Funny pictures of sexps

[bo198214]

I'm not so certain. Again looking at the logarithm, there are branches, so if we recenter, how do we know which branch we are within?

Right this is the question if you have a function with singularities (and only in this case the radius is not infinity and hence different continuations as you described may lead to different values).

However my recentering took place at the entire function \( e^x \).

Even though it might not be explicitly obvious, I suspect the same problem is hidden in the steps you have taken.

It may be that something similar is here in play. Indeed the development at 1.5 is though solvable but inaccurate:

   

Edit: On the other hand it seems that only the additive constant needs to be adapted. The recentering process merely guaranties that the result is again an Abel function but does not guaranty the value -1 at 0.