On 4/29/10, Steve Witham <sw@tiac.net> wrote:

So, my superexponential function turned out like the Minkowski question mark function, i.e. fractally lumpy. I wanted a flattish area around zero (super negative exponentials) but no other lumps. ...

I haven't been paying much attention to the discussion, so this stuff may well be hopelessly off-message --- in which case, apologies in advance. But some of the features here remind me of my investigation in 2004 into blending pairs of functions for graphics purposes. The problem boils down to finding a blending function f(x) such that f(0) = 0, f(1) = 1; (d^n/dx^n) f(x) = 0 at x = 0,1, for all integer n > 0; and f(x) has continuous derivatives of all orders n throughout the unit interval --- and for all x, as things turned out. In addition, it's desirable (in some sense) to minimise the growth of the derivatives with n. What may be relevant is that the eventual construction is everywhere non-analytic, and its effective numerical computation involves a completely unconventional (though efficient) algorithm. As usual, I never got around even to submitting it for publication --- but if anybody is interested, I'm happy to forward an account! Fred Lunnon