[.. CHOP ..] And Mma 10 is *still* missing the numerics!
In[3]:= D[EllipticTheta[s, t, q], {t, 2}] ==
-4 q D[EllipticTheta[s, t, q], q]
Out[3]= (0,1,0) EllipticThetaPrime [s, t, q] ==
(0,0,1) -4 q EllipticTheta [s, t, q]
In[4] Pi N[% /. t -> -- /. s -> 1 /. q -> 0.69] 4
Out[4]= (0,0,1) 6.91574 == -2.76 EllipticTheta [1., 0.785398, 0.69]
Vs my old Macsyma package: (c171) DIFF(THETA[S](T,Q),T,2) = -4*Q*DIFF(THETA[S](T,Q),Q);
(d171) thetaderiv (t, 2, q) = - 4 q thetaderiv (t, 0, q, 1) s s [In my GMail composition window, the two esses in the preceding
line are indented to be subscripts of the thetaderivs.]
(c172) SUBST([T = DFLOAT(%PI/4),S = 1,Q = 0.69],%);
(d172) 6.91573338862223d0 = 6.91573338862223d0
Huge thanks to NeilB, Ashok Anumandla, and Babu Lella for (so far, partially) bringing up Macsyma 2.4 under XP under Yosemite! (But I'm still wondering why (d171) no longer auto-simplifies to a tautology.)
If it weren't for the missing numerics, Mma would have numerous ways to exactly and manipulably express Halphen's and similar constants:
InverseFunction[EllipticTheta^(0,0,1),3,3][1,\[Pi]/4,0] InverseFunction[EllipticTheta^(0,0,1),3,3][2,\[Pi]/4,0] InverseFunction[EllipticThetaPrime^(0,1,0),3,3][1,\[Pi]/4,0] InverseFunction[EllipticThetaPrime^(0,1,0),3,3][2,\[Pi]/4,0]
Only two of these work: In[311]:= N[%, 22] Out[311]= (0,0,1) {InverseFunction[EllipticTheta , 3, 3][
1.000000000000000000000, 0.7853981633974483096157, 0]
(0,0,1) , InverseFunction[EllipticTheta , 3, 3][
2.000000000000000000000, 0.7853981633974483096157, 0]
, 0.3281065668749782530255, 0.3281065668749782530255}
(Notice how it floated the exact subscripts 1 and 2, and now foolishly regards them as approximate.)
The mail server's memory is nearly as bad as mine. I recovered the May 2011 text only thanks to a bounce of a copy sent to Steven Finch. More distressing: That mail also contained a bunch of denestings apparently debuted here, but half of them I don't even recognize:
rwg>Apropos radical denestings, do you think these from math fun startled anybody? [Surreptitiously trashed (by GMail?). Wysiwyg my tush.]
gosper.org/denests.pdf
Five-termers are quite sparse. I've never been able to find a six-termer. --Bill
Wow, who found these? I'd like to meet him! --rwg
I just fixed a typo in the Macsyma notebook exported by Mma as http://gosper.org/thetpak.html ,(c575): The last thetaderiv_1 should be thetaderiv_2. This is hard to fix because Mma 10 can no longer read the Mma notebook full of bitmaps pasted in from the Macsyma notebook!
(Macsyma never learned how to display formulas in html.)
It is a scandal how many thetpak.html examples continue to elude Mma 10.
--rwg