08/25/2009, 09:55 PM
(08/25/2009, 09:43 PM)jaydfox Wrote: Henryk, I've found some rather bizarre behvior with your method of shifting the Abel function's center. For example, it seems fairly well behaved for real shifts, but even a small imaginary shift seems to produce garbage results (i.e., as I increase the matrix size, I don't get convergence, but rather rapid divergence).Well its not shifting of the Abel function (see my post before).
But this is indeed strange, the only thing I can say that I would expect the complex shift yielding complex values on the real axis, thatswhy I didnt try it.
Quote:I can use complex math with f(x)=e*(x+x0)-x0 to get the power series for log(x+x0), except the leading constant term of course. This holds true for each value of x0 I have tried so far, including complex values or pure imaginary values.And now prove it!
I could already show that the result does not depend on x0 (hopefully its on the forum), but not that it is log.
Quote:However, I was unable to successfully recenter the islog_e to x=0.25*i, which is a very small shift, relatively speaking. Why should we be unable to shift away from the real axis, if we are not limited by a radius of convergence?As I said, its not a shift of the Abel function but the result may be a different Abel function. I dont know about the mechanisms of its convergence. Proof of convergence of the intuitive method is anyway still open.