08/12/2010, 03:35 PM
(This post was last modified: 08/12/2010, 03:41 PM by sheldonison.)

(08/11/2010, 06:11 PM)sheldonison Wrote: update, it occurs to me that this is a general principle. Lets say g(z) is a function with real values at the real axis, and f(z) is an approximation, over a unit length, such that f(0)=g(0) and f(1)=g(1). Then there is some 1-cyclic theta fourier series function defining f(z) over the entire real axis.

Now, theta(z) is probably not even an analytic function. But we could wrap the real valued theta(z) around the unit circle. It would have a laurent series. We could throw out all of the terms in the laurent series with a_n*z^-n. Now theta_2(z) is still 1-cyclic, but it is complex valued at the real axis.

You might take this new f_2 function, and use it to generate another function, f_3. Generate the Taylor series over a half circle of f_2(z), using the complex conjugate for the other half of the circle where imag(z)<0. At this point, you should see the connection between the sequence of functions, f(z), f_2(z), f_3(z), and the iterative algorithm for the sexp(z) function I have described, where the sequence of f, f_1, f_2 ... converge to the desired g(z) function.

Another quick post, with some ideas I had when I was coming up with this algorithm to generate sexp(z). There may be a general case Riemann mapping algorithm where there is a real valued (at the real axis) auxiliary function, f(z+theta(z)). In our case, the sequence of f(z) functions is desired to converge to sexp(z) which is real valued at the real axis for z>-2.

One important note. It only works if the desired f(z) is analytic, and real valued at the real axis. f(z) is the initial approximation, and we're trying to come up with theta(z), which is exactly related to a Riemann mapping, but theta(z) may have a singularity, making convergence even more difficult.

- Sheldon