Finding radious of convergence of a series

From: Carlos Felippa (carlos_at_colorado.edu)
Date: 09/25/04 

Date: 24 Sep 2004 20:46:01 -0700

Consider the real function, with x>=0 and k a real parameter in [0,1],
and its Taylor series (*)

   f(x) = -2 k^2 / (2+k^2-2cosh(k sqrt(x))) = A_0 + A_1 x + A_2 x^2 + ... (*)

The coefficients A_j in (*) are polynomials in k^2.
Symbolic computations suggest that as j-> oo

  A_(j-1) / A_j -> k^(-2) (arccosh(1+k^2/2))^2 = 4 k^(-2) (arcsinh(k/2))^2

and that the convergence is uniform (in fact, extremely fast) for k in [0,1].
To prove it analytically I assumed that

(1) A_(j-1) / A_j -> d, the radius of convergence of (*) for fixed k
(2) d is obtained by finding the distance from x=0 to the closest
      pole of f(x) = closest zero of 2+k^2-cosh(k sqrt(x)) allowing
      x to be complex

The result checks out here, but is the procedure (1)-(2) always correct?

Relevant Pages

  • Re: tetration and logaritms
    ... and from looking at the first terms of this taylor series we see that we can get series reversion which of course gives exp- 1. ... finally 5/8 and 13/30 in the upper-left corner. ... to find the polynomials in h. ... We simply take the coefficients ...
    (sci.math)
  • Nonrenormalization vs Renormalization 12: Taylor and Laurent Series
    ... The Taylor series of a real function fwith real coefficients that ... Here the coefficients of the x - c powers are f^/n! ... The Laurent series is the complex analog which includes negative powers ... of and is especially useful in the Residue Theorem according to ...
    (sci.physics)
  • Re: fft on square wave
    ... % plot the coefficients (should be a real function) ... Tomas, since you're dealing with an even function, the FFT will be real if you have an odd number N of samples and if t=0 is one of them. ...
    (comp.soft-sys.matlab)