How to integrate $int_0^{+infty} frac{x^2ln x}{x^4-x^3+1},mathrm{d}x$?
1
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First I think it may use the method of contour integration,but I have no idea how to create a contour.And is there some normal method to solve it? $$int_0^{+infty} frac{x^2ln x}{x^4-x^3+1},mathrm{d}x$$
calculus special-functions contour-integration
asked Dec 13 '18 at 2:04
tcs459163616tcs459163616
293
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add a comment |
1
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First I think it may use the method of contour integration,but I have no idea how to create a contour.And is there some normal method to solve it? $$int_0^{+infty} frac{x^2ln x}{x^4-x^3+1},mathrm{d}x$$
calculus special-functions contour-integration
asked Dec 13 '18 at 2:04
tcs459163616tcs459163616
293
$endgroup$
$begingroup$
The standard way is express the real integral in terms of a contour integral involving $log(-z)^2$ with +ve x-axis as branch cut over a keyhole contour $C : infty - iepsilon to -iepsilon to -epsilon to i epsilon to infty + iepsilon $. $$int_0^infty frac{x^2log x}{x^4 - x^3 + 1}dx = frac{i}{4pi}int_C frac{z^2log(-z)^2 dz}{z^4 -z^3 + 1}$$ One then evaluate the contour integral by a partial fraction decomposition of the rational part of its integrand and taking residues.
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– achille hui
Dec 13 '18 at 2:48
add a comment |
1
1
1
$begingroup$
First I think it may use the method of contour integration,but I have no idea how to create a contour.And is there some normal method to solve it? $$int_0^{+infty} frac{x^2ln x}{x^4-x^3+1},mathrm{d}x$$
calculus special-functions contour-integration
asked Dec 13 '18 at 2:04
tcs459163616tcs459163616
293
$endgroup$
First I think it may use the method of contour integration,but I have no idea how to create a contour.And is there some normal method to solve it? $$int_0^{+infty} frac{x^2ln x}{x^4-x^3+1},mathrm{d}x$$
calculus special-functions contour-integration
calculus special-functions contour-integration
asked Dec 13 '18 at 2:04
tcs459163616tcs459163616
293
asked Dec 13 '18 at 2:04
tcs459163616tcs459163616
293
asked Dec 13 '18 at 2:04
tcs459163616tcs459163616
293
asked Dec 13 '18 at 2:04
tcs459163616tcs459163616
293
asked Dec 13 '18 at 2:04
tcs459163616tcs459163616
293
293
$begingroup$
The standard way is express the real integral in terms of a contour integral involving $log(-z)^2$ with +ve x-axis as branch cut over a keyhole contour $C : infty - iepsilon to -iepsilon to -epsilon to i epsilon to infty + iepsilon $. $$int_0^infty frac{x^2log x}{x^4 - x^3 + 1}dx = frac{i}{4pi}int_C frac{z^2log(-z)^2 dz}{z^4 -z^3 + 1}$$ One then evaluate the contour integral by a partial fraction decomposition of the rational part of its integrand and taking residues.
$endgroup$
– achille hui
Dec 13 '18 at 2:48
add a comment |
$begingroup$
The standard way is express the real integral in terms of a contour integral involving $log(-z)^2$ with +ve x-axis as branch cut over a keyhole contour $C : infty - iepsilon to -iepsilon to -epsilon to i epsilon to infty + iepsilon $. $$int_0^infty frac{x^2log x}{x^4 - x^3 + 1}dx = frac{i}{4pi}int_C frac{z^2log(-z)^2 dz}{z^4 -z^3 + 1}$$ One then evaluate the contour integral by a partial fraction decomposition of the rational part of its integrand and taking residues.
$endgroup$
– achille hui
Dec 13 '18 at 2:48
$begingroup$
The standard way is express the real integral in terms of a contour integral involving $log(-z)^2$ with +ve x-axis as branch cut over a keyhole contour $C : infty - iepsilon to -iepsilon to -epsilon to i epsilon to infty + iepsilon $. $$int_0^infty frac{x^2log x}{x^4 - x^3 + 1}dx = frac{i}{4pi}int_C frac{z^2log(-z)^2 dz}{z^4 -z^3 + 1}$$ One then evaluate the contour integral by a partial fraction decomposition of the rational part of its integrand and taking residues.
$endgroup$
– achille hui
Dec 13 '18 at 2:48
$begingroup$
The standard way is express the real integral in terms of a contour integral involving $log(-z)^2$ with +ve x-axis as branch cut over a keyhole contour $C : infty - iepsilon to -iepsilon to -epsilon to i epsilon to infty + iepsilon $. $$int_0^infty frac{x^2log x}{x^4 - x^3 + 1}dx = frac{i}{4pi}int_C frac{z^2log(-z)^2 dz}{z^4 -z^3 + 1}$$ One then evaluate the contour integral by a partial fraction decomposition of the rational part of its integrand and taking residues.
$endgroup$
– achille hui
Dec 13 '18 at 2:48
add a comment |
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$begingroup$
The standard way is express the real integral in terms of a contour integral involving $log(-z)^2$ with +ve x-axis as branch cut over a keyhole contour $C : infty - iepsilon to -iepsilon to -epsilon to i epsilon to infty + iepsilon $. $$int_0^infty frac{x^2log x}{x^4 - x^3 + 1}dx = frac{i}{4pi}int_C frac{z^2log(-z)^2 dz}{z^4 -z^3 + 1}$$ One then evaluate the contour integral by a partial fraction decomposition of the rational part of its integrand and taking residues.
$endgroup$
– achille hui
Dec 13 '18 at 2:48