I'm going to adjoin another topic here, per Ember Edison's questions. We're going to look at \( b = e^{-e} \) which is a very anomalous point. We're going to start by constructing the infinite composition,
\(
\gamma_\lambda(s) = \Omega_{j=1}^\infty \frac{e^{-ez}}{e^{\lambda(j-s)} + 1}\,\bullet z\\
\)
Which satisfies the functional equation,
\(
\gamma_\lambda(s+1) = e^{-e\gamma_\lambda(s)}/(e^{-\lambda s} + 1)\\
\)
And satisfies the similar family structure that \( \beta_\lambda \) and \( \varphi_\lambda \) did. This function is holomorphic every where \( \lambda(j-s) \neq (2k+1) \pi i \) for \( j \ge 1, j,k\in\mathbb{Z} \).
We're going to focus on \( \lambda =1 \); which is \( 2 \pi i \)-periodic and real valued. We want to insert the same sequence which guesses how close we are to tetration. These will be,
\(
\mu^0 = 0\\
\mu^{n+1}(s) = \log(1+e^{-s})/e - \log(1+\frac{\mu^n(s+1)}{\gamma_1(s+1)})/e\\
\)
We're going to stick to \( \mu^{100} \), which should give a decent amount of accuracy (it's giving about 50 digits on the real line, and 20 or so for complex arguments). Keeping that in mind, here is \( \gamma_1(x) + \mu^{100}(x) \) over \( -1 < x < 3 \) (remember we aren't normalized yet):
And this satisfies \( f(s+1) = e^{-e f(s)} \) to at least 20 digits.
And here's \( [\Re(\gamma_1(x+i) + \mu^{100}(x+i)),\Im(\gamma_1(x+i) + \mu^{100}(x+i))] \) over \( -1 < x < 3 \)
And this satisfies \( f(s+1) = e^{-e f(s)} \) to at least 20 digits. But remember, this tetration will have a period 2 pi i and singularities along \( \Im(s) = \pi \), so it's not the one we want. We want to grow the period to infinity, which again, I'm not certain how to do with this case like I was with \( b = e \).
I'll attach here the patch together code. It's similar to \( b = 1/2 \); it runs much slower, and it requires a new initialization file INIT_NEUTRAL.dat which I've included. I'm going to make a complex plane graph and see how this looks.
Abel_neutral.zip (Size: 979.41 KB / Downloads: 584)
Again, I'm very confident infinite compositions will not run into trouble on the real positive line mathematically. Coding it may produce odd anomalies though.
Here's a graph of the taylor series of \( f(z) = \gamma_1(z) + \mu^{100}(z) \) about \( z = 0 \) for \( |\Re(z)|, |\Im(z)| \le 1 \); this is accurate to about 30 digits (\( e^{-ef(z)} = f(z+1) \) to about 30 digits) where it's converging:
This function equals \( 1/e \) to about two decimal places everywhere (why it's such a monotone colour); and just sort of wobbles around. I've attached, to show the functional equation is being satisfied, a graph of \( f(z) - \exp(-\exp(1)*f(z-1)) \) over \( |\Re(z)| \le 1 \) and \( |\Im(z)| \le 1 \):
It's mostly just black because that's how well the taylor series satisfies the functional equation.
I'm still wary of how you'd normalize this, so that \( f(0) = 1 \); but as a super function, this is perfectly viable. It looks like an almost periodic mess at the moment. But the point I'm trying to make is that it still produces an analytic function. Everything I've been saying still holds--but there are many more questions to be asked about the infinite composition method for other bases. I am trying to flush out \( b = e \) perfectly, before I move onto other bases. But so far so good!
\(
\gamma_\lambda(s) = \Omega_{j=1}^\infty \frac{e^{-ez}}{e^{\lambda(j-s)} + 1}\,\bullet z\\
\)
Which satisfies the functional equation,
\(
\gamma_\lambda(s+1) = e^{-e\gamma_\lambda(s)}/(e^{-\lambda s} + 1)\\
\)
And satisfies the similar family structure that \( \beta_\lambda \) and \( \varphi_\lambda \) did. This function is holomorphic every where \( \lambda(j-s) \neq (2k+1) \pi i \) for \( j \ge 1, j,k\in\mathbb{Z} \).
We're going to focus on \( \lambda =1 \); which is \( 2 \pi i \)-periodic and real valued. We want to insert the same sequence which guesses how close we are to tetration. These will be,
\(
\mu^0 = 0\\
\mu^{n+1}(s) = \log(1+e^{-s})/e - \log(1+\frac{\mu^n(s+1)}{\gamma_1(s+1)})/e\\
\)
We're going to stick to \( \mu^{100} \), which should give a decent amount of accuracy (it's giving about 50 digits on the real line, and 20 or so for complex arguments). Keeping that in mind, here is \( \gamma_1(x) + \mu^{100}(x) \) over \( -1 < x < 3 \) (remember we aren't normalized yet):
And this satisfies \( f(s+1) = e^{-e f(s)} \) to at least 20 digits.
And here's \( [\Re(\gamma_1(x+i) + \mu^{100}(x+i)),\Im(\gamma_1(x+i) + \mu^{100}(x+i))] \) over \( -1 < x < 3 \)
And this satisfies \( f(s+1) = e^{-e f(s)} \) to at least 20 digits. But remember, this tetration will have a period 2 pi i and singularities along \( \Im(s) = \pi \), so it's not the one we want. We want to grow the period to infinity, which again, I'm not certain how to do with this case like I was with \( b = e \).
I'll attach here the patch together code. It's similar to \( b = 1/2 \); it runs much slower, and it requires a new initialization file INIT_NEUTRAL.dat which I've included. I'm going to make a complex plane graph and see how this looks.
Abel_neutral.zip (Size: 979.41 KB / Downloads: 584)
Again, I'm very confident infinite compositions will not run into trouble on the real positive line mathematically. Coding it may produce odd anomalies though.
Here's a graph of the taylor series of \( f(z) = \gamma_1(z) + \mu^{100}(z) \) about \( z = 0 \) for \( |\Re(z)|, |\Im(z)| \le 1 \); this is accurate to about 30 digits (\( e^{-ef(z)} = f(z+1) \) to about 30 digits) where it's converging:
This function equals \( 1/e \) to about two decimal places everywhere (why it's such a monotone colour); and just sort of wobbles around. I've attached, to show the functional equation is being satisfied, a graph of \( f(z) - \exp(-\exp(1)*f(z-1)) \) over \( |\Re(z)| \le 1 \) and \( |\Im(z)| \le 1 \):
It's mostly just black because that's how well the taylor series satisfies the functional equation.
I'm still wary of how you'd normalize this, so that \( f(0) = 1 \); but as a super function, this is perfectly viable. It looks like an almost periodic mess at the moment. But the point I'm trying to make is that it still produces an analytic function. Everything I've been saying still holds--but there are many more questions to be asked about the infinite composition method for other bases. I am trying to flush out \( b = e \) perfectly, before I move onto other bases. But so far so good!
