# Smooth superation

Smooth superation is generalization of operation superation to non-integer number of superations.

## Definitions

Assume, some holomorphic function \( f \) is given.

Consider holomorphic function \(a \) of two variables such that

\( a(0,x)=f(x) \)

\( a(n,0)=f(0) \)

\( a(n,x+1)= a(n-1,a(n,x)) \)

Then:

\(a \) is ackermann to function \( f \).

\( a(n+1,) \) is superation of function \( a(n,.) \).

\( a(n,) \) is supation of function \( a(n+1,.) \).

\( a(n,.) \) is \( n \)th superation of function \( f \).

\( f \) is seed function with respect to aclermann \( a \).

The 0th superation of function \( f \) is \(f\).

The 0th supation of function \( f \) is \(f\).

## Uniqueness

The solution \( a \) is not unique.

The additional requirements are necessary to provide the uniqueness.

Perhaps, for some functions $f$, the "true" ackermann should be choosen to extend the range of holomorphism and validity of the equations above.

For a given range of validity and hlomorfism the "true" ackermann should have slowest growth (or fastest decay) in the direction of the imaginary axes of the arguments.

In general case, even these requirements may be not sufficient to provide the uniqueness.

## Example

addition to base \(b\), multiplication to base \(b\), exponentiation to base \(b\), tetration to base \(b\), pentation to base \(b\), hexation to base \(b\),

can be considered as native ackermanns to seed addition to base \(b\).

The special efforts are necessary to adjust the terminology, to establish the correct numeration, in order that tetration happen to be 4th operation, pentation to be 5th operation, etc.

## References

https://zh.wikipedia.org/wiki/超运算
超运算可通过递归进行定义，對於所有正整數*a*，正整數*b*和正整數*n*：
\(a [1] b = a + b, \ \text{for} \ n > 1, \ a [n] b = \underbrace{a [n-1] (a [n-1] (a [n-1] \cdots (a [n-1] (a [n-1] a}_b)) \cdots ))\) ..

## Keywords

Abelfunction, Ackermann, Ackermann function, Iterate, Supation, Superation, Superfunction, Superfunctions, Tetration