When you said `ultraflat', I thought you were referring to the property of all derivatives being zero at that point. Obviously, complex-differentiable functions (such as yours) cannot have this property (except for constant functions); however, there are infinitely differentiable examples over the reals such as f(x) = exp(-1/x^2). A function with this property is considered here: http://cp4space.wordpress.com/2013/02/28/radical-tauism/ Sincerely, Adam P. Goucher

----- Original Message ----- From: Bill Gosper Sent: 04/04/13 05:13 AM To: math-fun@mailman.xmission.com Subject: [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? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun