Iterability of exp(x)-1

[andydude]

I've been looking at some graphs of the root-test: \( \left(\lim_{k \rightarrow \infty} |c_k|^{1/k}\right) \) for the coefficients of fractional-iterates of the natural decremented exponential, and I'm starting to believe Baker over Walker, i.e., that it does in fact diverge for non-integers.

Here are the graphs, (using \( DE(x) = e^x - 1 \)):

• DE^[-3/2](x)
  
• DE^[-1](x)
  
• DE^[-1/2](x)
  
• x (omitted)
  
• DE^[1/2](x)
  
• DE(x)
  
• DE^[3/2](x)
  

In order for these functions to converge, the root-test must be bounded, and as you can see the non-integer root-tests seem to be unbounded, supporting Baker.

Andrew Robbins