03/10/2013, 06:35 AM
(This post was last modified: 03/10/2013, 06:36 AM by sheldonison.)
(03/09/2013, 08:25 AM)sheldonison Wrote: I'm not looking for an exact 2-periodic solution; but the solution I'm looking for has limiting behavior that looks more and more like a periodic function as imag(z) goes to infinity.
- Sheldon
Here is a better three term approximation for the superfunction for base=(1/e)^e. I don't have a closed form. From the upper fixed point, the superfunction of z is approximately as follows; this approximation has smaller and smaller error has imag(z) gets larger.
superfunction(z) \( \approx \frac{1}{e} + \exp(\pi i z) + 0.679570457114761308840071867838\exp(2\pi i z) -0.615754674910887518935868955048 z\times \exp(3\pi i z) \)
Notice the third coefficient isn't your standard three periodic function, since we multiply the periodic function by z. These coefficients are half the coefficients of b^(1/e + z). The solution isn't exactly 2-periodic, but that's because the base function is rationally indifferent with a pseudo period=2. I don't have a closed form, and I think the coefficients get even more complicated for the fourth term and above. The error term for this function is roughly \( \exp(4\pi z) \). As you iterate b^z, or its inverse, the function changes with each double iteration, and becomes less well behaved the farther you get from the imaginary axis.
- Sheldon
