} + c_{n-2} x^{n-2} + dots + c_0}$ for some real coefficients $b_i, c_j$'s.




      Expanding the right-hand side and multiplying the numerator-denominator on both sides, then equating the coefficients of each term, it seems to me this is possible. But I am wondering whether there is a generalized theorem on meromorphic function which can tell me this directly.







      complex-analysis meromorphic-functions






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      share|cite|improve this question










      asked Jan 16 at 7:58









      user1101010user1101010

      9011830




      9011830






















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