Non-integral number of compositions
Yesterday in Complex Analysis I proved that an increasing function from ![[0,1]](/cz/latexrender/pictures/ccfcd347d0bf65dc77afe01a3306a96b.png "[0,1]")
onto is continuous and h as a continuous inverse. Dr. Johnson gave me an interesting problem to work on afterwards. If 
is such a function, you know what 
is, and also what 
is, but what about 
? How would you make sense of that?
Possibly relevant posts:
- More on non-integral numbers of compositions (3/14/2005)
- Differentiable Flows (3/26/2005)
- Bolzano-Weierstrass theorem (2/6/2005)
What about pi as a series?
Comment by D. Trebbien — 3/12/2005 @ 6:28 pm
Defining a rational power of a function turns out not to be that hard— unless you want to guarantee existence and uniqueness, but I think the fact that it is increasing from one compact interval to another guarantees those. But you don’t have for instance
, unless 
are equal or integers. Without that, I think the series idea fails. But if you could show that if 
is a Cauchy sequence in 
then 
is a Cauchy sequence, then you would have a way of defining 
.
Comment by Alex — 3/14/2005 @ 10:44 am