06/03/2010, 01:47 PM
(This post was last modified: 06/03/2010, 11:52 PM by sheldonison.)
(06/03/2010, 08:00 AM)bo198214 Wrote:Yeah, exactly that. I like that \( b^x-b^{-x} \) is on odd function with a fixed point of zero. The function grows exponentially both in the positive and negative directions. This seems to lead to a nice super function definition that shares some of the characteristics of exponentiation in the complex plane. Also, I think the superfunction of 2sinh is entire; it has no singularities. The math is much easier than the complex fixed point of base e, followed by a Riemann mapping...(06/02/2010, 04:47 PM)sheldonison Wrote: Could one extend this 2sinh superfunction definition to other real bases? Which ones? My quick initial guess is that it would be limited to bases greater than e^(0.5)?You mean base \( b \) in \( b^x-b^{-x} \) which then would approach \( b^x \) for \( x\to\infty \)?
My secret wish was that for the \( b^x-b^{-x} \) superfunction, the base change function would work without a wobble. I generated the superfunction for \( f(x)=2^x-2^{-x} \) with a base change back to the superfunction for 2sinh, \( f(x)=e^x-e^{-x} \). As expected, there was a small (magnitude of +/-0.02%) 1-cyclic wobble...
- Sheldon