Uniqueness summary and idea

[jaydfox]

then use j integer iterations using i_j terms to get back to the desired input.

What do you mean by this?

I suppose it depends on how you plan to do the fraction iteration.

For example:

\( \text{Given: }f(z)=e^z-1\\
\\[10pt]

\\
f^{\circ \small 0.3}(z)\ =\ f^{\circ \small 20.3}\left(f^{\circ \small -20}(z)\right)
\)

In that case, the statement about using j integer iterations would not apply.

On the other hand, if you attempted to solve:
\(
f^{\circ \small 0.3}(z)\ =\ f^{\circ \small 20}\left(f^{\circ \small 0.3}\left(f^{\circ \small -20}(z)\right)\right)
\)

then you would use one integer iteration (with the integer being 20). And if you attempted to solve:

\(
f^{\circ \small 0.3}(z)\ =\ f^{\circ \small 4}\left(f^{\circ \small 16}\left(f^{\circ \small 0.3}\left(f^{\circ \small -20}(z)\right)\right)\right)
\)

then you would use two integer iterations (16 and 4, in that order). The same thing could be said about the negative iterations for the innermost term. It depends on whether you use the IDM to reduce several integer iterations to a single power series, or whether you actually perform the power series for a single iteration, an integer number of times.

Just depends on how you think it's best to implement. In the theoretic sense, I suppose the first version, using no supplemental integer iterations, is best. I'm not sure which I'll use long term, and they're equivalent, so I left the door open by including the additional integer iterations to step the number back out.