On Sat, Nov 17, 2012 at 3:46 PM, Bill Gosper <billgosper@gmail.com> wrote:

What, exactly, is an "elliptic integral singular value"? Weisstein (http://mathworld.wolfram.com/EllipticIntegralSingularValue.html) when converted from modulus- to parameterspeak, says it's a value m=ModularLambda[√r] such that EllipticK[1-m]/EllipticK[m] ==√r, where r seems to be restricted to a positive rational. E.g., "the seventh singular value":

ModularLambda[I Sqrt[7]] == 1/16 (8 - 3 Sqrt[7])

EllipticK[1/16 (8 - 3 Sqrt[7])] == Gamma[1/7] Gamma[2/7] Gamma[4/7]/(4 7^(1/4) π)

EllipticK[1/2 + 3 Sqrt[7]/16]/ EllipticK[1/16 (8 - 3 Sqrt[7])] == Sqrt[7]

Why rational r? Why √ ? Why positive? E.g.,

ModularLambda[2 I E^(-I π/6)] == -7 + 4 Sqrt[3]

EllipticK[-7 + 4 Sqrt[3]] == 3^(1/4) Sqrt[2 + Sqrt[3]] Gamma[1/3]^3/(8 2^(1/3) π)

EllipticK[8 - 4 Sqrt[3]]/EllipticK[-7 + 4 Sqrt[3]] == 2 E^(-I π/6)

Why real? E.g., EllipticK[E^(i π/3)]/EllipticK[E^(-(i π/3))] == E^(i π/6)

In:= ModularLambda[I*%[[2]]] == %[[1, 2, 1]] Out= ModularLambda[I E^(i π/6)] == E^(i π/3) Or EllipticK[-2 (1 + Sqrt[2])] == Sqrt[(2 - Sqrt[2])/π] Gamma[3/8] Gamma[9/8] EllipticK[1 + 2 (1 + Sqrt[2])]/ EllipticK[-2 (1 + Sqrt[2])] == -I + 1/Sqrt[2] But things can get weird:

This weirdness is from the remarkable identity ModularLambda[τ]==ModularLambda[τ+2]==ModularLambda[τ/(1+2 τ)] Ah, so that's why they named it Modular!

ModularLambda[4 (-1)^(1/3)] == (-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4

This is the same m as ModularLambda[2 I Sqrt[3]] == (-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4

i.e., the "twelfth singular value" instead of period ratio 4 E^(-I π/6).

<undersimplified>

(Not quite the same as yesterday's.)

Actually, there are four (pairs) of these: {EllipticK[(2 + Sqrt[2] + Sqrt[3] + Sqrt[6])^4] == -(( 3^(1/4) Sqrt[13] (1 - 2 Sqrt[2] + Sqrt[3]) E^(-I ArcTan[2 Sqrt[3]]) Gamma[1/3]^3)/(16 2^(5/6) π)), EllipticK[1 - (2 + Sqrt[2] + Sqrt[3] + Sqrt[6])^4]/ EllipticK[(2 + Sqrt[2] + Sqrt[3] + Sqrt[6])^4] == 2 Sqrt[3/13] E^(I ArcTan[2 Sqrt[3]]), EllipticK[(2 + Sqrt[2] - Sqrt[3] - Sqrt[6])^4] == ( 3^(3/4) (-1 + 2 Sqrt[2] + Sqrt[3]) Gamma[1/3]^3)/(16 2^(5/6) π), EllipticK[1 - (2 + Sqrt[2] - Sqrt[3] - Sqrt[6])^4]/ EllipticK[(2 + Sqrt[2] - Sqrt[3] - Sqrt[6])^4] == 2/Sqrt[3], EllipticK[(2 - Sqrt[2] + Sqrt[3] - Sqrt[6])^4] == ( 3^(1/4) (1 + 2 Sqrt[2] + Sqrt[3]) Gamma[1/3]^3)/(16 2^(5/6) π), EllipticK[1 - (2 - Sqrt[2] + Sqrt[3] - Sqrt[6])^4]/ EllipticK[(2 - Sqrt[2] + Sqrt[3] - Sqrt[6])^4] == 2 Sqrt[3], EllipticK[(2 - Sqrt[2] - Sqrt[3] + Sqrt[6])^4] == ( 3^(1/4) Sqrt[7] (1 + 2 Sqrt[2] - Sqrt[3]) E^(-I ArcTan[2/Sqrt[3]]) Gamma[1/3]^3)/(16 2^(5/6) π), EllipticK[1 - (2 - Sqrt[2] - Sqrt[3] + Sqrt[6])^4]/ EllipticK[(2 - Sqrt[2] - Sqrt[3] + Sqrt[6])^4] == ( 2 E^(I ArcTan[2/Sqrt[3]]))/Sqrt[7]} And probably four (pairs) like EllipticK[(-3 + 2 Sqrt[2] + 2 Sqrt[3] - Sqrt[6])^2] == 1/8 Sqrt[(Gamma[1/24] Gamma[5/24] Gamma[7/24] Gamma[11/24])/( 2 (-1 + Sqrt[2]) (2 - Sqrt[3]) (Sqrt[2] + Sqrt[3]) π)] EllipticK[1 - (-3 + 2 Sqrt[2] + 2 Sqrt[3] - Sqrt[6])^2]/ EllipticK[(-3 + 2 Sqrt[2] + 2 Sqrt[3] - Sqrt[6])^2] == Sqrt[2/3] EllipticK[(3 + 2 Sqrt[2] + 2 Sqrt[3] + Sqrt[6])^2] == 1/8 E^(-I ArcTan[Sqrt[6]]) Sqrt[( 7 Gamma[1/24] Gamma[5/24] Gamma[7/24] Gamma[11/24])/( 6 (1 + Sqrt[2]) (2 + Sqrt[3]) (Sqrt[2] + Sqrt[3]) π)] EllipticK[1 - (3 + 2 Sqrt[2] + 2 Sqrt[3] + Sqrt[6])^2]/ EllipticK[(3 + 2 Sqrt[2] + 2 Sqrt[3] + Sqrt[6])^2] == Sqrt[6/7] E^(I ArcTan[Sqrt[6]]) There are too many to tabulate. We need an algorithm. But what is so important about (K'/K)^2 being an integer? Only the 3rd of the above six pairs exhibits a "singular value" (the 12th).

Even weirder: ModularLambda[I 3 E^(-I π/6)] == 1/2 (1 - Sqrt[1 - 4 (16001 - 6350 2^(1/3) (1 - I Sqrt[3]) - 5040 2^(2/3) (1 + I Sqrt[3]))])

EllipticK[1/2 (1 - Sqrt[ 1 - 4 (16001 - 6350 2^(1/3) (1 - I Sqrt[3]) - 5040 2^(2/3) (1 + I Sqrt[3]))])] == ((-1)^(1/12) 3^(1/4) Sqrt[5/9 + (2 2^(1/3))/9 - (2 I 2^(1/3))/(3 Sqrt[3])] Gamma[1/3]^3)/(4 2^(1/3) π)

EllipticK[ 1/2 + 1/2 Sqrt[1 - 4 (16001 - 6350 2^(1/3) (1 - I Sqrt[3]) - 5040 2^(2/3) (1 + I Sqrt[3]))]]/ EllipticK[ 1/2 - 1/2 Sqrt[1 - 4 (16001 - 6350 2^(1/3) (1 - I Sqrt[3]) - 5040 2^(2/3) (1 + I Sqrt[3]))]] == Sqrt[7] E^(I ArcCot[3 Sqrt[3]])

instead of 3 E^(-I π/6) !

Not weird. Just modular. --rwg Still wondering if EllipticK[E^(±i π/3)] == (3^(1/4) E^(±i π/12) Gamma[1/3]^3)/(4 2^(1/3) π) is actually new.

And sadly, ModularLambda[I GoldenRatio] seems transcendental. --rwg

Elsewhere I said *rwg>I'm having no luck finding a table of special values of EllipticE. This may be due to a simpler formula than above to produce them from EllipticK. It looks like they're something K + something/K.*

Indeed, see (40) and (41) of http://mathworld.wolfram.com/EllipticIntegralSingularValue.html . This requires the elliptic α function. But closed forms require finding 𝜗' for particular q, which is equivalent to the problem I was attacking with "logderiveta".