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Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*...
#5
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
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Messages In This Thread
RE: Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... - by tommy1729 - 03/13/2013, 10:52 PM

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