12/12/2025, 07:49 PM
we start by constructing the base e product tetration of x which is denoted as \(\text{prodtet}_{e}(x)\)
the base e product tetration has these recursive properties:
\(\text{prodtet}_{e}(x)=\text{prodtet}_{e}(x-1)*e^{\text{prodtet}_{e}(x-1)}\)
\(\text{prodtet}_{e}(x)=\text{W}(\text{prodtet}_{e}(x+1))\)
also, product tetration has this following graph: https://www.desmos.com/calculator/zkjiz8jwh0
if you're wondering how i extended product tetration to real numbers, i used lagrange interpolation on the very far negative side of product tetration and applied recursive formulas.
some example values of product tetration:
\(\text{prodtet}_{e}(-1)=\text{W}(1)=0.56714... \)
\(\text{prodtet}_{e}(0)=1\)
\(\text{prodtet}_{e}(1)=e\)
\(\text{prodtet}_{e}(2)=e^{e+1}\)
and one accepted non-integer value for product tetration:
\(\text{prodtet}_{e}(1/2)=1.5134...\)
and we can express product tetration to different bases by introducing the inverse function of product tetration which will be called the product super logarithm.
the product super logarithm will be denoted as \(\text{prodslog}_{e}(x)\)
the product super logarithm also has some noteworthy recursive properties of its own:
\(\text{prodslog}_{e}(x)=\text{prodslog}_{e}(xe^{x})-1\)
\(\text{prodslog}_{e}(x)=\text{prodslog}_{e}(\text{W}(x))+1\)
the product super logarithm also has a noteworthy graph (sorry if it's slow): https://www.desmos.com/calculator/sxjnv2212j
now for the base change formula of both the product tetration and the product super logarithm:
\( \text{prodtet}_{a}(x)=\frac{\text{prodtet}_{e}(\text{prodslog}_{e}(\text{ln}(a))+x)}{\text{ln}(a)}\)
\(\text{prodslog}_{a}(x)=\text{prodslog}_{e}(x*\text{ln}(a))-\text{prodslog}_{e}(\text{ln}(a))\)
here, i give an example of the base change formulas with base pi: https://www.desmos.com/calculator/kwsqnpe4ef
the proof of the base change formulas is left as an exercise to the reader.
now, back to the main topic of this post.
we can literally extend normal tetration to real numbers by using product tetration because product tetration grows faster than normal tetration.
here's how the process goes:
we extend both the product tetration and the product super logarithm to real numbers using any interpolation technique we consider good. (in this case, i considered the lagrange interpolation good.)
we take the product super logarithm of normal tetration and interpolate the result we get by doing so. (the interpolation still has to be good.)
we finally take the product tetration of the above interpolation to get normal tetration.
here's an example of this process being applied to base 2 tetration: https://www.desmos.com/calculator/r26svg3olm
the numerical accuracy on desmos is bad for product tetration and the product super logarithm.
and desmos can only calculate numbers up to \(2^{1024}\).
i hope there is a calculator that functions like desmos but with greater numerical accuracy and the ability to calculate greater numbers than \(2^{1024}\).
the base e product tetration has these recursive properties:
\(\text{prodtet}_{e}(x)=\text{prodtet}_{e}(x-1)*e^{\text{prodtet}_{e}(x-1)}\)
\(\text{prodtet}_{e}(x)=\text{W}(\text{prodtet}_{e}(x+1))\)
also, product tetration has this following graph: https://www.desmos.com/calculator/zkjiz8jwh0
if you're wondering how i extended product tetration to real numbers, i used lagrange interpolation on the very far negative side of product tetration and applied recursive formulas.
some example values of product tetration:
\(\text{prodtet}_{e}(-1)=\text{W}(1)=0.56714... \)
\(\text{prodtet}_{e}(0)=1\)
\(\text{prodtet}_{e}(1)=e\)
\(\text{prodtet}_{e}(2)=e^{e+1}\)
and one accepted non-integer value for product tetration:
\(\text{prodtet}_{e}(1/2)=1.5134...\)
and we can express product tetration to different bases by introducing the inverse function of product tetration which will be called the product super logarithm.
the product super logarithm will be denoted as \(\text{prodslog}_{e}(x)\)
the product super logarithm also has some noteworthy recursive properties of its own:
\(\text{prodslog}_{e}(x)=\text{prodslog}_{e}(xe^{x})-1\)
\(\text{prodslog}_{e}(x)=\text{prodslog}_{e}(\text{W}(x))+1\)
the product super logarithm also has a noteworthy graph (sorry if it's slow): https://www.desmos.com/calculator/sxjnv2212j
now for the base change formula of both the product tetration and the product super logarithm:
\( \text{prodtet}_{a}(x)=\frac{\text{prodtet}_{e}(\text{prodslog}_{e}(\text{ln}(a))+x)}{\text{ln}(a)}\)
\(\text{prodslog}_{a}(x)=\text{prodslog}_{e}(x*\text{ln}(a))-\text{prodslog}_{e}(\text{ln}(a))\)
here, i give an example of the base change formulas with base pi: https://www.desmos.com/calculator/kwsqnpe4ef
the proof of the base change formulas is left as an exercise to the reader.
now, back to the main topic of this post.
we can literally extend normal tetration to real numbers by using product tetration because product tetration grows faster than normal tetration.
here's how the process goes:
we extend both the product tetration and the product super logarithm to real numbers using any interpolation technique we consider good. (in this case, i considered the lagrange interpolation good.)
we take the product super logarithm of normal tetration and interpolate the result we get by doing so. (the interpolation still has to be good.)
we finally take the product tetration of the above interpolation to get normal tetration.
here's an example of this process being applied to base 2 tetration: https://www.desmos.com/calculator/r26svg3olm
the numerical accuracy on desmos is bad for product tetration and the product super logarithm.
and desmos can only calculate numbers up to \(2^{1024}\).
i hope there is a calculator that functions like desmos but with greater numerical accuracy and the ability to calculate greater numbers than \(2^{1024}\).
