Hi.

I was playing around with this new tetration method based on the continuum sum.

The idea is to use the exp-series mentioned here:

http://math.eretrandre.org/tetrationforu...hp?tid=396

We have

\( f(z) = \sum_{n=0}^{\infty} a_n b^{nz} \)

or, even better

\( f(z) = \sum_{n=-\infty}^{\infty} a_n b^{nz} \).

Then,

\( \sum_{n=0}^{z-1} f(n) = \sum_{n=-\infty}^{\infty} a_n \frac{b^{nz} - 1}{b^n - 1} \)

with the term at \( n = 0 \) interpreted as \( a_0 z \), so,

\( \sum_{n=0}^{z-1} f(n) = \left(\sum_{n=-\infty}^{-1} \frac{a_n}{b^n - 1}\right) + \left(\sum_{n=1}^{\infty} \frac{a_n}{b^n - 1}\right) + a_0 z + \left(\sum_{n=-\infty}^{-1} \frac{a_n}{b^n - 1} b^{nz}\right) + \left(\sum_{n=1}^{\infty} \frac{a_n}{b^n - 1} b^{nz}\right) \).

However, when viewed in the complex plane, we see that exp-series are just Fourier series

\( f(z) = \sum_{n=-\infty}^{\infty} a_n e^{i \frac{2\pi}{P} n z} \).

which represent a periodic function with period \( P \).

Thus it would seem that only periodic functions can be continuum-summed this way. Tetration is not periodic, so how could this help?

Well, we could consider the possibility of continuum-summing an aperiodic function by taking a limit of a sequence of periodic functions that converge to it. The hypothesis I have is that if \( f_0, f_1, f_2, ... \) is a sequence of periodic analytic functions converging to a given \( f \), then their continuum sums, if they converge to anything, converge to the same thing, regardless of the sequence of functions.

Would you have any ideas to prove or refute this hypothesis?

An example. Let \( f(z) = z \), the identity function. We can't continuum-sum it with the exp-series directly. But now let \( f_u(z) = u \sinh\left(\frac{z}{u}\right) \), so that \( \lim_{u \rightarrow \infty} f_u(z) = f(z) \), a sequence of periodic (with imaginary period \( 2 \pi i u \)) entire functions converging to \( f(z) \). The continuum sum is, by using the exponential expansion of sinh giving \( f_u(z) = -\frac{u}{2} e^{-\frac{z}{u}} + \frac{u}{2} e^{\frac{z}{u}} \),

\( \sum_{n=0}^{z-1} f_u(n) = \frac{u}{2} \left(\frac{e^{\frac{z}{u}} - 1}{e^{\frac{1}{u}} - 1} - \frac{e^{-\frac{z}{u}} - 1}{e^{-\frac{1}{u}} - 1}\right) \).

Though I didn't bother to try to work it out by hand, instead using a computer math package, the limit is \( \frac{x(x-1)}{2} \) as \( u \rightarrow \infty \), agreeing with the result from Faulhaber's formula.

Another example is the function \( f(z) = \frac{1}{z} \), or better, \( f(z) = \frac{1}{z+1} \). We can construct periodic approximations like \( f_u(z) = \frac{1}{u \sinh\left(\frac{z}{u}\right) + 1} \), and take the limit at infinity. The terms in the Fourier series are even worse. If we use a numerical approximation of the series with period , I get the continuum sum from 0 to -1/2 as ~0.6137 (rounded, act. more like something over 0.61369) suggesting the process is recovering the digamma function, as can be seen by setting 1/2 in the canonical formula \( -\gamma + \Psi(x + 1) = \sum_{n=0}^{x-1} \frac{1}{x+1} \) yielding 0.61370563888011...

Trying it with \( log(1 + z) \), so as to attempt to evaluate \( z! = \exp\left(\sum_{n=0}^{z-1} \log(1+z)\right) \) yields values that agree with the gamma function, providing more evidence that gamma is the natural extension of the factorial function to the complex plane.

Thus it seems this continuum sum is recovering all the expected sums and extensions. So the question comes up: what happens if we use it on Tetration, to sum up Ansus' continuum sum formula

\( \log_b\left(\frac{\mathrm{tet}'_b(z)}{\mathrm{tet}'_b(0) \log(b)^z}\right) = \sum_{n=0}^{z-1} \mathrm{tet}_b(n) \)

?

I don't yet have a really fast and efficient numerical program ready to go, but the idea behind the algorithm I'm using and the current code I can post in the Computation forum if you'd like.

Trying it out, though, it seems to converge well. I can't get a lot of precision due to the amount of nodes required, but enough to make graphs and simple observations of the behavior is available.

For tetration \( ^{z} \sqrt{2} \), we get \( ^{1/2} \sqrt{2} \approx 1.243623 \), agreeing with the result from the regular iteration up to the expected numerical error of the algorithm (determined by the residual, or the difference of \( \log_b \) of the approximation at 0.5 and the value of the approximation at -0.5, so we can knock off the 3 and get 1.24362 which is near 1.2436216... (residual mag was approx 10^-6)).

If we try it now at the natural base, base \( e \), we get \( ^{1/2} e \approx 1.64635\ \mathrm{to}\ 1.64636 \), which agrees with the results from the Cauchy integral, the Abel iteration, Kneser iteration, etc. Indeed, if we look at the graph at the imaginary axis, we see this:

which looks equivalent to the result from the Cauchy integral. Toward \( \pm i \infty \), the function looks to settle to values of given by approximately \( 0.318 + 1.337i \) and its conjugate, which of course agrees with the fixed points of logarithm given by \( -W_{-1}(-1) = 0.31813150520476... - 1.33723570143069... i \) and \( -W_0(-1) = 0.31813150520476... + 1.33723570143069... i \).

Graph of \( ^{ix} e \):

Graph of \( ^x e \):

Complex Bases

The most exciting thing about this method is that it even can be used on complex bases outside the Shell-Thron convergent region. For example, I tried it on the base \( 2.28 + 1.35i \), located in the period-4 lobe of the tetration fractal coming off the convergent region. I don't know if anyone has even seen the graph of the tetration of a complex base like this before. It is really wacky, and no, this is not a numerical error. Here is \( ^{x} (2.33 + 1.28i) \):

Apparently the period-4 behavior is only in the integer tetrations. The continuous tetrational function is instead unbounded on the positive real axis \( x > 0 \). Also, it does not appear to be injective any more.

The graph at the imaginary axis looks like this (i.e. \( ^{ix} (2.33 + 1.28i) \)):

showing the decay to the fixed points of the logarithm for this base.

One interesting mathematical question raised by this is that it seems to go fine right across \( b = e^{1/e} \), despite looking like the regular iteration below this base. Could this mean that the regular iteration really does have an analytic continuation outside the Shell-Thron region and the STB is not actually a natural boundary? OF course, this initial experimental investigation is no substitute for rigorous mathematical proof.

Another interesting question is why does it appear so many seemingly disparate methods keep turning up this same function, despite the existence of infinitely many solutions to the basic tetration functional equation? Could it be that there is some "natural" uniqueness condition that all these methods happen to be compatible with?

I was playing around with this new tetration method based on the continuum sum.

The idea is to use the exp-series mentioned here:

http://math.eretrandre.org/tetrationforu...hp?tid=396

We have

\( f(z) = \sum_{n=0}^{\infty} a_n b^{nz} \)

or, even better

\( f(z) = \sum_{n=-\infty}^{\infty} a_n b^{nz} \).

Then,

\( \sum_{n=0}^{z-1} f(n) = \sum_{n=-\infty}^{\infty} a_n \frac{b^{nz} - 1}{b^n - 1} \)

with the term at \( n = 0 \) interpreted as \( a_0 z \), so,

\( \sum_{n=0}^{z-1} f(n) = \left(\sum_{n=-\infty}^{-1} \frac{a_n}{b^n - 1}\right) + \left(\sum_{n=1}^{\infty} \frac{a_n}{b^n - 1}\right) + a_0 z + \left(\sum_{n=-\infty}^{-1} \frac{a_n}{b^n - 1} b^{nz}\right) + \left(\sum_{n=1}^{\infty} \frac{a_n}{b^n - 1} b^{nz}\right) \).

However, when viewed in the complex plane, we see that exp-series are just Fourier series

\( f(z) = \sum_{n=-\infty}^{\infty} a_n e^{i \frac{2\pi}{P} n z} \).

which represent a periodic function with period \( P \).

Thus it would seem that only periodic functions can be continuum-summed this way. Tetration is not periodic, so how could this help?

Well, we could consider the possibility of continuum-summing an aperiodic function by taking a limit of a sequence of periodic functions that converge to it. The hypothesis I have is that if \( f_0, f_1, f_2, ... \) is a sequence of periodic analytic functions converging to a given \( f \), then their continuum sums, if they converge to anything, converge to the same thing, regardless of the sequence of functions.

Would you have any ideas to prove or refute this hypothesis?

An example. Let \( f(z) = z \), the identity function. We can't continuum-sum it with the exp-series directly. But now let \( f_u(z) = u \sinh\left(\frac{z}{u}\right) \), so that \( \lim_{u \rightarrow \infty} f_u(z) = f(z) \), a sequence of periodic (with imaginary period \( 2 \pi i u \)) entire functions converging to \( f(z) \). The continuum sum is, by using the exponential expansion of sinh giving \( f_u(z) = -\frac{u}{2} e^{-\frac{z}{u}} + \frac{u}{2} e^{\frac{z}{u}} \),

\( \sum_{n=0}^{z-1} f_u(n) = \frac{u}{2} \left(\frac{e^{\frac{z}{u}} - 1}{e^{\frac{1}{u}} - 1} - \frac{e^{-\frac{z}{u}} - 1}{e^{-\frac{1}{u}} - 1}\right) \).

Though I didn't bother to try to work it out by hand, instead using a computer math package, the limit is \( \frac{x(x-1)}{2} \) as \( u \rightarrow \infty \), agreeing with the result from Faulhaber's formula.

Another example is the function \( f(z) = \frac{1}{z} \), or better, \( f(z) = \frac{1}{z+1} \). We can construct periodic approximations like \( f_u(z) = \frac{1}{u \sinh\left(\frac{z}{u}\right) + 1} \), and take the limit at infinity. The terms in the Fourier series are even worse. If we use a numerical approximation of the series with period , I get the continuum sum from 0 to -1/2 as ~0.6137 (rounded, act. more like something over 0.61369) suggesting the process is recovering the digamma function, as can be seen by setting 1/2 in the canonical formula \( -\gamma + \Psi(x + 1) = \sum_{n=0}^{x-1} \frac{1}{x+1} \) yielding 0.61370563888011...

Trying it with \( log(1 + z) \), so as to attempt to evaluate \( z! = \exp\left(\sum_{n=0}^{z-1} \log(1+z)\right) \) yields values that agree with the gamma function, providing more evidence that gamma is the natural extension of the factorial function to the complex plane.

Thus it seems this continuum sum is recovering all the expected sums and extensions. So the question comes up: what happens if we use it on Tetration, to sum up Ansus' continuum sum formula

\( \log_b\left(\frac{\mathrm{tet}'_b(z)}{\mathrm{tet}'_b(0) \log(b)^z}\right) = \sum_{n=0}^{z-1} \mathrm{tet}_b(n) \)

?

I don't yet have a really fast and efficient numerical program ready to go, but the idea behind the algorithm I'm using and the current code I can post in the Computation forum if you'd like.

Trying it out, though, it seems to converge well. I can't get a lot of precision due to the amount of nodes required, but enough to make graphs and simple observations of the behavior is available.

For tetration \( ^{z} \sqrt{2} \), we get \( ^{1/2} \sqrt{2} \approx 1.243623 \), agreeing with the result from the regular iteration up to the expected numerical error of the algorithm (determined by the residual, or the difference of \( \log_b \) of the approximation at 0.5 and the value of the approximation at -0.5, so we can knock off the 3 and get 1.24362 which is near 1.2436216... (residual mag was approx 10^-6)).

If we try it now at the natural base, base \( e \), we get \( ^{1/2} e \approx 1.64635\ \mathrm{to}\ 1.64636 \), which agrees with the results from the Cauchy integral, the Abel iteration, Kneser iteration, etc. Indeed, if we look at the graph at the imaginary axis, we see this:

which looks equivalent to the result from the Cauchy integral. Toward \( \pm i \infty \), the function looks to settle to values of given by approximately \( 0.318 + 1.337i \) and its conjugate, which of course agrees with the fixed points of logarithm given by \( -W_{-1}(-1) = 0.31813150520476... - 1.33723570143069... i \) and \( -W_0(-1) = 0.31813150520476... + 1.33723570143069... i \).

Graph of \( ^{ix} e \):

Graph of \( ^x e \):

Complex Bases

The most exciting thing about this method is that it even can be used on complex bases outside the Shell-Thron convergent region. For example, I tried it on the base \( 2.28 + 1.35i \), located in the period-4 lobe of the tetration fractal coming off the convergent region. I don't know if anyone has even seen the graph of the tetration of a complex base like this before. It is really wacky, and no, this is not a numerical error. Here is \( ^{x} (2.33 + 1.28i) \):

Apparently the period-4 behavior is only in the integer tetrations. The continuous tetrational function is instead unbounded on the positive real axis \( x > 0 \). Also, it does not appear to be injective any more.

The graph at the imaginary axis looks like this (i.e. \( ^{ix} (2.33 + 1.28i) \)):

showing the decay to the fixed points of the logarithm for this base.

One interesting mathematical question raised by this is that it seems to go fine right across \( b = e^{1/e} \), despite looking like the regular iteration below this base. Could this mean that the regular iteration really does have an analytic continuation outside the Shell-Thron region and the STB is not actually a natural boundary? OF course, this initial experimental investigation is no substitute for rigorous mathematical proof.

Another interesting question is why does it appear so many seemingly disparate methods keep turning up this same function, despite the existence of infinitely many solutions to the basic tetration functional equation? Could it be that there is some "natural" uniqueness condition that all these methods happen to be compatible with?