(04/25/2010, 01:10 PM)bo198214 Wrote: ..Ah, I understand. It can be singular, pulling out any property you like, from the exponential tail. \( e^{b^x} \) may approach its limiting value, and
\( \eta=P \).
\( P(e^{b^x}) \) may do that you like, at least until the closest cutline. While within the range of the holomorphism, you may deform and squeese the superfunction as you want with the transform indicated; it may be good for the application, or for the representation, or for the evaluation; no other deep meaning, that would allow or prohibit some particular properties. Roughly, you can convert ANY real holomorphic monotonic function to ANY other real holomorphic monotonic function, at least until to reach a singularity or a cutline.