[math-fun] Brad's recent interest in sn|cn double angles
prompts me to t(rot )out these four old 𝜗 half-angle formulas: Out[419]= {EllipticTheta[1, x/2, q] == (1/(2^( 1/4)))((-EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] - EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[2, x/2, q] == (1/(2^( 1/4)))((EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] - EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[3, x/2, q] == (1/(2^( 1/4)))((EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] + EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[4, x/2, q] == (1/(2^( 1/4)))((-EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] + EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4))} Check: In[432]:= Assuming[0 < x < \[Pi]/2 && 0 < q < 1, FullSimplify@Series[%419, {q, 0, 22}, {x, 0, 22}]] // tim During evaluation of In[432]:= 419.850403,4 (*Seven minutes*). Out[432]= {True, True, True, True} —rwg
Hi Bill, While I think it is a good idea to extend double or half angle identities to the theta function, my interest is not to further entrench the Jacobi formalism, nor to endlessly list identities without saying why they would matter, or what they could be used for. In my opinion, elliptic function angle doubling formulae are not as exciting considered in and of themselves, but more exciting when paired with known function values on a small disk around the origin and a known period ratio. What is the shortest proof that Harold Edwards elliptic function restricted to a purely imaginary period ratio is essentially the same as Jacobi's sn? One common tactic is to compare poles and zeros, which can easily be done after Part III of "A normal form for elliptic curves". The other option, after Part II, is just to observe that the doubling formulae are the same, and that up to scale, the two functions agree on a small disk around the origin. Once it is well understood how the angle doubling formulae define the generic elliptic function psi, then Part III S.22 of Edward's article finally begins to make constructive sense. However, part of the algorithm is still confusing because multi-valuedness enters through use of double-angle formula. A half-angle, were it to exist, would be better. So here is a practical question for you Bill: Your half angle formulae, can they be used to simplify the algorithm for computing period ratio tau from known initial data? Cheers --Brad On Tue, Nov 19, 2019 at 9:34 PM Bill Gosper <billgosper@gmail.com> wrote:
prompts me to t(rot )out these four old 𝜗 half-angle formulas: Out[419]= {EllipticTheta[1, x/2, q] == (1/(2^( 1/4)))((-EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] - EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[2, x/2, q] == (1/(2^( 1/4)))((EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] - EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[3, x/2, q] == (1/(2^( 1/4)))((EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] + EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[4, x/2, q] == (1/(2^( 1/4)))((-EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] + EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4))} Check: In[432]:= Assuming[0 < x < \[Pi]/2 && 0 < q < 1, FullSimplify@Series[%419, {q, 0, 22}, {x, 0, 22}]] // tim
During evaluation of In[432]:= 419.850403,4 (*Seven minutes*).
Out[432]= {True, True, True, True} —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
See Also: https://demonstrations.wolfram.com/EdwardssSolutionOfPendulumOscillation/ Updated today, thank you Ed Pegg! If you are not interested in mathematica code, there is documentation, actually a math-phys paper, linked in references. Here it is again: https://github.com/bradklee/Dissertation/tree/master/SimplePendulum Hold your laughter, please--as far as I'm concerned, v1.0 is inline for publication after I double check the technical content. Now is a time for Naysayers to speak up if they have any legitimate reason why it shouldn't be published. Actually, any comments or criticism are welcome. Thanks again! Cheers --Brad ( PS. The reviews for another pendulum article on arxiv were horribly negative. I stand by the initial attempt, but have done as the refs asked and completed a total rewrite, which will ultimately replace the version on arxiv, hopefully. ) On Wed, Nov 20, 2019 at 12:09 PM Brad Klee <bradklee@gmail.com> wrote:
Hi Bill,
While I think it is a good idea to extend double or half angle identities to the theta function, my interest is not to further entrench the Jacobi formalism, nor to endlessly list identities without saying why they would matter, or what they could be used for.
In my opinion, elliptic function angle doubling formulae are not as exciting considered in and of themselves, but more exciting when paired with known function values on a small disk around the origin and a known period ratio.
What is the shortest proof that Harold Edwards elliptic function restricted to a purely imaginary period ratio is essentially the same as Jacobi's sn?
One common tactic is to compare poles and zeros, which can easily be done after Part III of "A normal form for elliptic curves". The other option, after Part II, is just to observe that the doubling formulae are the same, and that up to scale, the two functions agree on a small disk around the origin.
Once it is well understood how the angle doubling formulae define the generic elliptic function psi, then Part III S.22 of Edward's article finally begins to make constructive sense. However, part of the algorithm is still confusing because multi-valuedness enters through use of double-angle formula. A half-angle, were it to exist, would be better.
So here is a practical question for you Bill: Your half angle formulae, can they be used to simplify the algorithm for computing period ratio tau from known initial data?
Cheers --Brad
On Tue, Nov 19, 2019 at 9:34 PM Bill Gosper <billgosper@gmail.com> wrote:
prompts me to t(rot )out these four old 𝜗 half-angle formulas: Out[419]= {EllipticTheta[1, x/2, q] == (1/(2^( 1/4)))((-EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] - EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[2, x/2, q] == (1/(2^( 1/4)))((EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] - EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[3, x/2, q] == (1/(2^( 1/4)))((EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] + EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[4, x/2, q] == (1/(2^( 1/4)))((-EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] + EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4))} Check: In[432]:= Assuming[0 < x < \[Pi]/2 && 0 < q < 1, FullSimplify@Series[%419, {q, 0, 22}, {x, 0, 22}]] // tim
During evaluation of In[432]:= 419.850403,4 (*Seven minutes*).
Out[432]= {True, True, True, True} —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
-
Bill Gosper -
Brad Klee