Timeline for A closed expression for definite integral of a rational function
Current License: CC BY-SA 4.0
8 events
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| Oct 15, 2022 at 15:57 | comment | added | Dispersion | @MattF. I mean solely expressing the roots as an algebraic function of the coefficients. I did not say that the integral could be expressed this way. Of course by the residue theorem, you have a factor of $2\pi i$ that pops up, but modulo that factor, the residues are rational functions of the roots. | |
| Oct 15, 2022 at 11:53 | comment | added | user44143 | @Zachary, an algebraic function will almost never be possible: even in the simplest case, $\int_{-\infty}^\infty dx/(x^2+1)=\pi$ | |
| Oct 15, 2022 at 1:46 | comment | added | Dispersion | So you are essentially asking if you can express the roots in terms of the coefficients? By Abel-Ruffini, if $\operatorname{deg} Q\ge 5$, then there is no such expression as an algebraic function of the coefficients. There might be more complicated expression for the roots, however. | |
| Oct 15, 2022 at 1:00 | comment | added | Troshkin Michael | @GeraldEdgar Of course, one could do it, but my hope was that we could avoid it somehow, and evaluate the integral without first finding zeroes of the denominator. | |
| Oct 15, 2022 at 0:49 | comment | added | Gerald Edgar | I think to get an "expression" for that integral, you would first need an "expression" for the complex zeros of the denominator? Once you have the zeros of the denominator, you can do partial fractions (with complex coefficients). | |
| Oct 15, 2022 at 0:49 | comment | added | LSpice | What sort of expression do you seek? Since the integral already depends only on the coefficients, the question is just what kinds of expressions are allowed. | |
| Oct 15, 2022 at 0:48 | history | edited | LSpice | CC BY-SA 4.0 |
ration -> rational
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| Oct 15, 2022 at 0:39 | history | asked | Troshkin Michael | CC BY-SA 4.0 |