How can we show that $int_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over...
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Motivated by this paper.
Conjecture:
$$int_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x ,mathrm dx=k,tag1$$
where $k$ is a real number.
Making an attempt:
$u=e^x+1implies ,mathrm du=e^x,mathrm dx$ and let $k=1$ for simplification, then (1) becomes
$$int_{1}^{infty}{u^3over pi^2+(u-x)^2}cdot{ln(u-1)over pi^2+(u+x)^2}cdot{2mathrm duover u-1}.tag2$$
I have no idea where to go from here! I don't think substitution work here, probably using contour integration.
How can we prove (1)?
calculus integration definite-integrals contour-integration
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|
show 15 more comments
$begingroup$
Motivated by this paper.
Conjecture:
$$int_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x ,mathrm dx=k,tag1$$
where $k$ is a real number.
Making an attempt:
$u=e^x+1implies ,mathrm du=e^x,mathrm dx$ and let $k=1$ for simplification, then (1) becomes
$$int_{1}^{infty}{u^3over pi^2+(u-x)^2}cdot{ln(u-1)over pi^2+(u+x)^2}cdot{2mathrm duover u-1}.tag2$$
I have no idea where to go from here! I don't think substitution work here, probably using contour integration.
How can we prove (1)?
calculus integration definite-integrals contour-integration
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Related: en.wikipedia.org/wiki/Gregory_coefficients and math.stackexchange.com/questions/45745/…
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– Jack D'Aurizio
Apr 19 '17 at 20:46
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how do you came up with this conjecture? (+1)
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– tired
Apr 19 '17 at 22:35
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Btw. what is the meaning of the $pm$ symbol? both integrals give the same value?
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– tired
Apr 19 '17 at 22:41
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Most likely the integral is given by the residue at $x=i pi $ plus/minus the residue at $x=infty$. Unluckily i don't have the time to dig in deeper (especially one has to show that all other residue contributions vanish) but maybe someone can take it from here
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– tired
Apr 19 '17 at 23:18
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Numerically the cancelations defintily happens, so this is the way to go
$endgroup$
– tired
Apr 19 '17 at 23:30
|
show 15 more comments
$begingroup$
Motivated by this paper.
Conjecture:
$$int_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x ,mathrm dx=k,tag1$$
where $k$ is a real number.
Making an attempt:
$u=e^x+1implies ,mathrm du=e^x,mathrm dx$ and let $k=1$ for simplification, then (1) becomes
$$int_{1}^{infty}{u^3over pi^2+(u-x)^2}cdot{ln(u-1)over pi^2+(u+x)^2}cdot{2mathrm duover u-1}.tag2$$
I have no idea where to go from here! I don't think substitution work here, probably using contour integration.
How can we prove (1)?
calculus integration definite-integrals contour-integration
$endgroup$
Motivated by this paper.
Conjecture:
$$int_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x ,mathrm dx=k,tag1$$
where $k$ is a real number.
Making an attempt:
$u=e^x+1implies ,mathrm du=e^x,mathrm dx$ and let $k=1$ for simplification, then (1) becomes
$$int_{1}^{infty}{u^3over pi^2+(u-x)^2}cdot{ln(u-1)over pi^2+(u+x)^2}cdot{2mathrm duover u-1}.tag2$$
I have no idea where to go from here! I don't think substitution work here, probably using contour integration.
How can we prove (1)?
calculus integration definite-integrals contour-integration
calculus integration definite-integrals contour-integration
edited Feb 10 '18 at 5:50
Saad
19.7k92352
19.7k92352
asked Apr 19 '17 at 20:42
gymbvghjkgkjkhgfklgymbvghjkgkjkhgfkl
1
1
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Related: en.wikipedia.org/wiki/Gregory_coefficients and math.stackexchange.com/questions/45745/…
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– Jack D'Aurizio
Apr 19 '17 at 20:46
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how do you came up with this conjecture? (+1)
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– tired
Apr 19 '17 at 22:35
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Btw. what is the meaning of the $pm$ symbol? both integrals give the same value?
$endgroup$
– tired
Apr 19 '17 at 22:41
$begingroup$
Most likely the integral is given by the residue at $x=i pi $ plus/minus the residue at $x=infty$. Unluckily i don't have the time to dig in deeper (especially one has to show that all other residue contributions vanish) but maybe someone can take it from here
$endgroup$
– tired
Apr 19 '17 at 23:18
$begingroup$
Numerically the cancelations defintily happens, so this is the way to go
$endgroup$
– tired
Apr 19 '17 at 23:30
|
show 15 more comments
$begingroup$
Related: en.wikipedia.org/wiki/Gregory_coefficients and math.stackexchange.com/questions/45745/…
$endgroup$
– Jack D'Aurizio
Apr 19 '17 at 20:46
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how do you came up with this conjecture? (+1)
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– tired
Apr 19 '17 at 22:35
$begingroup$
Btw. what is the meaning of the $pm$ symbol? both integrals give the same value?
$endgroup$
– tired
Apr 19 '17 at 22:41
$begingroup$
Most likely the integral is given by the residue at $x=i pi $ plus/minus the residue at $x=infty$. Unluckily i don't have the time to dig in deeper (especially one has to show that all other residue contributions vanish) but maybe someone can take it from here
$endgroup$
– tired
Apr 19 '17 at 23:18
$begingroup$
Numerically the cancelations defintily happens, so this is the way to go
$endgroup$
– tired
Apr 19 '17 at 23:30
$begingroup$
Related: en.wikipedia.org/wiki/Gregory_coefficients and math.stackexchange.com/questions/45745/…
$endgroup$
– Jack D'Aurizio
Apr 19 '17 at 20:46
$begingroup$
Related: en.wikipedia.org/wiki/Gregory_coefficients and math.stackexchange.com/questions/45745/…
$endgroup$
– Jack D'Aurizio
Apr 19 '17 at 20:46
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how do you came up with this conjecture? (+1)
$endgroup$
– tired
Apr 19 '17 at 22:35
$begingroup$
how do you came up with this conjecture? (+1)
$endgroup$
– tired
Apr 19 '17 at 22:35
$begingroup$
Btw. what is the meaning of the $pm$ symbol? both integrals give the same value?
$endgroup$
– tired
Apr 19 '17 at 22:41
$begingroup$
Btw. what is the meaning of the $pm$ symbol? both integrals give the same value?
$endgroup$
– tired
Apr 19 '17 at 22:41
$begingroup$
Most likely the integral is given by the residue at $x=i pi $ plus/minus the residue at $x=infty$. Unluckily i don't have the time to dig in deeper (especially one has to show that all other residue contributions vanish) but maybe someone can take it from here
$endgroup$
– tired
Apr 19 '17 at 23:18
$begingroup$
Most likely the integral is given by the residue at $x=i pi $ plus/minus the residue at $x=infty$. Unluckily i don't have the time to dig in deeper (especially one has to show that all other residue contributions vanish) but maybe someone can take it from here
$endgroup$
– tired
Apr 19 '17 at 23:18
$begingroup$
Numerically the cancelations defintily happens, so this is the way to go
$endgroup$
– tired
Apr 19 '17 at 23:30
$begingroup$
Numerically the cancelations defintily happens, so this is the way to go
$endgroup$
– tired
Apr 19 '17 at 23:30
|
show 15 more comments
2 Answers
2
active
oldest
votes
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Some integrals
- Let us prove that
$$boxed{I_0 = intlimits_{-infty}^{+infty}{dzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1 + ie^xsin y - iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1right)^2 - left(e^xsin y - yright)^2 + pi^2 = 0\
left(e^xcos y - x + 1right)left(e^xsin y - yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y = x - 1\
left|e^xsin y - yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
So,
$$I_0 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z-z+1right)^2+pi^2} = 2pi ilim_{ztopi i}{1over2left(e^z-z+1right)left(e^z-1right)} = {1over2}.$$
- Let us prove that
$$boxed{I_1 = intlimits_{-infty}^{+infty}{dzoverleft(e^z+z+1right)^2+pi^2} = {2over3}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1 + ie^xsin y + iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1right)^2 - left(e^xsin y + yright)^2 + pi^2 = 0\
left(e^xcos y + x + 1right)left(e^xsin y + yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y + x + 1 = 0\
left|e^xsin y + yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
Note that the point $z=pi i$ is a second-order pole, so
$$I_1 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z+z+1right)^2+pi^2} = 2pi ilim_{ztopi i} {dover dz}left({(z-pi i)^2overleft(e^z+z+1right)^2+pi^2}right) = {2over3}.$$
(see also Wolfram Alpha).
- Let us prove that
$$boxed{I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Really,
$$I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z-1overleft(e^z-z+1right)^2+pi^2},dz + I_0$$
$$ = {1overpi}left.arctan{e^z-z-1overpi}right|_{-infty}^{+infty} + {1over 2} = {1over2}.$$
- Let us prove that
$$boxed{I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2} = {1over3}}$$
Similarly,
$$I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z+1overleft(e^z+z+1right)^2+pi^2},dz - I_1$$
$$ = {1overpi}left.arctan{e^z+z-1overpi}right|_{-infty}^{+infty} - {2over 3} = {1over3}.$$
- Let us prove that
$$boxed{I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx = 0}$$
Really,
$$I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx$$
$$= intlimits_{-infty}^{+infty}{e^z+1over2}left({1overleft(e^z-z+1right)^2+pi^2} - {1overleft(e^z+z+1right)^2+pi^2}right),dx$$
$$= {I_2+I_0-I_3-I_1over2} = {1over2}left({1over2}+{1over2}-{2over3}-{1over3}right) = 0.$$
- Let us prove that
$$boxed{I_5 = intlimits_{-infty}^{+infty}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx =
1}$$
The denominator is
$$D(z) = left(left(e^z+1right)^2+z^2
+pi^2 - 2zleft(e^z+1right)right) left(left(e^z+1right)^2+z^2+pi^2 + 2zleft(e^z+1right)right)$$
$$= left(left(e^z+1right)^2+z^2+pi^2right)^2 - 4z^2left(e^z+1right)^2,$$
$$D'(z) = 4left(e^z+z+1right)left(left(e^z+1right)^2+z^2+pi^2right) -8zleft(e^z+1right)left(e^z+z+1right)$$
$$=4left(e^z+z+1right)left(left(e^z-z+1right)^2+pi^2right)$$
The point $z=pi i $ is the simple pole. So,
$$I_5 = 2pi i,mathrm{Res}_{z=pi i}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)}$$
$$ = 2pi i,lim_{ztopi i}{2ze^z(e^z+1)^2over D'(z)} = 1.$$
(see also Wolfram Alpha)
Final calculations
$$I = intlimits_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
$$= kI_5pm I_4 = k.$$
Finally,
$$boxed{boxed{I = k}}$$
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@ZaidAlyafeai Thank you. Fixed.
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– Yuri Negometyanov
May 12 '17 at 17:28
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Excellent here goes the (+1) .
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– Zaid Alyafeai
May 13 '17 at 1:56
add a comment |
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First note that considering
$$F(k)=int_{-infty}^{+infty}{(ke^xpm1)over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
Let $x to log(x)$
$$F(k)=int_{0}^{+infty}{(kxpm1) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx = k$$
By separating the integrals note that
$$I_1=int_{0}^{+infty}{1 over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx=0$$
I could prove it numerically using Matlab. Hence I only show
$$I_2=int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Consider the function
$$f(z) = frac{(z-1)^2}{(1-(z+log z))(1-(z-log(z))}$$
Integrated around a key-hole contour around the principle branch of the logarithm
$$log(z) = log|z|+imathrm{Arg}(z)$$
Hence the contour
By taking the limits the smaller circle and the bigger one go to zero hence
$$int_{-infty}^{0}frac{(x-1)^2}{(1-(x+log|x|+ipi ))(1-(x-log|x|-ipi)}dx+int_{0}^{-infty}frac{(x-1)^2}{(1-(x+log|x|-ipi ))(1-(x-log|x|+ipi)}dx = 2pi imathrm{Res}(f,1)$$
Convert to the positive limit
$$int_{0}^{infty}frac{(x+1)^2}{(1+x-log x-ipi )(1+x+log x+ipi)}-frac{(x+1)^2}{(1+x-log x+ipi )(1+x+log x-ipi)}dx = 2pi imathrm{Res}(f,1)$$
This magically reduces to our integral
$$int_{0}^{+infty}{4pi ,i log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = 2pi imathrm{Res}(f,1)$$
Note that
$$mathrm{Res}(f,1) = lim_{z to 1}frac{(z-1)^3}{(1-(z+log z))(1-(z-log(z))} = 1$$
Hence we finally get our result
$$int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Using the same approach we could show
$$int^infty_{-infty}frac{dx}{(e^x-x+1)^2+pi^2}=frac{1}{2}$$
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Aren't you missing an ${i}$ in your definition of complex logarithm?
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– Dmoreno
May 11 '17 at 21:24
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@Dmoreno, yesss, thanks.
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– Zaid Alyafeai
May 11 '17 at 21:25
add a comment |
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2 Answers
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2 Answers
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$begingroup$
Some integrals
- Let us prove that
$$boxed{I_0 = intlimits_{-infty}^{+infty}{dzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1 + ie^xsin y - iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1right)^2 - left(e^xsin y - yright)^2 + pi^2 = 0\
left(e^xcos y - x + 1right)left(e^xsin y - yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y = x - 1\
left|e^xsin y - yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
So,
$$I_0 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z-z+1right)^2+pi^2} = 2pi ilim_{ztopi i}{1over2left(e^z-z+1right)left(e^z-1right)} = {1over2}.$$
- Let us prove that
$$boxed{I_1 = intlimits_{-infty}^{+infty}{dzoverleft(e^z+z+1right)^2+pi^2} = {2over3}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1 + ie^xsin y + iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1right)^2 - left(e^xsin y + yright)^2 + pi^2 = 0\
left(e^xcos y + x + 1right)left(e^xsin y + yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y + x + 1 = 0\
left|e^xsin y + yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
Note that the point $z=pi i$ is a second-order pole, so
$$I_1 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z+z+1right)^2+pi^2} = 2pi ilim_{ztopi i} {dover dz}left({(z-pi i)^2overleft(e^z+z+1right)^2+pi^2}right) = {2over3}.$$
(see also Wolfram Alpha).
- Let us prove that
$$boxed{I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Really,
$$I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z-1overleft(e^z-z+1right)^2+pi^2},dz + I_0$$
$$ = {1overpi}left.arctan{e^z-z-1overpi}right|_{-infty}^{+infty} + {1over 2} = {1over2}.$$
- Let us prove that
$$boxed{I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2} = {1over3}}$$
Similarly,
$$I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z+1overleft(e^z+z+1right)^2+pi^2},dz - I_1$$
$$ = {1overpi}left.arctan{e^z+z-1overpi}right|_{-infty}^{+infty} - {2over 3} = {1over3}.$$
- Let us prove that
$$boxed{I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx = 0}$$
Really,
$$I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx$$
$$= intlimits_{-infty}^{+infty}{e^z+1over2}left({1overleft(e^z-z+1right)^2+pi^2} - {1overleft(e^z+z+1right)^2+pi^2}right),dx$$
$$= {I_2+I_0-I_3-I_1over2} = {1over2}left({1over2}+{1over2}-{2over3}-{1over3}right) = 0.$$
- Let us prove that
$$boxed{I_5 = intlimits_{-infty}^{+infty}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx =
1}$$
The denominator is
$$D(z) = left(left(e^z+1right)^2+z^2
+pi^2 - 2zleft(e^z+1right)right) left(left(e^z+1right)^2+z^2+pi^2 + 2zleft(e^z+1right)right)$$
$$= left(left(e^z+1right)^2+z^2+pi^2right)^2 - 4z^2left(e^z+1right)^2,$$
$$D'(z) = 4left(e^z+z+1right)left(left(e^z+1right)^2+z^2+pi^2right) -8zleft(e^z+1right)left(e^z+z+1right)$$
$$=4left(e^z+z+1right)left(left(e^z-z+1right)^2+pi^2right)$$
The point $z=pi i $ is the simple pole. So,
$$I_5 = 2pi i,mathrm{Res}_{z=pi i}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)}$$
$$ = 2pi i,lim_{ztopi i}{2ze^z(e^z+1)^2over D'(z)} = 1.$$
(see also Wolfram Alpha)
Final calculations
$$I = intlimits_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
$$= kI_5pm I_4 = k.$$
Finally,
$$boxed{boxed{I = k}}$$
$endgroup$
$begingroup$
@ZaidAlyafeai Thank you. Fixed.
$endgroup$
– Yuri Negometyanov
May 12 '17 at 17:28
$begingroup$
Excellent here goes the (+1) .
$endgroup$
– Zaid Alyafeai
May 13 '17 at 1:56
add a comment |
$begingroup$
Some integrals
- Let us prove that
$$boxed{I_0 = intlimits_{-infty}^{+infty}{dzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1 + ie^xsin y - iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1right)^2 - left(e^xsin y - yright)^2 + pi^2 = 0\
left(e^xcos y - x + 1right)left(e^xsin y - yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y = x - 1\
left|e^xsin y - yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
So,
$$I_0 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z-z+1right)^2+pi^2} = 2pi ilim_{ztopi i}{1over2left(e^z-z+1right)left(e^z-1right)} = {1over2}.$$
- Let us prove that
$$boxed{I_1 = intlimits_{-infty}^{+infty}{dzoverleft(e^z+z+1right)^2+pi^2} = {2over3}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1 + ie^xsin y + iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1right)^2 - left(e^xsin y + yright)^2 + pi^2 = 0\
left(e^xcos y + x + 1right)left(e^xsin y + yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y + x + 1 = 0\
left|e^xsin y + yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
Note that the point $z=pi i$ is a second-order pole, so
$$I_1 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z+z+1right)^2+pi^2} = 2pi ilim_{ztopi i} {dover dz}left({(z-pi i)^2overleft(e^z+z+1right)^2+pi^2}right) = {2over3}.$$
(see also Wolfram Alpha).
- Let us prove that
$$boxed{I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Really,
$$I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z-1overleft(e^z-z+1right)^2+pi^2},dz + I_0$$
$$ = {1overpi}left.arctan{e^z-z-1overpi}right|_{-infty}^{+infty} + {1over 2} = {1over2}.$$
- Let us prove that
$$boxed{I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2} = {1over3}}$$
Similarly,
$$I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z+1overleft(e^z+z+1right)^2+pi^2},dz - I_1$$
$$ = {1overpi}left.arctan{e^z+z-1overpi}right|_{-infty}^{+infty} - {2over 3} = {1over3}.$$
- Let us prove that
$$boxed{I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx = 0}$$
Really,
$$I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx$$
$$= intlimits_{-infty}^{+infty}{e^z+1over2}left({1overleft(e^z-z+1right)^2+pi^2} - {1overleft(e^z+z+1right)^2+pi^2}right),dx$$
$$= {I_2+I_0-I_3-I_1over2} = {1over2}left({1over2}+{1over2}-{2over3}-{1over3}right) = 0.$$
- Let us prove that
$$boxed{I_5 = intlimits_{-infty}^{+infty}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx =
1}$$
The denominator is
$$D(z) = left(left(e^z+1right)^2+z^2
+pi^2 - 2zleft(e^z+1right)right) left(left(e^z+1right)^2+z^2+pi^2 + 2zleft(e^z+1right)right)$$
$$= left(left(e^z+1right)^2+z^2+pi^2right)^2 - 4z^2left(e^z+1right)^2,$$
$$D'(z) = 4left(e^z+z+1right)left(left(e^z+1right)^2+z^2+pi^2right) -8zleft(e^z+1right)left(e^z+z+1right)$$
$$=4left(e^z+z+1right)left(left(e^z-z+1right)^2+pi^2right)$$
The point $z=pi i $ is the simple pole. So,
$$I_5 = 2pi i,mathrm{Res}_{z=pi i}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)}$$
$$ = 2pi i,lim_{ztopi i}{2ze^z(e^z+1)^2over D'(z)} = 1.$$
(see also Wolfram Alpha)
Final calculations
$$I = intlimits_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
$$= kI_5pm I_4 = k.$$
Finally,
$$boxed{boxed{I = k}}$$
$endgroup$
$begingroup$
@ZaidAlyafeai Thank you. Fixed.
$endgroup$
– Yuri Negometyanov
May 12 '17 at 17:28
$begingroup$
Excellent here goes the (+1) .
$endgroup$
– Zaid Alyafeai
May 13 '17 at 1:56
add a comment |
$begingroup$
Some integrals
- Let us prove that
$$boxed{I_0 = intlimits_{-infty}^{+infty}{dzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1 + ie^xsin y - iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1right)^2 - left(e^xsin y - yright)^2 + pi^2 = 0\
left(e^xcos y - x + 1right)left(e^xsin y - yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y = x - 1\
left|e^xsin y - yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
So,
$$I_0 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z-z+1right)^2+pi^2} = 2pi ilim_{ztopi i}{1over2left(e^z-z+1right)left(e^z-1right)} = {1over2}.$$
- Let us prove that
$$boxed{I_1 = intlimits_{-infty}^{+infty}{dzoverleft(e^z+z+1right)^2+pi^2} = {2over3}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1 + ie^xsin y + iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1right)^2 - left(e^xsin y + yright)^2 + pi^2 = 0\
left(e^xcos y + x + 1right)left(e^xsin y + yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y + x + 1 = 0\
left|e^xsin y + yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
Note that the point $z=pi i$ is a second-order pole, so
$$I_1 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z+z+1right)^2+pi^2} = 2pi ilim_{ztopi i} {dover dz}left({(z-pi i)^2overleft(e^z+z+1right)^2+pi^2}right) = {2over3}.$$
(see also Wolfram Alpha).
- Let us prove that
$$boxed{I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Really,
$$I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z-1overleft(e^z-z+1right)^2+pi^2},dz + I_0$$
$$ = {1overpi}left.arctan{e^z-z-1overpi}right|_{-infty}^{+infty} + {1over 2} = {1over2}.$$
- Let us prove that
$$boxed{I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2} = {1over3}}$$
Similarly,
$$I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z+1overleft(e^z+z+1right)^2+pi^2},dz - I_1$$
$$ = {1overpi}left.arctan{e^z+z-1overpi}right|_{-infty}^{+infty} - {2over 3} = {1over3}.$$
- Let us prove that
$$boxed{I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx = 0}$$
Really,
$$I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx$$
$$= intlimits_{-infty}^{+infty}{e^z+1over2}left({1overleft(e^z-z+1right)^2+pi^2} - {1overleft(e^z+z+1right)^2+pi^2}right),dx$$
$$= {I_2+I_0-I_3-I_1over2} = {1over2}left({1over2}+{1over2}-{2over3}-{1over3}right) = 0.$$
- Let us prove that
$$boxed{I_5 = intlimits_{-infty}^{+infty}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx =
1}$$
The denominator is
$$D(z) = left(left(e^z+1right)^2+z^2
+pi^2 - 2zleft(e^z+1right)right) left(left(e^z+1right)^2+z^2+pi^2 + 2zleft(e^z+1right)right)$$
$$= left(left(e^z+1right)^2+z^2+pi^2right)^2 - 4z^2left(e^z+1right)^2,$$
$$D'(z) = 4left(e^z+z+1right)left(left(e^z+1right)^2+z^2+pi^2right) -8zleft(e^z+1right)left(e^z+z+1right)$$
$$=4left(e^z+z+1right)left(left(e^z-z+1right)^2+pi^2right)$$
The point $z=pi i $ is the simple pole. So,
$$I_5 = 2pi i,mathrm{Res}_{z=pi i}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)}$$
$$ = 2pi i,lim_{ztopi i}{2ze^z(e^z+1)^2over D'(z)} = 1.$$
(see also Wolfram Alpha)
Final calculations
$$I = intlimits_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
$$= kI_5pm I_4 = k.$$
Finally,
$$boxed{boxed{I = k}}$$
$endgroup$
Some integrals
- Let us prove that
$$boxed{I_0 = intlimits_{-infty}^{+infty}{dzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1 + ie^xsin y - iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y - x + 1right)^2 - left(e^xsin y - yright)^2 + pi^2 = 0\
left(e^xcos y - x + 1right)left(e^xsin y - yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y = x - 1\
left|e^xsin y - yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
So,
$$I_0 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z-z+1right)^2+pi^2} = 2pi ilim_{ztopi i}{1over2left(e^z-z+1right)left(e^z-1right)} = {1over2}.$$
- Let us prove that
$$boxed{I_1 = intlimits_{-infty}^{+infty}{dzoverleft(e^z+z+1right)^2+pi^2} = {2over3}}$$
Roots of the denominator can be defined from the system
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1 + ie^xsin y + iyright)^2 + pi^2 = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
left(e^xcos y + x + 1right)^2 - left(e^xsin y + yright)^2 + pi^2 = 0\
left(e^xcos y + x + 1right)left(e^xsin y + yright) = 0,
end{cases}$$
$$begin{cases}
z=x+iy\
e^xcos y + x + 1 = 0\
left|e^xsin y + yright| = pi,
end{cases}$$
with the solutions $z=pmpi i$ (see also Wolfram Alpha).
Note that the point $z=pi i$ is a second-order pole, so
$$I_1 = 2pi i,mathrm{Res}_{z=pi i}{1overleft(e^z+z+1right)^2+pi^2} = 2pi ilim_{ztopi i} {dover dz}left({(z-pi i)^2overleft(e^z+z+1right)^2+pi^2}right) = {2over3}.$$
(see also Wolfram Alpha).
- Let us prove that
$$boxed{I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2} = {1over2}}$$
Really,
$$I_2 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z-z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z-1overleft(e^z-z+1right)^2+pi^2},dz + I_0$$
$$ = {1overpi}left.arctan{e^z-z-1overpi}right|_{-infty}^{+infty} + {1over 2} = {1over2}.$$
- Let us prove that
$$boxed{I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2} = {1over3}}$$
Similarly,
$$I_3 = intlimits_{-infty}^{+infty}{e^zdzoverleft(e^z+z+1right)^2+pi^2}= intlimits_{-infty}^{+infty}{e^z+1overleft(e^z+z+1right)^2+pi^2},dz - I_1$$
$$ = {1overpi}left.arctan{e^z+z-1overpi}right|_{-infty}^{+infty} - {2over 3} = {1over3}.$$
- Let us prove that
$$boxed{I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx = 0}$$
Really,
$$I_4 = intlimits_{-infty}^{+infty}{2z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx$$
$$= intlimits_{-infty}^{+infty}{e^z+1over2}left({1overleft(e^z-z+1right)^2+pi^2} - {1overleft(e^z+z+1right)^2+pi^2}right),dx$$
$$= {I_2+I_0-I_3-I_1over2} = {1over2}left({1over2}+{1over2}-{2over3}-{1over3}right) = 0.$$
- Let us prove that
$$boxed{I_5 = intlimits_{-infty}^{+infty}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)},dx =
1}$$
The denominator is
$$D(z) = left(left(e^z+1right)^2+z^2
+pi^2 - 2zleft(e^z+1right)right) left(left(e^z+1right)^2+z^2+pi^2 + 2zleft(e^z+1right)right)$$
$$= left(left(e^z+1right)^2+z^2+pi^2right)^2 - 4z^2left(e^z+1right)^2,$$
$$D'(z) = 4left(e^z+z+1right)left(left(e^z+1right)^2+z^2+pi^2right) -8zleft(e^z+1right)left(e^z+z+1right)$$
$$=4left(e^z+z+1right)left(left(e^z-z+1right)^2+pi^2right)$$
The point $z=pi i $ is the simple pole. So,
$$I_5 = 2pi i,mathrm{Res}_{z=pi i}{2ze^z(e^z+1)^2overleft(left(e^z-z+1right)^2+pi^2right)left(left(e^z+z+1right)^2+pi^2right)}$$
$$ = 2pi i,lim_{ztopi i}{2ze^z(e^z+1)^2over D'(z)} = 1.$$
(see also Wolfram Alpha)
Final calculations
$$I = intlimits_{-infty}^{+infty}{ke^xpm1over pi^2+(e^x-x+1)}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
$$= kI_5pm I_4 = k.$$
Finally,
$$boxed{boxed{I = k}}$$
edited Dec 5 '18 at 1:54
answered May 12 '17 at 13:50
Yuri NegometyanovYuri Negometyanov
11k1728
11k1728
$begingroup$
@ZaidAlyafeai Thank you. Fixed.
$endgroup$
– Yuri Negometyanov
May 12 '17 at 17:28
$begingroup$
Excellent here goes the (+1) .
$endgroup$
– Zaid Alyafeai
May 13 '17 at 1:56
add a comment |
$begingroup$
@ZaidAlyafeai Thank you. Fixed.
$endgroup$
– Yuri Negometyanov
May 12 '17 at 17:28
$begingroup$
Excellent here goes the (+1) .
$endgroup$
– Zaid Alyafeai
May 13 '17 at 1:56
$begingroup$
@ZaidAlyafeai Thank you. Fixed.
$endgroup$
– Yuri Negometyanov
May 12 '17 at 17:28
$begingroup$
@ZaidAlyafeai Thank you. Fixed.
$endgroup$
– Yuri Negometyanov
May 12 '17 at 17:28
$begingroup$
Excellent here goes the (+1) .
$endgroup$
– Zaid Alyafeai
May 13 '17 at 1:56
$begingroup$
Excellent here goes the (+1) .
$endgroup$
– Zaid Alyafeai
May 13 '17 at 1:56
add a comment |
$begingroup$
First note that considering
$$F(k)=int_{-infty}^{+infty}{(ke^xpm1)over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
Let $x to log(x)$
$$F(k)=int_{0}^{+infty}{(kxpm1) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx = k$$
By separating the integrals note that
$$I_1=int_{0}^{+infty}{1 over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx=0$$
I could prove it numerically using Matlab. Hence I only show
$$I_2=int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Consider the function
$$f(z) = frac{(z-1)^2}{(1-(z+log z))(1-(z-log(z))}$$
Integrated around a key-hole contour around the principle branch of the logarithm
$$log(z) = log|z|+imathrm{Arg}(z)$$
Hence the contour
By taking the limits the smaller circle and the bigger one go to zero hence
$$int_{-infty}^{0}frac{(x-1)^2}{(1-(x+log|x|+ipi ))(1-(x-log|x|-ipi)}dx+int_{0}^{-infty}frac{(x-1)^2}{(1-(x+log|x|-ipi ))(1-(x-log|x|+ipi)}dx = 2pi imathrm{Res}(f,1)$$
Convert to the positive limit
$$int_{0}^{infty}frac{(x+1)^2}{(1+x-log x-ipi )(1+x+log x+ipi)}-frac{(x+1)^2}{(1+x-log x+ipi )(1+x+log x-ipi)}dx = 2pi imathrm{Res}(f,1)$$
This magically reduces to our integral
$$int_{0}^{+infty}{4pi ,i log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = 2pi imathrm{Res}(f,1)$$
Note that
$$mathrm{Res}(f,1) = lim_{z to 1}frac{(z-1)^3}{(1-(z+log z))(1-(z-log(z))} = 1$$
Hence we finally get our result
$$int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Using the same approach we could show
$$int^infty_{-infty}frac{dx}{(e^x-x+1)^2+pi^2}=frac{1}{2}$$
$endgroup$
$begingroup$
Aren't you missing an ${i}$ in your definition of complex logarithm?
$endgroup$
– Dmoreno
May 11 '17 at 21:24
$begingroup$
@Dmoreno, yesss, thanks.
$endgroup$
– Zaid Alyafeai
May 11 '17 at 21:25
add a comment |
$begingroup$
First note that considering
$$F(k)=int_{-infty}^{+infty}{(ke^xpm1)over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
Let $x to log(x)$
$$F(k)=int_{0}^{+infty}{(kxpm1) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx = k$$
By separating the integrals note that
$$I_1=int_{0}^{+infty}{1 over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx=0$$
I could prove it numerically using Matlab. Hence I only show
$$I_2=int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Consider the function
$$f(z) = frac{(z-1)^2}{(1-(z+log z))(1-(z-log(z))}$$
Integrated around a key-hole contour around the principle branch of the logarithm
$$log(z) = log|z|+imathrm{Arg}(z)$$
Hence the contour
By taking the limits the smaller circle and the bigger one go to zero hence
$$int_{-infty}^{0}frac{(x-1)^2}{(1-(x+log|x|+ipi ))(1-(x-log|x|-ipi)}dx+int_{0}^{-infty}frac{(x-1)^2}{(1-(x+log|x|-ipi ))(1-(x-log|x|+ipi)}dx = 2pi imathrm{Res}(f,1)$$
Convert to the positive limit
$$int_{0}^{infty}frac{(x+1)^2}{(1+x-log x-ipi )(1+x+log x+ipi)}-frac{(x+1)^2}{(1+x-log x+ipi )(1+x+log x-ipi)}dx = 2pi imathrm{Res}(f,1)$$
This magically reduces to our integral
$$int_{0}^{+infty}{4pi ,i log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = 2pi imathrm{Res}(f,1)$$
Note that
$$mathrm{Res}(f,1) = lim_{z to 1}frac{(z-1)^3}{(1-(z+log z))(1-(z-log(z))} = 1$$
Hence we finally get our result
$$int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Using the same approach we could show
$$int^infty_{-infty}frac{dx}{(e^x-x+1)^2+pi^2}=frac{1}{2}$$
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Aren't you missing an ${i}$ in your definition of complex logarithm?
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– Dmoreno
May 11 '17 at 21:24
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@Dmoreno, yesss, thanks.
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– Zaid Alyafeai
May 11 '17 at 21:25
add a comment |
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First note that considering
$$F(k)=int_{-infty}^{+infty}{(ke^xpm1)over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
Let $x to log(x)$
$$F(k)=int_{0}^{+infty}{(kxpm1) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx = k$$
By separating the integrals note that
$$I_1=int_{0}^{+infty}{1 over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx=0$$
I could prove it numerically using Matlab. Hence I only show
$$I_2=int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Consider the function
$$f(z) = frac{(z-1)^2}{(1-(z+log z))(1-(z-log(z))}$$
Integrated around a key-hole contour around the principle branch of the logarithm
$$log(z) = log|z|+imathrm{Arg}(z)$$
Hence the contour
By taking the limits the smaller circle and the bigger one go to zero hence
$$int_{-infty}^{0}frac{(x-1)^2}{(1-(x+log|x|+ipi ))(1-(x-log|x|-ipi)}dx+int_{0}^{-infty}frac{(x-1)^2}{(1-(x+log|x|-ipi ))(1-(x-log|x|+ipi)}dx = 2pi imathrm{Res}(f,1)$$
Convert to the positive limit
$$int_{0}^{infty}frac{(x+1)^2}{(1+x-log x-ipi )(1+x+log x+ipi)}-frac{(x+1)^2}{(1+x-log x+ipi )(1+x+log x-ipi)}dx = 2pi imathrm{Res}(f,1)$$
This magically reduces to our integral
$$int_{0}^{+infty}{4pi ,i log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = 2pi imathrm{Res}(f,1)$$
Note that
$$mathrm{Res}(f,1) = lim_{z to 1}frac{(z-1)^3}{(1-(z+log z))(1-(z-log(z))} = 1$$
Hence we finally get our result
$$int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Using the same approach we could show
$$int^infty_{-infty}frac{dx}{(e^x-x+1)^2+pi^2}=frac{1}{2}$$
$endgroup$
First note that considering
$$F(k)=int_{-infty}^{+infty}{(ke^xpm1)over pi^2+(e^x-x+1)^2}cdot{(e^x+1)^2over pi^2+(e^x+x+1)^2}cdot 2x mathrm dx$$
Let $x to log(x)$
$$F(k)=int_{0}^{+infty}{(kxpm1) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx = k$$
By separating the integrals note that
$$I_1=int_{0}^{+infty}{1 over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot frac{2log(x)}{x} mathrm dx=0$$
I could prove it numerically using Matlab. Hence I only show
$$I_2=int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Consider the function
$$f(z) = frac{(z-1)^2}{(1-(z+log z))(1-(z-log(z))}$$
Integrated around a key-hole contour around the principle branch of the logarithm
$$log(z) = log|z|+imathrm{Arg}(z)$$
Hence the contour
By taking the limits the smaller circle and the bigger one go to zero hence
$$int_{-infty}^{0}frac{(x-1)^2}{(1-(x+log|x|+ipi ))(1-(x-log|x|-ipi)}dx+int_{0}^{-infty}frac{(x-1)^2}{(1-(x+log|x|-ipi ))(1-(x-log|x|+ipi)}dx = 2pi imathrm{Res}(f,1)$$
Convert to the positive limit
$$int_{0}^{infty}frac{(x+1)^2}{(1+x-log x-ipi )(1+x+log x+ipi)}-frac{(x+1)^2}{(1+x-log x+ipi )(1+x+log x-ipi)}dx = 2pi imathrm{Res}(f,1)$$
This magically reduces to our integral
$$int_{0}^{+infty}{4pi ,i log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = 2pi imathrm{Res}(f,1)$$
Note that
$$mathrm{Res}(f,1) = lim_{z to 1}frac{(z-1)^3}{(1-(z+log z))(1-(z-log(z))} = 1$$
Hence we finally get our result
$$int_{0}^{+infty}{log(x) over pi^2+(x-log(x)+1)^2}cdot{(x+1)^2over pi^2+(x+log(x)+1)^2}cdot mathrm dx = frac{1}{2}$$
Using the same approach we could show
$$int^infty_{-infty}frac{dx}{(e^x-x+1)^2+pi^2}=frac{1}{2}$$
edited May 12 '17 at 0:42
answered May 11 '17 at 21:15
Zaid AlyafeaiZaid Alyafeai
9,34622369
9,34622369
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Aren't you missing an ${i}$ in your definition of complex logarithm?
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– Dmoreno
May 11 '17 at 21:24
$begingroup$
@Dmoreno, yesss, thanks.
$endgroup$
– Zaid Alyafeai
May 11 '17 at 21:25
add a comment |
$begingroup$
Aren't you missing an ${i}$ in your definition of complex logarithm?
$endgroup$
– Dmoreno
May 11 '17 at 21:24
$begingroup$
@Dmoreno, yesss, thanks.
$endgroup$
– Zaid Alyafeai
May 11 '17 at 21:25
$begingroup$
Aren't you missing an ${i}$ in your definition of complex logarithm?
$endgroup$
– Dmoreno
May 11 '17 at 21:24
$begingroup$
Aren't you missing an ${i}$ in your definition of complex logarithm?
$endgroup$
– Dmoreno
May 11 '17 at 21:24
$begingroup$
@Dmoreno, yesss, thanks.
$endgroup$
– Zaid Alyafeai
May 11 '17 at 21:25
$begingroup$
@Dmoreno, yesss, thanks.
$endgroup$
– Zaid Alyafeai
May 11 '17 at 21:25
add a comment |
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Related: en.wikipedia.org/wiki/Gregory_coefficients and math.stackexchange.com/questions/45745/…
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– Jack D'Aurizio
Apr 19 '17 at 20:46
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how do you came up with this conjecture? (+1)
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– tired
Apr 19 '17 at 22:35
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Btw. what is the meaning of the $pm$ symbol? both integrals give the same value?
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– tired
Apr 19 '17 at 22:41
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Most likely the integral is given by the residue at $x=i pi $ plus/minus the residue at $x=infty$. Unluckily i don't have the time to dig in deeper (especially one has to show that all other residue contributions vanish) but maybe someone can take it from here
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– tired
Apr 19 '17 at 23:18
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Numerically the cancelations defintily happens, so this is the way to go
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– tired
Apr 19 '17 at 23:30