01/27/2010, 08:30 PM
(This post was last modified: 01/28/2010, 11:40 AM by sheldonison.)
(01/27/2010, 06:28 PM)mike3 Wrote: Ah. What do you mean by the Fourier series "exponentially decays to a constant as i increases"? Do you mean the coefficients of the series (in which case how could it converge?)? Do you mean the behavior toward imaginary infinity (if so, does it decay to the two principal fixed points of exp)?I mean the 1-cyclic repeating function \( (\theta(z)-z) \) exponentially decays to a constant as \( \Im(z) \) increases. After the Riemann mapping, it's individual Fourier series terms all decay to zero as imaginary increases. The terms grow as we approach the real axis, and at the real axis, it has singularities at integer values.
The complex sexp developed from the fixed point (the inverse Abel function), \( \psi^{-1}(z) \) already goes to the fixed point as \( \Im(z) \) increases, and as \( Re(z) \) decreases. It would be nice to have a graph of \( \psi^{-1}(z) \), which becomes super-exponential (complex only) as \( \Re(z) \) increases, and as \( \Im(z) \) decreases, and goes to the fixed point as \( \Im(z) \) increases, and as \( \Re(z) \) decreases.
Finally, \( \operatorname{sexp}_e(z)=\psi^{-1}(\theta(z)) \), so as as \( \Im(z) \) increases, the sexp_e(z) will converge to \( \psi^{-1}(z+k) \), where k is a small constant.
- Shel
