Re: [math-fun] DLMF fungous inflection
Indubitably. But the author seemed to wish to qualitatively distinguish the the 2nd half of the interval. Spoiler lowermost. es>At K(k), the quarter period, the even derivatives of cn(x) vanish by symmetry. The special thing that happens at k = 1/sqrt(2) is that the third derivative vanishes. Thus the graph of cn(x) differs from a straight line by terms of 5-th order. -- Gene
________________________________> From: Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>>>To: math-fun@mailman.xmission.com <http://gosper.org/webmail/src/compose.php?send_to=math-fun%40mailman.xmission.com> >Sent: Friday, December 23, 2011 12:12 PM>Subject: [math-fun] DLMF fungous inflection> >(This was too easy for the kids, but it feels good>when you find the words an Authority missed.)>>See http://dlmf.nist.gov/22.3.F2.mag and http://dlmf.nist.gov/22.3.F3.mag>"For cn(x,k) the curve for>k=1/sqrt2=0.70710… is a boundary between the curves>that have an inflection point in the interval>0≤x≤2K(k), and its translates, and those>that do not;...">>a) why don't they just say "at x=K(k)" vs "in the interval 0≤x≤2K(k)"?>>b) Huh? cn(x,k) always has an inflection point there! But *something*>happens at k=1/sqrt2. What should they have said?>--rwg>Caution: DLMF uses K(modulus) vs K(parameter), so if you plot this>with Macsyma or Mma, the magic k = 1/2, not 1/sqrt2. In either case,>K=4 (1/4)!^2/Sqrt[Pi] ~ 1.85407>(This K property is imperfectly analogous to the graphical definition of>Halphen's constant, about 69% into http://gosper.org/thetpak.html .)>_______________________________________________ For K > 4 (1/4)!^2/Sqrt[Pi], for x>K, the point of inflection of cn(x,k) splits into three! --rwg
Yes indeed, that's a nice observation. One can see this from cn''(x) = cn(x) (2 k^2 sn(x)^2 - 1) = 0. In addition to the trivial inflection points where cn(x) = 0, we have additional ones where sn(x)^2 = 1/(2 k^2). For real x, 0 <= sn(x)^2 <= 1, so these extra inflection points exist only when k >= 1/sqrt(2). And for k = 1/sqrt(2), sn(x) = +-1 exactly where cn(x) = 0, so three inflection points coincide. -- Gene
________________________________ From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Sent: Friday, December 23, 2011 4:52 PM Subject: Re: [math-fun] DLMF fungous inflection
Indubitably. But the author seemed to wish to qualitatively distinguish the the 2nd half of the interval. Spoiler lowermost.
es>At K(k), the quarter period, the even derivatives of cn(x) vanish by symmetry. The special thing that happens at k = 1/sqrt(2) is that the third derivative vanishes. Thus the graph of cn(x) differs from a straight line by terms of 5-th order.
-- Gene
________________________________> From: Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>>>To: math-fun@mailman.xmission.com <http://gosper.org/webmail/src/compose.php?send_to=math-fun%40mailman.xmission.com> >Sent: Friday, December 23, 2011 12:12 PM>Subject: [math-fun] DLMF fungous inflection> >(This was too easy for the kids, but it feels good>when you find the words an Authority missed.)>>See http://dlmf.nist.gov/22.3.F2.mag and http://dlmf.nist.gov/22.3.F3.mag>"For cn(x,k) the curve for>k=1/sqrt2=0.70710… is a boundary between the curves>that have an inflection point in the interval>0≤x≤2K(k), and its translates, and those>that do not;...">>a) why don't they just say "at x=K(k)" vs "in the interval 0≤x≤2K(k)"?>>b) Huh? cn(x,k) always has an inflection point there! But *something*>happens at k=1/sqrt2. What should they have said?>--rwg>Caution: DLMF uses K(modulus) vs K(parameter), so if you plot this>with Macsyma or Mma, the magic k = 1/2, not 1/sqrt2. In either case,>K=4 (1/4)!^2/Sqrt[Pi] ~ 1.85407>(This K property is imperfectly analogous to the graphical definition of>Halphen's constant, about 69% into http://gosper.org/thetpak.html .)>_______________________________________________ For K > 4 (1/4)!^2/Sqrt[Pi], for x>K, the point of inflection of cn(x,k) splits into three! --rwg
Sorry, due to the lingering effects of a headsplitting brainfreeze, my previous post made no sense. On Fri, Dec 23, 2011 at 4:52 PM, Bill Gosper <billgosper@gmail.com> wrote:
es>At K(k), the quarter period, the even derivatives of cn(x) vanish by symmetry. The special thing that happens at k = 1/sqrt(2) is that the third derivative vanishes. Thus the graph of cn(x) differs from a straight line by terms of 5-th order.
-- Gene
Indeed, but the author was mumbling about intervals 0<x<2K and k>1/sqrt2.
Spoiler lowermost.
________________________________ > From: Bill Gosper <billgosper@gmail.com<http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>
To: math-fun@mailman.xmission.com<http://gosper.org/webmail/src/compose.php?send_to=math-fun%40mailman.xmission.com> >Sent: Friday, December 23, 2011 12:12 PM >Subject: [math-fun] DLMF fungous inflection
(This was too easy for the kids, but it feels good>when you find the words an Authority missed.)>>See http://dlmf.nist.gov/22.3.F2.mag and http://dlmf.nist.gov/22.3.F3.mag>"For cn(x,k) the curve for>k=1/sqrt2=0.70710… is a boundary between the curves>that have an inflection point in the interval>0≤x≤2K(k), and its translates, and those>that do not;...">>a) why don't they just say "at x=K(k)" vs "in the interval 0≤x≤2K(k)"?>>b) Huh? cn(x,k) always has an inflection point there! But *something*>happens at k=1/sqrt2. What should they have said?>--rwg>Caution: DLMF uses K(modulus) vs K(parameter), so if you plot this>with Macsyma or Mma, the magic k = 1/2, not 1/sqrt2. In either case,>K=4 (1/4)!^2/Sqrt[Pi] ~ 1.85407>(This K property is imperfectly analogous to the graphical definition of>Halphen's constant, about 69% into http://gosper.org/thetpak.html .)
For k>1/sqrt2, (K(k) > 4 (1/4)!^2/Sqrt[Pi]), the point of inflection at x=K(k) of cn(x,k) splits into three! --rwg
(For the record, as I send this, the quoted original message has a > at the beginning of each line, but by Gene's reply, those lines had all run together. GMail again?)
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