Area[sQUIRKle] Hello, I like the subject that concerns the area of the circle and squirkle but this result is unknown to me Le dimanche 27 mai 2018 à 19:00:16 UTC+1, <math-fun-request@mailman.xmission.com> a écrit : Send math-fun mailing list submissions to     math-fun@mailman.xmission.com To subscribe or unsubscribe via the World Wide Web, visit     https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun or, via email, send a message with subject or body 'help' to     math-fun-request@mailman.xmission.com You can reach the person managing the list at     math-fun-owner@mailman.xmission.com When replying, please edit your Subject line so it is more specific than "Re: Contents of math-fun digest..." Today's Topics:   1. Area[sQUIRKle] (Bill Gosper) ---------------------------------------------------------------------- Message: 1 Date: Sat, 26 May 2018 15:59:31 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Area[sQUIRKle] Message-ID:     <CAA-4O0EbSsA+P3E3xuD30FJsRagn+qyHRYa13t5FKu2X+VmGrw@mail.gmail.com> Content-Type: text/plain; charset="UTF-8" Instead of just getting 4/Binomial[2/p,1/p] (or equivalent but messy ? functions) from Assuming[p > 0, 2 Integrate[(1 - Abs[x^p])^(1/p), {x, -1, 1}]], Assuming[p > 0, Area[ImplicitRegion[Abs[x^p] + Abs[y^p] < 1, {x, y}]]] does weird things that don't always work: 1/3    Area[ImplicitRegion[Abs[x]^(1/3)+Abs[y]^(1/3)<1,{x,y}]] 1/2    2/3 2/3    Area[ImplicitRegion[Abs[x]^(2/3)+Abs[y]^(2/3)<1,{x,y}]] 1      2 3/2    (2 2^(2/3) ?? Gamma[2/3])/Gamma[1/6]-(2^(2/3) ?(3 ?) Gamma[5/6])/Gamma[-(2/3)] 2      ? 5/2    - 5 ?(2/(5+?5)) ? Gamma[7/5])/(Gamma[-2/5] Gamma[4/5]))+(2^(1/5) ?? Gamma[7/5])/Gamma[9/10] 3      3 3^(1/3) AppellF1[4/3,-(1/3),-(1/3),7/3,-(1/(-1+(-1)^(2/3))),1/(1+(- 1)^(1/3))] 2 ?3  (2^(1-1/Sqrt[3]) Sqrt[\[Pi]/3] Gamma[1/(2 Sqrt[3])])/Gamma[1/6 (3+Sqrt[3])] 4      (8 Gamma[5/4]^2)/?? 9/2    (2^(5/9) ?? Gamma[11/9])/Gamma[13/18]+(? Csc[2 ?/9] Gamma[11/9])/(Gamma[4/9] Gamma[7/9]) 5      (2 2^(3/5) ?? Gamma[6/5])/Gamma[7/10] Notice the AppellF1 for p=3, which must equal 4/Binomial[2/3,1/3], unbeknownst to FullSimplify. Is it beknownst to anybody?  --rwg ------------------------------ Subject: Digest Footer _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ------------------------------ End of math-fun Digest, Vol 183, Issue 43 *****************************************