[math-fun] Tan[Sin[x]]-Sin[Tan[x]] puzzles
From Neil Bickford (under protest): Plot[Tan[Sin[x]]-Sin[Tan[x]], {x, 0, π/2}] is ultraflat near 0 and goes berserk at π/2. Ultraflat: x^7/30+(29 x^9)/756+(1913 x^11)/75600+O[x]^12 Berserk: Local minima and maxima cluster at π/2, vibrating between roughly .5 and 2.5. What limits, if any, do they approach? Note that by analytic continuation, an arbitrarily short segment of the flat part near zero completely predicts the berserk part west of π/2. Are all the terms of the series approximated above positive?
asin(tanh(x))-atan(sinh(x)) is even flatter! --- Quoting Bill Gosper <billgosper@gmail.com>:
From Neil Bickford (under protest): Plot[Tan[Sin[x]]-Sin[Tan[x]], {x, 0, π/2}] is ultraflat near 0 and goes berserk at π/2. Ultraflat: x^7/30+(29 x^9)/756+(1913 x^11)/75600+O[x]^12 Berserk: Local minima and maxima cluster at π/2, vibrating between roughly .5 and 2.5. What limits, if any, do they approach? Note that by analytic continuation, an arbitrarily short segment of the flat part near zero completely predicts the berserk part west of π/2. Are all the terms of the series approximated above positive?
Mike Hirschhorn also points out that this function is treated in the very first problem of http://www.math.upenn.edu/Arnold/Arnold-Trivium-1991.pdf Mike's solution reminded me of the possibility of expressing all the turning points (e.g., the nth) as a single series in n, in the manner that gosper.org/newsrope.pdf (with http://gosper.org/rope.png) finds the fixed points of tan x. This amounts to finding the zeroes of the derivative Cos[x]*(Sec[Sin[x]])^2 == (Sec[x])^2*Cos[Tan[x]]. With Macsyma, I found x==π/2 -1/(2 (1/4 + n) π)+ 1/( 24 (1/4 + n)^3 π^3) -(((5 + Cos[1]^2) Sec[1]^2)/( 160 (1/4 + n)^5 π^5))+ ( Sec[1]^3 (35 Cos[1] + 2 Cos[1]^3 + 14 Sin[1]))/(1792 (1/4 + n)^7 π^7) +... Plugging this into the derivative to get (approximately 0) Sec[1]^4 (342+279 Cos[1]^2+8 Cos[1]^4+234 Cos[1] Sin[1])/(2304 (1/4+n)^5 π^5)+ Sec[1]^6 (24-1368 Cos[1]^2+459 Cos[1]^4+16 Cos[1]^6-1224 Cos[1] Sin[1]-30 Cos[1]^3 Sin[1])/(18432 (1/4+n)^7 π^7)+... proved an unexpected challenge, even to Macsyma, which complained of an essential singularity in cos(tan(x)), since tan x blows up as n->oo. But it blows up in a special way! tan(x)==2 * (n + 1/4) * Pi - 1/(8 * Cos[1]^2 * (n + 1/4)^3 * Pi^3) +... so the cos can punt the 2 n Pi. On the Lisp Machines, I had Bill Dubuque's experimental Taylor, which smoothly handled essential singularities, including Rich's famous Screw Limit (c393) (%e^%e^%e^(x+%e^-(a+x+%e^x+%e^%e^x))-%e^%e^%e^x, 'limit(%%,x,inf) = tlimit(%%,x,inf)); Time= 7379 msec. x %e x - %e - %e - x - a %e + x x %e %e %e %e - a (d393) limit (%e - %e ) = %e x -> inf At the dawn of time, I suggested that SW hire Dubuque to add this capability to Mma, and he decided nobody would use it. I wish I had thought of this cos(tan x) example, although Mma at the time probably didn't even know about integers. Dubuque would have had a fit over how lame the rest of Series was, and insisted on rewriting it. One of the reason's I couldn't do this tan(sin) problem in Mma is that, even today, it can't expand in negative powers of y-1, e.g.: In[1275]:= Series[1/x+1/(1-x),{x,0,3}] Out[1275]= 1/x+1+x+x^2+x^3+O[x]^4 In[1276]:= InverseSeries[%,y] Out[1276]= 1/y+(1/y)^2+2/y^3+5/y^4+14/y^5+O[1/y]^6 This should have looked like Out[1278], below. Strongly hinting In[1277]:= InverseSeries[%%-1,y-1] Out[1277]= InverseSeries[1/x+x+x^2+x^3+O[x]^4,-1+y] fails. Shoving its nose in it In[1278]:= InverseSeries[%%%-1,z]/.z->y-1 Out[1278]= 1/(-1+y)+(1/(-1+y))^3+(1/(-1+y))^4+3/(-1+y)^5+O[1/(-1+y)]^6 appears to work, but it's bogus: In[1279]:= %-1/y Out[1279]= -(1/y)+(1/(-1+y)+(1/(-1+y))^3+(1/(-1+y))^4+3/(-1+y)^5+O[1/(-1+y)]^6) If you select %1276 and convert to inputform and then back to standardform (⇧⌘I, ⇧⌘N), you get (y-ComplexInfinity)+(y-ComplexInfinity)^2+2 (y-ComplexInfinity)^3+5 (y-ComplexInfinity)^4+14 (y-ComplexInfinity)^5+O[y-ComplexInfinity]^6 !--rwg On Wed, Apr 3, 2013 at 9:13 PM, Bill Gosper <billgosper@gmail.com> wrote:
From Neil Bickford (under protest): Plot[Tan[Sin[x]]-Sin[Tan[x]], {x, 0, π/2}] is ultraflat near 0 and goes berserk at π/2. Ultraflat: x^7/30+(29 x^9)/756+(1913 x^11)/75600+O[x]^12 Berserk: Local minima and maxima cluster at π/2, vibrating between roughly .5 and 2.5. What limits, if any, do they approach? Note that by analytic continuation, an arbitrarily short segment of the flat part near zero completely predicts the berserk part west of π/2. Are all the terms of the series approximated above positive?
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