Where does this “factorization” of a meromorphic function come from?
$begingroup$
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.
complex-analysis meromorphic-functions
asked Jan 16 at 7:58
user1101010user1101010
9011830
$endgroup$
add a comment |
$begingroup$
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.
complex-analysis meromorphic-functions
asked Jan 16 at 7:58
user1101010user1101010
9011830
$endgroup$
add a comment |
$begingroup$
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.
complex-analysis meromorphic-functions
asked Jan 16 at 7:58
user1101010user1101010
9011830
$endgroup$
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
complex-analysis meromorphic-functions
asked Jan 16 at 7:58
user1101010user1101010
9011830
asked Jan 16 at 7:58
user1101010user1101010
9011830
asked Jan 16 at 7:58
user1101010user1101010
9011830
asked Jan 16 at 7:58
user1101010user1101010
9011830
asked Jan 16 at 7:58
user1101010user1101010
9011830
9011830
add a comment |
add a comment |
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