Wow, a very long thread has formed in my absence, lol. I figure I should add a long thread in return, so buckle up, lol.
I'd like to add a couple details. Mphlee did a much better job at explaining the concerns I have with this notation. The more I'm rereading through this thread, I've come to the conclusion, in order for Leo's idea to make sense; there's no two ways around it; we need to introduce Riemann Surfaces. As Leo seems dead set on multi-valued functions; it's imperative we introduce a more rigorous framework. Leo makes mention of choosing the "worthy" iteration (lol, which I understand entirely; but he hasn't really given a way to find said worthy iteration; besides sticking to Schroder or Abel about a fixed point).
When making the leap to Riemann Surfaces; it's pretty much imperative to use differetial equations. So, I thought we'd take a little detour through differential equations and superfunctions. Let's assume we can develop an iteration,
\(
F(z,\xi)
\)
Such that \( F(z,F(z',\xi)) = F(z+z',\xi) \). A little unused theorem on this forum, is classifying this superfunction as a flow.
\(
\frac{d}{dz} F(z,\xi) = \lim_{h\to 0} \frac{F(z+h,\xi) - F(z,\xi)}{h} = \lim_{h\to 0} \frac{F(h,\xi) - \xi}{h} \bullet F(z,\xi) \bullet \xi\\
\)
I like to call this object,
\(
\log (F ; \xi) = \lim_{h\to 0} \frac{F(h,\xi) - \xi}{h}
\)
Which can be referred to as the logarithm of the super function; or traditionally it's known as a generator of a flow map. Now, for every super function there is one logarithm, and for every logarithm there is one superfunction. There is no obvious connection between the logarithm and the initial function \( F(1,\xi) = f(\xi) \); but you can derive a formula for it.
Call the function,
\(
h(\xi) = \log (F ; \xi) \\
\)
And we're reducing the problem to the differential equation,
\(
y' = h(y)\\
\)
with the initial condition \( y(0) = \xi \). For instance, a tetration function \( \text{tet} \) and its super logarithm \( \text{slog} \) are completely determined by the map,
\(
h(\xi) = \lim_{z\to 0} \frac{\text{tet}(z+\text{slog}(\xi)) - \xi}{z}\\
\)
Where, the solution of the differential equation in a neighborhood of \( 0 \);
\(
y'(z) = h(y)\\
y(0) = \xi\\
\text{implies}\\
y(z) = \text{tet}(z + \text{slog}(\xi))\\
\)
And vice versa; so this is an equivalence. Now, the manner I like to view this function is borrowed from my paper The Compositional Integral: The Narrow And The Complex Looking-Glass. If we call \( u(s,z) \) the solution to the differential equation,
\(
u'(s,z) = f(s,u)\,\,\text{and}\,\,u(a) = z\,\,\text{then write}\\
u(s,z) = \int_a^s f(s,z)\,ds\bullet z\\
\)
To better explain this notation, I refer to the paper; which can be found on arXiv here, https://arxiv.org/abs/2003.05280 But the gist of the notation comes from the Omega notation, which is;
Choose an arc \( \gamma \) connecting the points \( a,s \), and choose a partition of \( [0,1] \). We'll call this \( \{s_j\}_{j=0}^n \) in which \( 1 = s_0 \ge s_1 \ge s_2 ... \ge s_{n-1} \ge s_n = 0 \). Call the term \( \Delta \gamma_j = \gamma(s_{j}) - \gamma(s_{j+1}) \). And write,
\(
\int_\gamma f(s,z)\,ds\bullet z = \lim_{\Delta \to 0} \Omega_{j=0}^{n-1} z + f(\gamma_j,z)\Delta \gamma_j \bullet z\\
\)
Where, if we call the term \( p_{jn}(z) = z + f(\gamma_j,z)\Delta \gamma_j \), this just means,
\(
\Omega_{j=0}^{n-1} z + f(\gamma_j,z)\Delta \gamma_j \bullet z = p_{0n}(p_{1n}(...p_{(n-1)n}(z)))\\
\)
Where we are just taking the limit \( n\to\infty \). To make a long story short; this thing converges well enough (at least for the purposes of this post). And we get a domain \( \mathcal{D} \subset \mathbb{C} \) for which \( u(s,z) : \mathcal{U} \times \mathcal{D} \to \mathbb{C} \); and \( \mathcal{U} \) is a neighborhood of \( a \) and \( \overline{\mathcal{D}} = \mathbb{C} \). And this function satisfies the differential equation,
\(
u'(s,z) = f(s,u)\\
u(a,z) = z\\
\)
So what does this have to do with super-functions? Much of the paper was written to design a formalism; and that formalism isn't quite complete yet; but you can do some crazy stuff with this notation; and its specific to differential equations/superfunctions. Now; if you'll remember your basics of Riemann Surfaces; a lot of the time we can create a Riemann surface from a differential equation. For that, think of the log map \( y' = \frac{1}{s} \); where as we go around and around we spiral around a Riemann surface.
The paper I linked to, does not delve into the Riemann surface aspect at all; but it gives a type of language that is very beneficial which would be too hard to explain here. But I'll give a run through. And I swear, it'll get to the point where we define something like a conjugate class \( [f,g] \); which I furiously tried to formalize in that paper. Define the function,
\(
y(s,\xi) = \int_0^s h(\xi)\,ds\bullet \xi
\)
Then,
\(
y(0,\xi) = \xi\\
y(s,y(s',\xi)) = y(s+s',\xi)\\
\)
And this is the equation of a semi-group. And further more; all semi-groups have to look like this for some \( h \) (at least analytic ones). But there's more hidden to this notation.
The Compositional Integral Theorem;
Suppose that \( \mathcal{S} \subseteq \mathbb{C} \) is simply connected. Suppose that \( \phi(s,z) : \mathcal{S} \times \mathbb{C} \to \mathbb{C} \) is a holomorphic function. For all Jordan curves \( \gamma \subset \mathcal{S} \), the following identity is true,
\(
\int_\gamma \phi(s,z)\,ds\bullet z = z\\
\)
\\
Which this means; as with Cauchy's integral theorem; a closed integration about a holomorphic function on a simply connected domain is an identity. Now, suppose we have a pole at \( s=0 \) of \( f(s,z) \); and lets take the domain,
\(
\mathcal{S}_{\delta,\delta'} = \{s \in \mathbb{C}\,|\, \delta' \le |s| \le \delta,\, s \not \in [-\delta,-\delta']\}\\
\)
Which looks like an annulus about zero with a cut along the negative real axis. This domain is simply connected, call the outer circle \( \gamma_\delta \) oriented positively; the inner circle \( \gamma_{\delta'}^{-} \) oriented negatively; and the arcs \( \rho \) and \( \rho^- \) the line \( [-\delta,-\delta'] \); where they're oriented positively and negatively respectively.
Then,
\(
\int_{\partial S} f(s,z)\,ds\bullet z = z\\
\)
By The compositional integral theorem. But the path \( \partial S = \gamma_{\delta} \bullet \rho \bullet \gamma_{\delta'}^- \bullet \rho^- \). And using the bullet notation,
\(
z = \int_{\gamma_\delta} f(s,z) ds \bullet \int_{\rho} f(s,z) ds \bullet \int_{\gamma_{\delta'}^-} f(s,z)\,ds\bullet \int_{\rho^{-}} f(s,z)\,ds\bullet z\\
\)
And you'll begin to see where I'm going with this by the inversion rule. The functional inverse of \( \int_\gamma f(s,z)\,ds\bullet z \) is just the integration \( \int_\tau f(s,z)\,ds\bullet z \); where \( \tau \) is just the opposite orientation of \( \gamma \). So what we wind up with, is,
\(
\int_{\gamma_\delta} f(s,z) ds\bullet z = \int_{\rho^-} \bullet \int_{\gamma_{\delta'}} \bullet \int_{\rho}\\
\)
So the circle \( |s| \le \delta \) and the circle \( |s| \le \delta' \); are conjugate similar. Or rather.
\(
F(z) = \int_{\gamma_\delta} f(s,z) ds\bullet z\\
G(z) = \int_{\gamma_{\delta'}} f(s,z)\,ds\bullet z\\
\text{then there exists a function}\,P(z)\\
F(z) = P(G(P^{-1}(z)))\\
\)
Which is how miraculous, \( P \in [F,G] \) as Mphlee described.
Is \( P \) unique? Absolutely not. Because we haven't quite described what \( F \) is perfectly. So to that, I introduce The Residual Class.
The Residual Class of a Meromorphic function
Let \( \mathcal{S}\subseteq \mathbb{C} \) be a simply connected domain. Suppose \( f(s,z) : \mathcal{S}/\{\zeta\} \times \mathbb{C} \to \mathbb{C} \) is holomorphic with a singularity at \( s = \zeta \). The residual class \( \text{Rsd} \) of the function \( f \) about the singularity \( \zeta \) is given as,
\(
\text{Rsd}(f, \zeta) \ni \int_\gamma f(s,z) \,ds \bullet z
\)
For all \( \gamma \subset \mathcal{S} \), a Jordan curve containing \( \zeta \).
And the fact is,
\(
\forall F,G \in \text{Rsd}(f,\zeta)\,\,\exists P\,\,\text{s.t.}\,\,F(z) = P(G(P^{-1}(z)))\\
\)
And \( P \) can be expressed as an arc \( \tau \subset \mathcal{S} \),
\(
P(z) = \int_\tau f(s,z)\,ds\bullet z\\
\)
The paper then goes on to use this frantically at developing a formal language; which has its place in Riemann surfaces/differential equations. But note, Leo; I've never once used an abel function or a superfunction to construct these conjugates. And personally, I feel they arise much more naturally than with Abel functions or Schroder functions; which to me seems artificial. Especially if you are going to broach the fields of mechanics and physics; a differential equation will always come out supreme to the physics pedagogy than an iterative process at infinity. It's just the symptom of physics, lol.
Anyway; if you want to see how I construct this in much more depth, please read the paper. There isn't much on super-functions; but I definitely graze it often enough. It's written a little loosely; but the rigor is all there. And it's written in a fairly elementary way if you've studied lots of complex analysis. I spent about a year on it; all last year; before and during quarantine, lol. At least corona was good for something.
Whew, that was a long post. I'm going to go back to work now. I hope this makes sense,
Regards, James
EDIT:
I will add, I haven't proven this to its limit yet; as it's fairly tricky; but I've proven things around it. But if we have a function \( h \) from before, where this generates a semi-group/iteration/super-function; and we have a function \( \phi(s,z) \) which satisfies \( \phi(\zeta,z) = h(z) \); then,
\(
2\pi i h(z) = \lim_{\delta \to 0} \frac{\int_{\gamma_\delta} \frac{\phi(s,z)}{s-\zeta}\,ds\bullet z - z}{\delta}\\
\)
But this is tricky to use this how we want... manipulating how you're describing. I do it in a limited manner in that paper. And this gives us a way of relating superfunctions to a differential relationship. But it doesn't do what you're asking your theory to do. That sounds super hard. Especially if you keep with this multi-valued language; you're going to have to talk about riemann surfaces somewhere in your discussion.
Regards.
I'd like to add a couple details. Mphlee did a much better job at explaining the concerns I have with this notation. The more I'm rereading through this thread, I've come to the conclusion, in order for Leo's idea to make sense; there's no two ways around it; we need to introduce Riemann Surfaces. As Leo seems dead set on multi-valued functions; it's imperative we introduce a more rigorous framework. Leo makes mention of choosing the "worthy" iteration (lol, which I understand entirely; but he hasn't really given a way to find said worthy iteration; besides sticking to Schroder or Abel about a fixed point).
When making the leap to Riemann Surfaces; it's pretty much imperative to use differetial equations. So, I thought we'd take a little detour through differential equations and superfunctions. Let's assume we can develop an iteration,
\(
F(z,\xi)
\)
Such that \( F(z,F(z',\xi)) = F(z+z',\xi) \). A little unused theorem on this forum, is classifying this superfunction as a flow.
\(
\frac{d}{dz} F(z,\xi) = \lim_{h\to 0} \frac{F(z+h,\xi) - F(z,\xi)}{h} = \lim_{h\to 0} \frac{F(h,\xi) - \xi}{h} \bullet F(z,\xi) \bullet \xi\\
\)
I like to call this object,
\(
\log (F ; \xi) = \lim_{h\to 0} \frac{F(h,\xi) - \xi}{h}
\)
Which can be referred to as the logarithm of the super function; or traditionally it's known as a generator of a flow map. Now, for every super function there is one logarithm, and for every logarithm there is one superfunction. There is no obvious connection between the logarithm and the initial function \( F(1,\xi) = f(\xi) \); but you can derive a formula for it.
Call the function,
\(
h(\xi) = \log (F ; \xi) \\
\)
And we're reducing the problem to the differential equation,
\(
y' = h(y)\\
\)
with the initial condition \( y(0) = \xi \). For instance, a tetration function \( \text{tet} \) and its super logarithm \( \text{slog} \) are completely determined by the map,
\(
h(\xi) = \lim_{z\to 0} \frac{\text{tet}(z+\text{slog}(\xi)) - \xi}{z}\\
\)
Where, the solution of the differential equation in a neighborhood of \( 0 \);
\(
y'(z) = h(y)\\
y(0) = \xi\\
\text{implies}\\
y(z) = \text{tet}(z + \text{slog}(\xi))\\
\)
And vice versa; so this is an equivalence. Now, the manner I like to view this function is borrowed from my paper The Compositional Integral: The Narrow And The Complex Looking-Glass. If we call \( u(s,z) \) the solution to the differential equation,
\(
u'(s,z) = f(s,u)\,\,\text{and}\,\,u(a) = z\,\,\text{then write}\\
u(s,z) = \int_a^s f(s,z)\,ds\bullet z\\
\)
To better explain this notation, I refer to the paper; which can be found on arXiv here, https://arxiv.org/abs/2003.05280 But the gist of the notation comes from the Omega notation, which is;
Choose an arc \( \gamma \) connecting the points \( a,s \), and choose a partition of \( [0,1] \). We'll call this \( \{s_j\}_{j=0}^n \) in which \( 1 = s_0 \ge s_1 \ge s_2 ... \ge s_{n-1} \ge s_n = 0 \). Call the term \( \Delta \gamma_j = \gamma(s_{j}) - \gamma(s_{j+1}) \). And write,
\(
\int_\gamma f(s,z)\,ds\bullet z = \lim_{\Delta \to 0} \Omega_{j=0}^{n-1} z + f(\gamma_j,z)\Delta \gamma_j \bullet z\\
\)
Where, if we call the term \( p_{jn}(z) = z + f(\gamma_j,z)\Delta \gamma_j \), this just means,
\(
\Omega_{j=0}^{n-1} z + f(\gamma_j,z)\Delta \gamma_j \bullet z = p_{0n}(p_{1n}(...p_{(n-1)n}(z)))\\
\)
Where we are just taking the limit \( n\to\infty \). To make a long story short; this thing converges well enough (at least for the purposes of this post). And we get a domain \( \mathcal{D} \subset \mathbb{C} \) for which \( u(s,z) : \mathcal{U} \times \mathcal{D} \to \mathbb{C} \); and \( \mathcal{U} \) is a neighborhood of \( a \) and \( \overline{\mathcal{D}} = \mathbb{C} \). And this function satisfies the differential equation,
\(
u'(s,z) = f(s,u)\\
u(a,z) = z\\
\)
So what does this have to do with super-functions? Much of the paper was written to design a formalism; and that formalism isn't quite complete yet; but you can do some crazy stuff with this notation; and its specific to differential equations/superfunctions. Now; if you'll remember your basics of Riemann Surfaces; a lot of the time we can create a Riemann surface from a differential equation. For that, think of the log map \( y' = \frac{1}{s} \); where as we go around and around we spiral around a Riemann surface.
The paper I linked to, does not delve into the Riemann surface aspect at all; but it gives a type of language that is very beneficial which would be too hard to explain here. But I'll give a run through. And I swear, it'll get to the point where we define something like a conjugate class \( [f,g] \); which I furiously tried to formalize in that paper. Define the function,
\(
y(s,\xi) = \int_0^s h(\xi)\,ds\bullet \xi
\)
Then,
\(
y(0,\xi) = \xi\\
y(s,y(s',\xi)) = y(s+s',\xi)\\
\)
And this is the equation of a semi-group. And further more; all semi-groups have to look like this for some \( h \) (at least analytic ones). But there's more hidden to this notation.
The Compositional Integral Theorem;
Suppose that \( \mathcal{S} \subseteq \mathbb{C} \) is simply connected. Suppose that \( \phi(s,z) : \mathcal{S} \times \mathbb{C} \to \mathbb{C} \) is a holomorphic function. For all Jordan curves \( \gamma \subset \mathcal{S} \), the following identity is true,
\(
\int_\gamma \phi(s,z)\,ds\bullet z = z\\
\)
\\
Which this means; as with Cauchy's integral theorem; a closed integration about a holomorphic function on a simply connected domain is an identity. Now, suppose we have a pole at \( s=0 \) of \( f(s,z) \); and lets take the domain,
\(
\mathcal{S}_{\delta,\delta'} = \{s \in \mathbb{C}\,|\, \delta' \le |s| \le \delta,\, s \not \in [-\delta,-\delta']\}\\
\)
Which looks like an annulus about zero with a cut along the negative real axis. This domain is simply connected, call the outer circle \( \gamma_\delta \) oriented positively; the inner circle \( \gamma_{\delta'}^{-} \) oriented negatively; and the arcs \( \rho \) and \( \rho^- \) the line \( [-\delta,-\delta'] \); where they're oriented positively and negatively respectively.
Then,
\(
\int_{\partial S} f(s,z)\,ds\bullet z = z\\
\)
By The compositional integral theorem. But the path \( \partial S = \gamma_{\delta} \bullet \rho \bullet \gamma_{\delta'}^- \bullet \rho^- \). And using the bullet notation,
\(
z = \int_{\gamma_\delta} f(s,z) ds \bullet \int_{\rho} f(s,z) ds \bullet \int_{\gamma_{\delta'}^-} f(s,z)\,ds\bullet \int_{\rho^{-}} f(s,z)\,ds\bullet z\\
\)
And you'll begin to see where I'm going with this by the inversion rule. The functional inverse of \( \int_\gamma f(s,z)\,ds\bullet z \) is just the integration \( \int_\tau f(s,z)\,ds\bullet z \); where \( \tau \) is just the opposite orientation of \( \gamma \). So what we wind up with, is,
\(
\int_{\gamma_\delta} f(s,z) ds\bullet z = \int_{\rho^-} \bullet \int_{\gamma_{\delta'}} \bullet \int_{\rho}\\
\)
So the circle \( |s| \le \delta \) and the circle \( |s| \le \delta' \); are conjugate similar. Or rather.
\(
F(z) = \int_{\gamma_\delta} f(s,z) ds\bullet z\\
G(z) = \int_{\gamma_{\delta'}} f(s,z)\,ds\bullet z\\
\text{then there exists a function}\,P(z)\\
F(z) = P(G(P^{-1}(z)))\\
\)
Which is how miraculous, \( P \in [F,G] \) as Mphlee described.
Is \( P \) unique? Absolutely not. Because we haven't quite described what \( F \) is perfectly. So to that, I introduce The Residual Class.
The Residual Class of a Meromorphic function
Let \( \mathcal{S}\subseteq \mathbb{C} \) be a simply connected domain. Suppose \( f(s,z) : \mathcal{S}/\{\zeta\} \times \mathbb{C} \to \mathbb{C} \) is holomorphic with a singularity at \( s = \zeta \). The residual class \( \text{Rsd} \) of the function \( f \) about the singularity \( \zeta \) is given as,
\(
\text{Rsd}(f, \zeta) \ni \int_\gamma f(s,z) \,ds \bullet z
\)
For all \( \gamma \subset \mathcal{S} \), a Jordan curve containing \( \zeta \).
And the fact is,
\(
\forall F,G \in \text{Rsd}(f,\zeta)\,\,\exists P\,\,\text{s.t.}\,\,F(z) = P(G(P^{-1}(z)))\\
\)
And \( P \) can be expressed as an arc \( \tau \subset \mathcal{S} \),
\(
P(z) = \int_\tau f(s,z)\,ds\bullet z\\
\)
The paper then goes on to use this frantically at developing a formal language; which has its place in Riemann surfaces/differential equations. But note, Leo; I've never once used an abel function or a superfunction to construct these conjugates. And personally, I feel they arise much more naturally than with Abel functions or Schroder functions; which to me seems artificial. Especially if you are going to broach the fields of mechanics and physics; a differential equation will always come out supreme to the physics pedagogy than an iterative process at infinity. It's just the symptom of physics, lol.
Anyway; if you want to see how I construct this in much more depth, please read the paper. There isn't much on super-functions; but I definitely graze it often enough. It's written a little loosely; but the rigor is all there. And it's written in a fairly elementary way if you've studied lots of complex analysis. I spent about a year on it; all last year; before and during quarantine, lol. At least corona was good for something.
Whew, that was a long post. I'm going to go back to work now. I hope this makes sense,
Regards, James
EDIT:
I will add, I haven't proven this to its limit yet; as it's fairly tricky; but I've proven things around it. But if we have a function \( h \) from before, where this generates a semi-group/iteration/super-function; and we have a function \( \phi(s,z) \) which satisfies \( \phi(\zeta,z) = h(z) \); then,
\(
2\pi i h(z) = \lim_{\delta \to 0} \frac{\int_{\gamma_\delta} \frac{\phi(s,z)}{s-\zeta}\,ds\bullet z - z}{\delta}\\
\)
But this is tricky to use this how we want... manipulating how you're describing. I do it in a limited manner in that paper. And this gives us a way of relating superfunctions to a differential relationship. But it doesn't do what you're asking your theory to do. That sounds super hard. Especially if you keep with this multi-valued language; you're going to have to talk about riemann surfaces somewhere in your discussion.
Regards.