Hi.
I was wondering about the tetration of bases in the range \( 0 < b < e^{-e} \) via the continuum sum formula.
Here's how it goes. I would expect, that for at least some towers/heights/superexponents \( x \), \( ^{x + n} b \) (presumably a complex-valued function of a real number) for increasing integer \( n \) should approach the two oscillation points \( M_{high} \) and \( M_{low} \) where \( M_{high} > M_{low} \) due to the recurrent equation \( \mathrm{tet}_b(z+1) = b^{\mathrm{tet}_b(z)} \). These points solve \( b^{M_{high}} = M_{low} \) and \( b^{M_{low}} = M_{high} \).
So I'd propose that for all \( x \) within a sufficiently small interval \( (n - \epsilon, n + \epsilon) \) of an integer \( n \), \( ^x b\ \rightarrow\ ^n b \) as \( n \rightarrow \infty \) and also that \( \frac{d}{dx} ^x b \rightarrow 0 \).
If this is so, then we can take the difference \( ^x b - Sq(x) \), where \( Sq(x) \) is the "square wave function"
\(
Sq(x) = \begin{cases}
M_{high},\ \mathrm{if}\ \mathrm{nint}(x)\ \mathrm{is\ even} \\
M_{low},\ \mathrm{if}\ \mathrm{nint}(x)\ \mathrm{is\ odd}
\end{cases}
\)
where "nint" is the nearest-integer or round function. Thus for \( x \) a sufficiently small distance from an integer, \( ^x b \rightarrow Sq(x) \) as \( x \rightarrow \infty \), and so \( ^x b - Sq(x) \rightarrow 0 \). This means we can continuum-sum \( f(x)\ =\ ^x b - Sq(x) \) via Mueller's formula
\( \sum_{n=0}^{x-1} f(n) = f(0) + \sum_{n=1}^{\infty} f(n) - f(n + x - 1) \).
Note that this only requires the values at integers and at x stepped by an integer, so all evaluation points involved are within the necessary small intervals of integers and the approximation is assumed to hold. Then if we add the continuum sum of \( Sq(x) \) to the result, we have continuum-summed the tetration and we can apply the Ansus continuum sum formula. There is in fact such a continuum sum for \( Sq(x) \): we derive it from the Fourier series of the square wave and sum that with Faulhaber's formula on the trig. It yields a discontinuous function that jumps between two linear functions (like how the square wave jumps between two constant functions), though I don't have the formula on hand right now.
The only problem here is that we can only work in a small interval (in order for the approximation to hold): this means we cannot evaluate at, say, both -1 and 0 as we would need to find the correct normalization constants and the definite integral for the formula. If however the function is analytic, we should be able to reach those places via analytic continuation, yet analytically continuing an arbitrary Taylor series would probably require something like, you guessed it, the Mittag-Leffler formula. (Plus we need the continuation anyways to fill a whole length-1 interval). And why are the coefficients so gosh darned difficult to find for that puppy?!
What do you think of my theory? Also, can you give a graph of the regular superexponential of such a base developed the repelling real fixed point? Even though it's not the tetrational we want, it might nonetheless help to get a general idea of what the behavior may be like. Helms posted a graph for \( b = 0.04 \) once, but it was a parametric plot, not a plot along the x-axis.
I was wondering about the tetration of bases in the range \( 0 < b < e^{-e} \) via the continuum sum formula.
Here's how it goes. I would expect, that for at least some towers/heights/superexponents \( x \), \( ^{x + n} b \) (presumably a complex-valued function of a real number) for increasing integer \( n \) should approach the two oscillation points \( M_{high} \) and \( M_{low} \) where \( M_{high} > M_{low} \) due to the recurrent equation \( \mathrm{tet}_b(z+1) = b^{\mathrm{tet}_b(z)} \). These points solve \( b^{M_{high}} = M_{low} \) and \( b^{M_{low}} = M_{high} \).
So I'd propose that for all \( x \) within a sufficiently small interval \( (n - \epsilon, n + \epsilon) \) of an integer \( n \), \( ^x b\ \rightarrow\ ^n b \) as \( n \rightarrow \infty \) and also that \( \frac{d}{dx} ^x b \rightarrow 0 \).
If this is so, then we can take the difference \( ^x b - Sq(x) \), where \( Sq(x) \) is the "square wave function"
\(
Sq(x) = \begin{cases}
M_{high},\ \mathrm{if}\ \mathrm{nint}(x)\ \mathrm{is\ even} \\
M_{low},\ \mathrm{if}\ \mathrm{nint}(x)\ \mathrm{is\ odd}
\end{cases}
\)
where "nint" is the nearest-integer or round function. Thus for \( x \) a sufficiently small distance from an integer, \( ^x b \rightarrow Sq(x) \) as \( x \rightarrow \infty \), and so \( ^x b - Sq(x) \rightarrow 0 \). This means we can continuum-sum \( f(x)\ =\ ^x b - Sq(x) \) via Mueller's formula
\( \sum_{n=0}^{x-1} f(n) = f(0) + \sum_{n=1}^{\infty} f(n) - f(n + x - 1) \).
Note that this only requires the values at integers and at x stepped by an integer, so all evaluation points involved are within the necessary small intervals of integers and the approximation is assumed to hold. Then if we add the continuum sum of \( Sq(x) \) to the result, we have continuum-summed the tetration and we can apply the Ansus continuum sum formula. There is in fact such a continuum sum for \( Sq(x) \): we derive it from the Fourier series of the square wave and sum that with Faulhaber's formula on the trig. It yields a discontinuous function that jumps between two linear functions (like how the square wave jumps between two constant functions), though I don't have the formula on hand right now.
The only problem here is that we can only work in a small interval (in order for the approximation to hold): this means we cannot evaluate at, say, both -1 and 0 as we would need to find the correct normalization constants and the definite integral for the formula. If however the function is analytic, we should be able to reach those places via analytic continuation, yet analytically continuing an arbitrary Taylor series would probably require something like, you guessed it, the Mittag-Leffler formula. (Plus we need the continuation anyways to fill a whole length-1 interval). And why are the coefficients so gosh darned difficult to find for that puppy?!
What do you think of my theory? Also, can you give a graph of the regular superexponential of such a base developed the repelling real fixed point? Even though it's not the tetrational we want, it might nonetheless help to get a general idea of what the behavior may be like. Helms posted a graph for \( b = 0.04 \) once, but it was a parametric plot, not a plot along the x-axis.
