I'm curious: What programs and methods are you using to compute this? Esp. how do you compute the solution in reasonable time?
Also, can you try to expand, say, the \( \exp_{\sqrt{2}}^{1/2}(z) \) generated by this method about the fixed point 2? If it's not the same as the regular iteration then it should fail to be holomorphic there, right? (as that's the thing that characterizes the regular iteration -- that it's differentiable at the fixed point, no?) If not, then what happens if you try comparing coefficients to those obtained from the regular iteration and looking for the difference?
Also, can you try to expand, say, the \( \exp_{\sqrt{2}}^{1/2}(z) \) generated by this method about the fixed point 2? If it's not the same as the regular iteration then it should fail to be holomorphic there, right? (as that's the thing that characterizes the regular iteration -- that it's differentiable at the fixed point, no?) If not, then what happens if you try comparing coefficients to those obtained from the regular iteration and looking for the difference?