[math-fun] Siddarth Ghoshal has written an amazing identity
https://www.wolframcloud.com/objects/d0cafba7-0635-40d3-83af-c71cd08ca033 and possibly seminal paper <http://gosper.org/total_term_computations_case2.pdf> . "While this work stands alone as a very interesting identity, it suggests more deeply that the lacunary series above CAN be extended to a function on the entire complex plane, through some advanced generalization of analytic continuation." —rwg
<< At this point, symbolic algebra programs reach their limit and we need to progress by hand to evaluate this ... >> Good to know that we haven't managed to make ourselves completely redundant yet, then! WFL On 3/27/19, Bill Gosper <billgosper@gmail.com> wrote:
https://www.wolframcloud.com/objects/d0cafba7-0635-40d3-83af-c71cd08ca033 and possibly seminal paper <http://gosper.org/total_term_computations_case2.pdf> . "While this work stands alone as a very interesting identity, it suggests more deeply that the lacunary series above CAN be extended to a function on the entire complex plane, through some advanced generalization of analytic continuation." —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Strange timing … I just concluded a short course with this function (and hadn’t realized it has a history and is called “lacunary”): http://uuuuuu.lassp.cornell.edu/courses/physics_7654_asymptotic_analysis We stayed on the real axis, but the plot on the website, that covers the interior of the unit disk, suggests the boundary behavior at e^(\pm i2pi/3) is special. Is anything known about that? -Veit
On Mar 26, 2019, at 10:05 PM, Bill Gosper <billgosper@gmail.com> wrote:
https://www.wolframcloud.com/objects/d0cafba7-0635-40d3-83af-c71cd08ca033 and possibly seminal paper <http://gosper.org/total_term_computations_case2.pdf> . "While this work stands alone as a very interesting identity, it suggests more deeply that the lacunary series above CAN be extended to a function on the entire complex plane, through some advanced generalization of analytic continuation." —rwg
Veit, The boundary behaviour at exp(2 pi i / 3) might be explained on the basis that 1/3 has the worst approximations by dyadic rationals. (This is entirely analogous to the fact that phi has the worst approximations by rationals, and indeed ?(1/phi) = 2/3 where ?() is the Minkowski question mark function.) Best wishes, Adam P. Goucher
Sent: Wednesday, March 27, 2019 at 11:58 AM From: "Veit Elser" <ve10@cornell.edu> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Siddarth Ghoshal has written an amazing identity
Strange timing … I just concluded a short course with this function (and hadn’t realized it has a history and is called “lacunary”):
http://uuuuuu.lassp.cornell.edu/courses/physics_7654_asymptotic_analysis
We stayed on the real axis, but the plot on the website, that covers the interior of the unit disk, suggests the boundary behavior at e^(\pm i2pi/3) is special. Is anything known about that?
-Veit
On Mar 26, 2019, at 10:05 PM, Bill Gosper <billgosper@gmail.com> wrote:
https://www.wolframcloud.com/objects/d0cafba7-0635-40d3-83af-c71cd08ca033 and possibly seminal paper <http://gosper.org/total_term_computations_case2.pdf> . "While this work stands alone as a very interesting identity, it suggests more deeply that the lacunary series above CAN be extended to a function on the entire complex plane, through some advanced generalization of analytic continuation." —rwg
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Here's a short note of Noam Elkies about the behavior of x - x^2 + x^4 - x^8 + ... as x approaches 1 from below: http://www.math.harvard.edu/~elkies/Misc/sol8.html On Wed, Mar 27, 2019 at 8:30 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
Veit,
The boundary behaviour at exp(2 pi i / 3) might be explained on the basis that 1/3 has the worst approximations by dyadic rationals. (This is entirely analogous to the fact that phi has the worst approximations by rationals, and indeed ?(1/phi) = 2/3 where ?() is the Minkowski question mark function.)
Best wishes,
Adam P. Goucher
Sent: Wednesday, March 27, 2019 at 11:58 AM From: "Veit Elser" <ve10@cornell.edu> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Siddarth Ghoshal has written an amazing identity
Strange timing … I just concluded a short course with this function (and hadn’t realized it has a history and is called “lacunary”):
http://uuuuuu.lassp.cornell.edu/courses/physics_7654_asymptotic_analysis
We stayed on the real axis, but the plot on the website, that covers the interior of the unit disk, suggests the boundary behavior at e^(\pm i2pi/3) is special. Is anything known about that?
-Veit
On Mar 26, 2019, at 10:05 PM, Bill Gosper <billgosper@gmail.com> wrote:
https://www.wolframcloud.com/objects/d0cafba7-0635-40d3-83af-c71cd08ca033
and possibly seminal paper <http://gosper.org/total_term_computations_case2.pdf> . "While this work stands alone as a very interesting identity, it suggests more deeply that the lacunary series above CAN be extended to a function on the entire complex plane, through some advanced generalization of analytic continuation." —rwg
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Mar 27, 2019, at 8:29 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Veit,
The boundary behaviour at exp(2 pi i / 3) might be explained on the basis that 1/3 has the worst approximations by dyadic rationals. (This is entirely analogous to the fact that phi has the worst approximations by rationals, and indeed ?(1/phi) = 2/3 where ?() is the Minkowski question mark function.)
possibly related: For which z does one get the 'shortest proof' that f(z)=z-f(z^2) is inconsistent on the boundary?
Best wishes,
Adam P. Goucher
participants (5)
-
Adam P. Goucher -
Bill Gosper -
Fred Lunnon -
Veit Elser -
Victor Miller