I'm going to prove to you an entire expression for tetration in one post for all b greater than eta. Is this valid?
We start by defining:
 = \frac{1}{\Gamma(-s)} \int_0^\infty f(-u) u^{-s-1}\, \partial u)
We call this the exponential derivative because:
 = \frac{d^s f(t)}{dt^s} |_{t=0})
Where this complex derivative is evaluated according to the Riemann-Liouville differ-integral where exponentiation is the fix point. By this definition we find:
 = \ln(b)^s)
Now we define a new multiplication across FUNCTIONS; i.e. you cannot plug in numbers into this multiplication. We refer to
as the complex linear span of 
.
We now define:
 \times h = f \times (g \times h))
 = (f \times g) + (f \times h))
With this we define
:
 (^\omega e)^s) \times(\Gamma(\nu + 1) (^\nu e)^s) = \Gamma(\omega + \nu + 1) (^{\omega+ \nu } e)^s)
This result gives us a beautiful isomorphism
:
^s, (^3 e)^s, ..., (^N e)^s,...\})
 = \alpha \psi f + \beta \psi g)
 = (\psi f) \cdot (\psi g))
 = \frac{d \psi(f)}{ds})
^s) = \frac{s^N}{N!})
Now thanks to Newton we have:
}{\Gamma(\omega - N + 1)N!} (s - 1)^N)
}{\Gamma(\omega - N + 1)N!} \sum_{k=0}^N \frac{N!}{(N-k)! k!}(-1)^{N-k}s^k)
Now we apply the isomorphism to get the result
:
 \phi_\omega(s) = \sum_{N=0}^{\infty} \frac{\Gamma(\omega + 1)}{\Gamma(\omega - N + 1)N!} \sum_{k=0}^N \frac{N!}{(N-k)! k!}(-1)^{N-k}k!(^k e)^s)
We can reduce this equation to:
 = \sum_{N=0}^{\infty} \frac{\sum_{k=0}^{N} \frac{(-1)^{N-k}}{(N-k)!}(^k e)^s}{\Gamma(\omega - N +1)})
We know these functions satisfy the equations:
}{\Gamma(\omega + 1)\Gamma(\nu + 1)} \phi_{\omega + \nu}(s))
There is only one function which satisfies this:
 = (^\omega e)^s)
We can do the same procedure to arrive at the more general expression:
^s = \sum_{N=0}^{\infty} \frac{\sum_{k=0}^{N} \frac{(-1)^{N-k}}{(N-k)!}(^k b)^s}{\Gamma(\omega - N +1)})
This is probably the fastest way to derive tetration. It's analytic, entire for
when we let  < 0 )
which is easy to show by the ratio test since tetration grows ridiculously faster than the Gamma function.
Anyone see any mistakes I'm making? Or did I just solve tetration in fifteen minutes of work. LOL
Being quick I'll write the glorious formula;
:
^{N-k}}{(N-k)!(^k b)}}{ \Gamma(\omega - N +1)})
I'm currently finishing a paper on
and this result just happened to fall in my lap. I'm wondering if it's valid so that I can keep it in the paper.
We start by defining:
We call this the exponential derivative because:
Where this complex derivative is evaluated according to the Riemann-Liouville differ-integral where exponentiation is the fix point. By this definition we find:
Now we define a new multiplication across FUNCTIONS; i.e. you cannot plug in numbers into this multiplication. We refer to
We now define:
With this we define
This result gives us a beautiful isomorphism
Now thanks to Newton we have:
Now we apply the isomorphism to get the result
We can reduce this equation to:
We know these functions satisfy the equations:
There is only one function which satisfies this:
We can do the same procedure to arrive at the more general expression:
This is probably the fastest way to derive tetration. It's analytic, entire for
Anyone see any mistakes I'm making? Or did I just solve tetration in fifteen minutes of work. LOL
Being quick I'll write the glorious formula;
I'm currently finishing a paper on
