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edit approved Apr 24, 2016 at 6:29
Amir Sagiv
edit approved Apr 24, 2016 at 6:29
Amir Sagiv
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This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues:

$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ Motivated mainly by curiosty, I have painstakingly spent hours trying to prove this nifty result without the advances of complex analysis - to no avail. I have also extensively searched the net for such a solution without success. Now of course the integrand is naturally underpinned by a complex expression, so such a solution would probably be a bit outlandish - but it would be interesting to see whether it is feasible.

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues:

$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ Motivated mainly by curiosty, I have painstakingly spent hours trying to prove this nifty result without the advances of complex analysis - to no avail. Now of course the integrand is naturally underpinned by a complex expression, so such a solution would probably be a bit outlandish - but it would be interesting to see whether it is feasible.

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues:

$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ Motivated mainly by curiosty, I have painstakingly spent hours trying to prove this nifty result without the advances of complex analysis - to no avail. I have also extensively searched the net for such a solution without success. Now of course the integrand is naturally underpinned by a complex expression, so such a solution would probably be a bit outlandish - but it would be interesting to see whether it is feasible.

Source Link
ocg
ocg
  • 453
  • 2
  • 8

Evaluating an integral using real methods

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues:

$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ Motivated mainly by curiosty, I have painstakingly spent hours trying to prove this nifty result without the advances of complex analysis - to no avail. Now of course the integrand is naturally underpinned by a complex expression, so such a solution would probably be a bit outlandish - but it would be interesting to see whether it is feasible.