[math-fun] Gosper's EllipticK formula
Bill Gosper: Then I bashed on it so hard that it got unnervingly ungrotesque: (*) EllipticK[1/4 (2 + (-2 + t)/Sqrt[1 - t])] == (1 - t)^(1/4) EllipticK[t] How could anything this simple be new? WDS: Let s=Sqrt[1-t]. Then (*) is rewritten as EllipticK[ 1/2 - (1+s^2)/(4s) ] == EllipticK[ -(1-s)^2/(4s) ] == s^(1/2) EllipticK[1-s^2] Now let s=1-m. Then this is EllipticK[ -m^2/(4(1-m)) ] == (1-m)^(1/2) EllipticK[(2-m)m]. E.Salamin: Complains WDS formula is wrong. --WDS: arg. Well, I just started with Gosper's "ungrotesque" formula taking its correctness on faith, then worked by hand to produce mine. Doing same thing over again now: Gosper: EllipticK[1/4 (2 + (-2 + t)/Sqrt[1 - t])] == (1 - t)^(1/4) EllipticK[t] Let t=1-s^2: EllipticK[1/4 (2 - (1+s^2)/s)] == (s)^(1/2) EllipticK[1-s^2] EllipticK[(2s-1-s^2)/(4s)] == (s)^(1/2) EllipticK[1-s^2] EllipticK[-(1-s)^2/(4s)] == (s)^(1/2) EllipticK[1-s^2] which is same as I got before. Now let s=1-m: EllipticK[-m^2/(4(1-m))] == (1-m)^(1/2) EllipticK[1-(1-m)^2] EllipticK[-m^2/(4(1-m))] == (1-m)^(1/2) EllipticK[2m-m^2] EllipticK[-m^2/(4(1-m))] == (1-m)^(1/2) EllipticK[(2-m)m] which is same as I got before, OK? So either I made same error over again because my brain is truly decaying, or Gosper was wrong, or Salamin was wrong, you figure out which. I was just trying to produce a nicer equivalent version. --------------------- Looks like a branch bug. Empirically, yours only works for m west of 1: gosper.org/braunch.pdf Would it remain nicer if repaired? --rwg
On Mon, May 25, 2015 at 3:25 PM, Bill Gosper <billgosper@gmail.com> wrote:
Bill Gosper: Then I bashed on it so hard that it got unnervingly ungrotesque: (*) EllipticK[1/4 (2 + (-2 + t)/Sqrt[1 - t])] == (1 - t)^(1/4) EllipticK[t] How could anything this simple be new?
WDS: Let s=Sqrt[1-t]. Then (*) is rewritten as EllipticK[ 1/2 - (1+s^2)/(4s) ] == EllipticK[ -(1-s)^2/(4s) ] == s^(1/2) EllipticK[1-s^2] Now let s=1-m. Then this is EllipticK[ -m^2/(4(1-m)) ] == (1-m)^(1/2) EllipticK[(2-m)m]. E.Salamin: Complains WDS formula is wrong.
--WDS: arg. Well, I just started with Gosper's "ungrotesque" formula taking its correctness on faith, then worked by hand to produce mine. Doing same thing over again now:
Gosper: EllipticK[1/4 (2 + (-2 + t)/Sqrt[1 - t])] == (1 - t)^(1/4) EllipticK[t] Let t=1-s^2: EllipticK[1/4 (2 - (1+s^2)/s)] == (s)^(1/2) EllipticK[1-s^2] EllipticK[(2s-1-s^2)/(4s)] == (s)^(1/2) EllipticK[1-s^2] EllipticK[-(1-s)^2/(4s)] == (s)^(1/2) EllipticK[1-s^2] which is same as I got before.
Now let s=1-m: EllipticK[-m^2/(4(1-m))] == (1-m)^(1/2) EllipticK[1-(1-m)^2] EllipticK[-m^2/(4(1-m))] == (1-m)^(1/2) EllipticK[2m-m^2] EllipticK[-m^2/(4(1-m))] == (1-m)^(1/2) EllipticK[(2-m)m] which is same as I got before,
OK? So either I made same error over again because my brain is truly decaying, or Gosper was wrong, or Salamin was wrong, you figure out which. I was just trying to produce a nicer equivalent version. --------------------- Looks like a branch bug. Empirically, yours only works for m west of 1: gosper.org/braunch.pdf Would it remain nicer if repaired? --rwg
t->1-z is safe and really nice: EllipticK[- ( t^(1/4) - t^(-1/4))^2/4]=== EllipticK[1/2 - 1/(4 Sqrt[z]) - Sqrt[z]/4] == z^(1/4) EllipticK[1 - z] Tempting now is z->u^4, but (u^4)^(1/4) -> u will be wrong outside the wedge |Arg[u]|<pi/4. Similarly for (u^4)^(1/2) -> u^2. --rwg This week-old steamed flounder seems to lack shelf-life. The flounder farmers should've recombined some tomato genes. Today my trousers top button failed, resulting in droppage. So I put on my underpants over them and told people I was a victim of noncommutativity.
On 5/26/15, Bill Gosper <billgosper@gmail.com> wrote: << Today my trousers top button failed, resulting in droppage. So I put on my underpants over them and told people I was a victim of noncommutativity. >> Now you can keep Superman and John Major (as immortalised by cartoonist Steve Bell) company. WFL
* Fred Lunnon <fred.lunnon@gmail.com> [May 26. 2015 11:22]:
On 5/26/15, Bill Gosper <billgosper@gmail.com> wrote: << Today my trousers top button failed, resulting in droppage. So I put on my underpants over them and told people I was a victim of noncommutativity. >>
Now you can keep Superman and John Major (as immortalised by cartoonist Steve Bell) company. WFL
OT: reader of the Graun? Am I the only person to find the new tile-based layout horrific (enough to reduce my visits to a small fraction of what it used to be)? My word for that is "Kachelpest", hope this goes away really soon.
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I indulge in the hardcopy Grauniad, reasoning that I already spend enough time staring at a screens. But that does cost substantially more than my broadband connection. Speaking of broad bands, couldn't RWG just wear a van Allen? Regardless, I suspect that he was more likely a victim of steamed flounder, the shelf life of which might incidentally be extended by avoiding consumption at a single sitting! WFL On 5/26/15, Joerg Arndt <arndt@jjj.de> wrote:
* Fred Lunnon <fred.lunnon@gmail.com> [May 26. 2015 11:22]:
On 5/26/15, Bill Gosper <billgosper@gmail.com> wrote: << Today my trousers top button failed, resulting in droppage. So I put on my underpants over them and told people I was a victim of noncommutativity. >>
Now you can keep Superman and John Major (as immortalised by cartoonist Steve Bell) company. WFL
OT: reader of the Graun? Am I the only person to find the new tile-based layout horrific (enough to reduce my visits to a small fraction of what it used to be)? My word for that is "Kachelpest", hope this goes away really soon.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (3)
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Bill Gosper -
Fred Lunnon -
Joerg Arndt