Is is true that $\int_0^1\sin(\pi x)\,x^x\,(1-x)^{1-x}\,dx\ =\ \frac1{4!}e\pi$?

Asked Apr 01 '17 at 14:29

Active Apr 01 '17 at 14:29

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Question: $\ $Prove or disprove that $$\int_0^1\sin(\pi x)\,x^x\,(1-x)^{1-x}\ dx\ =\ \frac{e\pi}{4!}$$

The numerical integration by Matlab showed that $$\left|\ \int_0^1\sin(\pi x)\,x^x\,(1-x)^{1-x}\ dx\ - \ \frac{e\pi}{4!}\ \right|\ <\ 10^{-15}$$ which made me kind of believe that the equality holds.

Therefore, I tried to prove it by contour integration, integration by parts and all other methods I could come up with. However, it seems that none of them would work.

Could anyone help me with this question? Any hint will be much appreciated.

asked Apr 01 '17 at 14:29

Mengchun Zhang

2,557

a numerical method shows that this could be true – Dr. Sonnhard Graubner Apr 01 '17 at 14:40
@Dr.SonnhardGraubner Thanks! But do you think it is possible to prove it non-numerically? – Mengchun Zhang Apr 01 '17 at 14:44
I think it is possible at this time I have no idea – Dr. Sonnhard Graubner Apr 01 '17 at 14:46
According to WA, this is correct up to the 157th digit. No idea how to prove it either – Bananach Apr 01 '17 at 18:36
You've got much higher precision than I did, so now this is very likely to be true :) – Mengchun Zhang Apr 01 '17 at 19:03
I ran these out in Wolfram Alpha.It was unable to do the indefinite integral, but for the definite integral it gave a different result than $e\pi/4!$, starting at about the $15^{th}$ place. Where did you get this in the first place? – Cye Waldman Apr 01 '17 at 19:40
@ZaidAlyafeai Thanks! I will check that out. – Mengchun Zhang Apr 01 '17 at 20:22
@MengchunZhang, actually I thought of the same method as in the link I provided but I was stuck at evaluating the residue at infinity which is unfortunately not shown in the link I provided. – Zaid Alyafeai Apr 01 '17 at 20:23
@ZaidAlyafeai I was also stuck at that place, so I think I am going to study the residue at infinity first and then go back to the link you provided. Thanks again :) – Mengchun Zhang Apr 01 '17 at 20:37
@CyeWaldman Thanks, actually I got this question in a Chinese math BBS in which nobody could prove it, so I posted it here. Also, the proof for this is in the link provided by ZaidAlyafeai. – Mengchun Zhang Apr 01 '17 at 20:41
This is one of a few integrals that seems impossible to solve using real analysis methods. – Zaid Alyafeai Apr 01 '17 at 23:07

Is is true that $\int_0^1\sin(\pi x)\,x^x\,(1-x)^{1-x}\,dx\ =\ \frac1{4!}e\pi$?

0 Answers0