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:

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 as the complex linear span of .

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 when we let 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; :

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 as the complex linear span of .

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 when we let 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; :

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.