Appended below is "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." Borwein & Borwein, (2.3.17) E = K - pi/K Theta4Dot/Theta4 where Theta4Dot:= d/ds Theta(q), s:= -log(q)/pi Two years after my logderivetas remark below, I don't understand it. Here is how to get DedekindEta'[x] in radicals from DedekindEta[x] in radicals. Empirically, for rational x^2: Out[131]= \[Alpha] -> 1/(4 x) + Derivative[1][DedekindEta][x]/ DedekindEta[x])/(\[Pi] DedekindEta[x]^4 is algebraic. E.g., for x = 2 Sqrt[-3] In[181]:= %131 /. x -> Sqrt[-3]*2 Out[181]= \[Alpha] -> (-(I/(8 Sqrt[3])) + Derivative[1][DedekindEta][2 I Sqrt[3]]/ DedekindEta[2 I Sqrt[3]])/(\[Pi] DedekindEta[2 I Sqrt[3]]^4) In[182]:= RootApproximant[%[[2]]] Out[182]= Root[-4 + 5 #1^2 - 10 #1^4 - 2 #1^6 - 6 #1^8 + 8 #1^10 - 5 #1^12 + 9 #1^14 + 2 #1^16 &, 10] Insufficient precision. In[184]:= RootApproximant[N[%181[[2]], 669]] Out[184]= Root[1 + 143822848 #1^6 + 1048576 #1^12 &, 8] In[185]:= ToRadicals[%] Out[185]= ((-1)^(1/3) (-70226 - 40545 Sqrt[3])^(1/6))/(2 2^(2/3)) In[186]:= Strad[%] During evaluation of In[186]:= PossibleZeroQ::ztest1: Unable to decide whether numeric quantity Sqrt[26+15 Sqrt[3]]-(70226+40545 Sqrt[3])^(1/6) is equal to zero. Assuming it is. >> Out[186]= (I Sqrt[26 + 15 Sqrt[3]])/(2 2^(2/3)) In[187]:= Strad[%] Out[187]= (I (5 + 3 Sqrt[3]))/(4 2^(1/6)) Since there are various means to find DedekindEta[2 I Sqrt[3]] -> ((3 - Sqrt[3])^(1/4) Gamma[1/3]^(3/2))/(2 2^(7/8) Pi) , we have Derivative[1][DedekindEta][2 I Sqrt[3]] == -(( I (3 - Sqrt[3])^(1/4) Gamma[1/3]^( 3/2) (-1 - (Sqrt[3] (3 - Sqrt[3]) (5 + 3 Sqrt[3]) Gamma[1/3]^6)/( 64 2^(2/3) Pi^3)))/(16 2^(7/8) Sqrt[3] Pi)) and finally 2 EllipticE[(Sqrt[3] - 1)^2/8] == pi[(Sqrt@3 - 1)/Sqrt@8] == 2 ((2^(1/3) Pi^2)/( 3^(3/4) Gamma[1/3]^3) + ((3 + Sqrt[3]) Gamma[1/3]^3)/( 8 2^(1/3) 3^(3/4) Pi)) where pi[e]:= circumference of ellipse of unit diameter and eccentricity e: In[238]:= pi[e_] := 2 EllipticE[e^2] In[239]:= pi[0] Out[239]= Pi In[240]:= pi[1] Out[240]= 2 It would now be almost routine to produce tables of special values of EllipticK and EllipticE from my table of DedekindEta[complexsurd]. The obstacles are knowing if you've used enough precision in RootApproximant, and taming indigestible surdbergs. --rwg On Wed, Nov 7, 2012 at 3:05 PM, Bill Gosper <billgosper@gmail.com> wrote: The ("complete") first kind elliptic integral K(m) and parameter m are both expressible in 𝜗 constants, and those three+ 𝜗 constants are all expressible in terms of Dedekind η. So Out[840]= EllipticK[(16 η[q]^8 η[q^4]^16)/η[q^2]^24] == (π η[q^2]^10)/(2 η[q]^4 η[q^4]^4) Crude but effective: d/dq and eliminate EllipticK to get EllipticE[(16 \[Eta][q^4]^8)/(\[Eta][q]^8 + 16 \[Eta][q^4]^8)] == -( 1/(\[Eta][q]^8 + 16 \[Eta][q^4]^8))(-((\[Pi] \[Eta][q]^4 \[Eta][q^2]^10)/( 2 \[Eta][ q^4]^4)) - (\[Pi] \[Eta][ q^2]^9 (6 Sqrt[3] q^(3/2) logderiveta[q] \[Eta][q] \[Eta][q^2] \[Eta][q^4] - 30 Sqrt[3] q^(5/2) logderiveta[q^2] \[Eta][q] \[Eta][q^2] \[Eta][q^4] + 24 Sqrt[3] q^(9/2) logderiveta[q^4] \[Eta][q] \[Eta][q^2] \[Eta][ q^4]) (\[Eta][q]^8 + 16 \[Eta][q^4]^8))/(2 q^( 1/3) \[Eta][q]^4 \[Eta][ q^4]^4 (6 Sqrt[3] q^(7/6) logderiveta[q] \[Eta][q] \[Eta][q^4] - 24 Sqrt[3] q^(25/6) logderiveta[q^4] \[Eta][q] \[Eta][q^4])) The remaining problem is those logderivetas, but lo, ((1/(4*x) - ((logderiveta[(1/(E^x))]))/(E^x))/((DedekindEta[((x*I)/(2*Pi))])^4)) is algebraic for (x/π)^2 rational. Unfortunately, (x/π)^2= (i/2-1/√12)^2 (as in the q for k=(-1)^(1/3)) isn't rational, so EllipticE[(-1)^(1/3)] == E^(-(I π/12)) ((2 2^(1/3) π^2)/(3^(3/4) Gamma[1/3]^3) + Gamma[1/3]^3/(4 2^(1/3) 3^(1/4) π)) required Legendre's relation plus numerical empiricism. This, with the K((-1)^(1/3)) result gives closed forms for all 2F1[a+1/2,b+1/2,c;(-1)^(1/3)], a,b,c integers. Likewise for every k for which we find both K(k) and E(k), which are contiguous 2F1[k]. BtW, if you accept MatProd[mat[k],{k,2,a}] as closed form, then I can write 2F1[a+1/2,b+1/2,c;k] in closed form. --rwg Added 2012-11-11 13:22 And for such x, we have many DedekindEta[((x*I)/(2*Pi)) in our collection. In fact, people.math.carleton.ca/~*williams*/papers/*pdf*/*299*.*pdf claims to give a complete solution. Early on, it appears to omit half the odd denominators, but I assume this is remedied later in the derivation. Anyway, as a fairly strenuous example: EllipticE[1/(1 + (-1 + GoldenRatio^(3/2))^8/(16 GoldenRatio^4))] == ( 2 5^(3/4) GoldenRatio^4 π^(3/2))/( Sqrt[1 + Sqrt[5]] (1525 + 682 Sqrt[5])^(1/4) Gamma[1/20] Gamma[9/ 20]) + ((-7 - 3 Sqrt[5] + Sqrt[190 + 85 Sqrt[5]]) Gamma[1/ 20] Gamma[9/20])/( 8 (1525 + 682 Sqrt[5])^(1/4) Sqrt[5 (1 + Sqrt[5]) π]) 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. At the other extreme, has anyone seen *EllipticK[(-1)^(1/3)] == ((-1)^(1/12) 3^(1/4) Gamma[1/3]^3)/( 4 2^(1/3) π) ? --rwg