Well alas, logarithmic semi operators have paid off and have given a beautiful smooth curve over domain . This solution for rational operators is given by :

Which extends the ackerman function to domain real (given the restrictions provided).

the upper superfunction of is used (i.e: the cheta function).

Logarithmic semi-operators contain infinite rings and infinite abelian groups. In so far as {t} and {t-1} always form a ring and {t-1} is always an abelian group (therefore any operator greater than {1} is not commutative and is not abelian). There is an identity function S(t), however its values occur below e and are therefore still unknown for operators less than {1} (except at negative integers where it is a variant of infinity (therefore difficult to play with) and at 0 where it is 0). Greater than {1} operators have identity 1.

The difficulty is, if we use the lower superfunction of to define values less than e we get a hump in the middle of our transformation from . Therefore we have difficulty in defining an inverse for rational exponentiation. however, we still have a piecewise formula:

therefore rational roots, the inverse of rational exponentiation is defined so long as and .

rational division and rational subtraction is possible if and .

Here are some graphs, I'm sorry about their poor quality but I'm rather new to pari-gp so I don't know how to draw graphs using it. I'm stuck using python right now. Nonetheless here are the graphs.

the window for these ones is xmin = -1, xmax = 2, ymin = 0, ymax = 100

If there's any transformation someone would like to see specifically, please just ask me. I wanted to do the transformation of as we slowly raise t, but the graph doesn't look too good since x > e.

Some numerical values:

(I know I'm not supposed to be able to calculate the second one, but that's the power of recursion)

I'm very excited by this, I wonder if anyone has any questions comments?

for more on rational operators in general, see the identities they follow on this thread http://math.eretrandre.org/tetrationforu...hp?tid=546

thanks, James

PS: thanks go to Sheldon for the taylor series approximations of cheta and its inverse which allowed for the calculations.

Which extends the ackerman function to domain real (given the restrictions provided).

the upper superfunction of is used (i.e: the cheta function).

Logarithmic semi-operators contain infinite rings and infinite abelian groups. In so far as {t} and {t-1} always form a ring and {t-1} is always an abelian group (therefore any operator greater than {1} is not commutative and is not abelian). There is an identity function S(t), however its values occur below e and are therefore still unknown for operators less than {1} (except at negative integers where it is a variant of infinity (therefore difficult to play with) and at 0 where it is 0). Greater than {1} operators have identity 1.

The difficulty is, if we use the lower superfunction of to define values less than e we get a hump in the middle of our transformation from . Therefore we have difficulty in defining an inverse for rational exponentiation. however, we still have a piecewise formula:

therefore rational roots, the inverse of rational exponentiation is defined so long as and .

rational division and rational subtraction is possible if and .

Here are some graphs, I'm sorry about their poor quality but I'm rather new to pari-gp so I don't know how to draw graphs using it. I'm stuck using python right now. Nonetheless here are the graphs.

the window for these ones is xmin = -1, xmax = 2, ymin = 0, ymax = 100

If there's any transformation someone would like to see specifically, please just ask me. I wanted to do the transformation of as we slowly raise t, but the graph doesn't look too good since x > e.

Some numerical values:

(I know I'm not supposed to be able to calculate the second one, but that's the power of recursion)

I'm very excited by this, I wonder if anyone has any questions comments?

for more on rational operators in general, see the identities they follow on this thread http://math.eretrandre.org/tetrationforu...hp?tid=546

thanks, James

PS: thanks go to Sheldon for the taylor series approximations of cheta and its inverse which allowed for the calculations.