08/13/2007, 10:00 PM

bo198214 Wrote:"nonsense" is a bit harsh, and anyway, I made explicit I was talking about reals greater than 0. ln(5) is very well-defined, but the series defined around x=1 diverges for any x value greater than 2. My point, which should have been very clear, is that a function can be continued by analytic extension, even if its power series fails to converge for inputs that should be non-singular values. So long as a function has a non-zero radius of convergence for the Taylor series defined at a certain point, and so long as analytic extension is possible and well-defined for all non-singular points, then we can find the general solution.jaydfox Wrote:The logarithm is "entire" for reals > 0 (not quite "entire", but you know what I mean), but it's standard power series diverges outside the range (0, 2]. Analytic extension is used outside that range. So this function might only converge for x<=1.Dont tell nonsense. Log has a singularity at 0, thatswhy it is not entire (thatswhy its radius of convergence is 1, if developed at 1).

Thats the whole thing about entire functions, they have no singularities and hence their radius of convergence is infinity.

Analytic continuation is necessary only if there are complex singularities.

And I see now where you specified that your series was for the half-iteration of x+x^2. I was reading too quickly, I guess...

~ Jay Daniel Fox