10/31/2010, 01:56 AM

The analytic extension of Pentation wobbles.

pentation(-2)=-1

pentation(-1)=0

pentation(0)=1

Here, I'm graphing d/dx(pentation(x)), which wobbles in the neighborhood of z=-2..z=0, with two inflection points where the derivative has a local maxima, and a local minima. I calculated the Taylor series around -1 in the previous post, pentaylor(-1,1,0), which converges very nicely in the neighborhood of pent(-2)..pent(0), with approximately 21 digits of accuracy. Here, I am plotting the derivative of the paprx(x) Taylor series. So the pentation Taylor series apparently doesn't have the inherent beauty of the sexp Taylor series, for which all odd derivatives are positive for all real numbers>-2.

By the way, the wobble is quite a bit worst for smaller bases. Here is the graph for pentation base=1.7, from -6 to 6, where the wobble is visible without taking the first derivative. I think this has been pointed out before, but there is a lower limit base, for which pentation has a parabolic upper fixed point, in addition to the lower fixed point. That base is somewhere near B=1.6355. Again, the graph for pentation base 1.7 from -6 to 6.

- Sheldon

pentation(-2)=-1

pentation(-1)=0

pentation(0)=1

Here, I'm graphing d/dx(pentation(x)), which wobbles in the neighborhood of z=-2..z=0, with two inflection points where the derivative has a local maxima, and a local minima. I calculated the Taylor series around -1 in the previous post, pentaylor(-1,1,0), which converges very nicely in the neighborhood of pent(-2)..pent(0), with approximately 21 digits of accuracy. Here, I am plotting the derivative of the paprx(x) Taylor series. So the pentation Taylor series apparently doesn't have the inherent beauty of the sexp Taylor series, for which all odd derivatives are positive for all real numbers>-2.

By the way, the wobble is quite a bit worst for smaller bases. Here is the graph for pentation base=1.7, from -6 to 6, where the wobble is visible without taking the first derivative. I think this has been pointed out before, but there is a lower limit base, for which pentation has a parabolic upper fixed point, in addition to the lower fixed point. That base is somewhere near B=1.6355. Again, the graph for pentation base 1.7 from -6 to 6.

- Sheldon