03/13/2013, 10:52 PM
Another trivial proof.
Lets start with demystifying where the constant comes from.
exp(x)/M = x * f(x) * f^[2](x)*...
It was already shown that f was x - ln(x).
As usual we search for fixpoints.
1 is a fixpoint of x - ln(x) since 1 - ln(1) = 1.
We also know that f^[n](x) must approach 1 in the limit.
Hence if we plug in x=1 on the RHS we get 1 * 1 * 1 * ... = 1
Therefore we get
exp(x)/M = 1
=> exp(1)/M = 1
=> e/M = 1
=> M = e
Q.E.D.
This thread reminds me a bit of an idea I had not so long ago
http://math.eretrandre.org/tetrationforu...hp?tid=768
Im not sure where you want to go next with this. Like how to build tetration from it , or other intresting properties. Afterall we know it is analytic.
regards
tommy1729
Lets start with demystifying where the constant comes from.
exp(x)/M = x * f(x) * f^[2](x)*...
It was already shown that f was x - ln(x).
As usual we search for fixpoints.
1 is a fixpoint of x - ln(x) since 1 - ln(1) = 1.
We also know that f^[n](x) must approach 1 in the limit.
Hence if we plug in x=1 on the RHS we get 1 * 1 * 1 * ... = 1
Therefore we get
exp(x)/M = 1
=> exp(1)/M = 1
=> e/M = 1
=> M = e
Q.E.D.
This thread reminds me a bit of an idea I had not so long ago
http://math.eretrandre.org/tetrationforu...hp?tid=768
Im not sure where you want to go next with this. Like how to build tetration from it , or other intresting properties. Afterall we know it is analytic.
regards
tommy1729