Update on 2021.9.29 10p.m.
III. Restriction/Principle on any iterative method to make superfunctions converge or merge
In many articles already, the Abel function generated at first pair of fixed points of natural exponential function, is claimed as only defined in halfplane, whether upper or lower. I pointed out that these functions can exactly be extended to the whole complex plane, and if prefered, to arbitrary branch cuts. This is observed by some author way before than me myself's observation, but in the last post thread I pledged to open a brand new thread to making clarification about them. (And no examples temporarily)
First principle (Most parts are already well-known):
If one gets a series expansion or some appropriate initial guess of a solution \( \zeta(z) \) in \( \zeta(f(z))=g(\zeta(z)) \) at some point asymptotically, like a formal series of a schroder function of f(z)=2z+z^3 generated at the fixed point 0,
then he has 2 ways to make the initial guess or a series expansion with the first few terms converge, denoted seperately as \( MI:\zeta(z)=\lim_{n\to\infty}{g^n(\zeta_0(f^{-n}(z))};MJ:\zeta(z)=\lim_{n\to\infty}{g^{-n}(\zeta_0(f^{n}(z))} \).
The choice of using which to make it converge only depends on which of \( \lim_{n\to\infty}{f^{-n}(z)},\lim_{n\to\infty}{f^{n}(z)} \) converges asymptotically to the point where the series originally generated from.
However, the Julia equation is slightly different from these, you can use both.
Also, whenever f or g function is periodically-iterative, or whatever the terms, or if there lies a nonzero real constant t that \( f^t(z)=z \), neither of such reccurence will lead to a full convergence upon the whole complex plane, unless both f and g are conjugate to some 2 linear functions.
Second Principle:
Any iterations of noninteger order of any analytic functions are multivalued.
Mark the iterative base as \( f(z) \).
First scenario, \( f \) is conjugate to a linear function. Since a linear functional iteration has a closed form: \( (z\to{az+b})^t=(z\to{a^tz+b\frac{a^t-1}{a-1}}) \)(take limit when a=1), we see that the function \( a^t \) is exactly a multivalued function whenever t is noninteger, this proves the statement in this scenario.
Second scenario, whenever \( f \) is not conjugate to a linear function but is singlevalued, its iterations have no closed form. Also we can know that there must lie some point \( f_0 \) in the riemann sphere that f mapped at least two different points \( z_1,z_2 \) both to \( f_0 \), and f can not have branch cuts, so \( f^{-1} \) must be multivalued. Thus due to the recurrence, \( \zeta \) must be multivalued, and so follows \( f^t \).
Third scenario, whenever \( f \) is multivalued, is easy to check.
Third Principle:
This follows the second principle.
If one uses any technique to make some \( \zeta(z) \) converge, or maybe merge 2 branches, with the use of \( MI \), and the inverse function has many branch cuts:
Let \( f_P^{-1}(z) \) be the proper branch cuts which leads to a full convergence (on the whole complex plane) of \( \zeta(z) \), or for all z, \( f_P^{-1}(z),f_P^{-2}(z),f_P^{-3}(z),\dots \) converges to the same point from which we generate the \( \zeta \) function.
Then the principle branches can be reached by only adjusting the mostly insider branch \( f_i^{-1}(z) \):
\( \zeta_i(z)=\lim_{n\to\infty}g^n(\zeta_{InitialGuess}(f_P^{-n+1}(f_i^{-1}(z)))) \)
But to generate all branch cuts of \( \zeta \), all possible combinations of branch cuts should be taken:
\( \zeta_{i_1,i_2,i_3,\dots}(z)=\lim_{n\to\infty}g^n(\zeta_{InitialGuess}(\cdots(f_{i_3}^{-1}(f_{i_2}^{-1}(f_{i_1}^{-1}(z)))\cdots)) \)
A simple derivation is that \( f_P^{-1}(z) \) must map the fixed point L of f to L itself. Or to say, the branch must contain the same fixed point of which we used to generate \( \zeta \).
This should also be considered in theta mapping.
Forth Principle:
If \( \zeta_1,\zeta_2 \) behaves asymptotically similarly at some point, \( MI,MJ \) will apply to both.
So it follows if \( MI,MJ \) dowsn't apply to both, then \( \zeta_1,\zeta_2 \) are not asymptotically similar, or at least one of them doesn't exist whenever considered as a branch cut.
This principle tells that the nonconstructable cases are only solvable when there lies a symmetry applicable, or to consider nonconstructable cases as branch cuts of some parabolic case. So nonconstructable cases are rarely solvable.
Fifth Principle:
A superfunction of an operator is never constructed whenever considered as singlevalued.
Here we give an example only to clarify the statement. Consider the \( A \) operator, its inverse, \( A_z^{-1}[f(z)]=f(f^{-1}(z)+1) \), answers "whose superfunction is f(z)". Then both \( A \) and \( A^{-1} \) are multivalued. Since A answers "what is the superfunction of f(z)", it follows instantly the multivalued-ity of tetration, so A must be multivalued. Also consider two singlevalued branch such that \( f_1(f_2^{-1}(z))=z \), hence that \( A[f_1]=A[f_2]=g \), so \( A^{-1}[g]=\{f_1,f_2\} \), is multivalued.
Also, it's easy to check the successor function \( P(z)=z+1 \) is a fixed point of \( A,A^{-1} \). So any technique using cauchy's integral to extend the definition of some sequence is non-self-consistent. Because if one uses \( a_j=\frac{1}{2\pi{i}}\int_C{\frac{\sum_{n=0}^\infty{a_nt^n}}{t^{j+1}}\mathrm{d}t} \) to generalize to all j, the result must be the same when using \( a_{-j}=\frac{1}{2\pi{i}}\int_C{\frac{\sum_{n=0}^\infty{a_{-n}t^n}}{t^{j+1}}\mathrm{d}t} \). So in the hyperoperator case, we have for all negative integer -n, the hyperoperator is the successor function following the successor function is the fixed point of the operator \( A^{-1} \), then it's not available to generalize hyperoperators in the usage of Cauchy's integral formula, otherwise any non-0th-1st-or-2nd hyperoperators should be considered multivalued.
III. Restriction/Principle on any iterative method to make superfunctions converge or merge
In many articles already, the Abel function generated at first pair of fixed points of natural exponential function, is claimed as only defined in halfplane, whether upper or lower. I pointed out that these functions can exactly be extended to the whole complex plane, and if prefered, to arbitrary branch cuts. This is observed by some author way before than me myself's observation, but in the last post thread I pledged to open a brand new thread to making clarification about them. (And no examples temporarily)
First principle (Most parts are already well-known):
If one gets a series expansion or some appropriate initial guess of a solution \( \zeta(z) \) in \( \zeta(f(z))=g(\zeta(z)) \) at some point asymptotically, like a formal series of a schroder function of f(z)=2z+z^3 generated at the fixed point 0,
then he has 2 ways to make the initial guess or a series expansion with the first few terms converge, denoted seperately as \( MI:\zeta(z)=\lim_{n\to\infty}{g^n(\zeta_0(f^{-n}(z))};MJ:\zeta(z)=\lim_{n\to\infty}{g^{-n}(\zeta_0(f^{n}(z))} \).
The choice of using which to make it converge only depends on which of \( \lim_{n\to\infty}{f^{-n}(z)},\lim_{n\to\infty}{f^{n}(z)} \) converges asymptotically to the point where the series originally generated from.
However, the Julia equation is slightly different from these, you can use both.
Also, whenever f or g function is periodically-iterative, or whatever the terms, or if there lies a nonzero real constant t that \( f^t(z)=z \), neither of such reccurence will lead to a full convergence upon the whole complex plane, unless both f and g are conjugate to some 2 linear functions.
Second Principle:
Any iterations of noninteger order of any analytic functions are multivalued.
Mark the iterative base as \( f(z) \).
First scenario, \( f \) is conjugate to a linear function. Since a linear functional iteration has a closed form: \( (z\to{az+b})^t=(z\to{a^tz+b\frac{a^t-1}{a-1}}) \)(take limit when a=1), we see that the function \( a^t \) is exactly a multivalued function whenever t is noninteger, this proves the statement in this scenario.
Second scenario, whenever \( f \) is not conjugate to a linear function but is singlevalued, its iterations have no closed form. Also we can know that there must lie some point \( f_0 \) in the riemann sphere that f mapped at least two different points \( z_1,z_2 \) both to \( f_0 \), and f can not have branch cuts, so \( f^{-1} \) must be multivalued. Thus due to the recurrence, \( \zeta \) must be multivalued, and so follows \( f^t \).
Third scenario, whenever \( f \) is multivalued, is easy to check.
Third Principle:
This follows the second principle.
If one uses any technique to make some \( \zeta(z) \) converge, or maybe merge 2 branches, with the use of \( MI \), and the inverse function has many branch cuts:
Let \( f_P^{-1}(z) \) be the proper branch cuts which leads to a full convergence (on the whole complex plane) of \( \zeta(z) \), or for all z, \( f_P^{-1}(z),f_P^{-2}(z),f_P^{-3}(z),\dots \) converges to the same point from which we generate the \( \zeta \) function.
Then the principle branches can be reached by only adjusting the mostly insider branch \( f_i^{-1}(z) \):
\( \zeta_i(z)=\lim_{n\to\infty}g^n(\zeta_{InitialGuess}(f_P^{-n+1}(f_i^{-1}(z)))) \)
But to generate all branch cuts of \( \zeta \), all possible combinations of branch cuts should be taken:
\( \zeta_{i_1,i_2,i_3,\dots}(z)=\lim_{n\to\infty}g^n(\zeta_{InitialGuess}(\cdots(f_{i_3}^{-1}(f_{i_2}^{-1}(f_{i_1}^{-1}(z)))\cdots)) \)
A simple derivation is that \( f_P^{-1}(z) \) must map the fixed point L of f to L itself. Or to say, the branch must contain the same fixed point of which we used to generate \( \zeta \).
This should also be considered in theta mapping.
Forth Principle:
If \( \zeta_1,\zeta_2 \) behaves asymptotically similarly at some point, \( MI,MJ \) will apply to both.
So it follows if \( MI,MJ \) dowsn't apply to both, then \( \zeta_1,\zeta_2 \) are not asymptotically similar, or at least one of them doesn't exist whenever considered as a branch cut.
This principle tells that the nonconstructable cases are only solvable when there lies a symmetry applicable, or to consider nonconstructable cases as branch cuts of some parabolic case. So nonconstructable cases are rarely solvable.
Fifth Principle:
A superfunction of an operator is never constructed whenever considered as singlevalued.
Here we give an example only to clarify the statement. Consider the \( A \) operator, its inverse, \( A_z^{-1}[f(z)]=f(f^{-1}(z)+1) \), answers "whose superfunction is f(z)". Then both \( A \) and \( A^{-1} \) are multivalued. Since A answers "what is the superfunction of f(z)", it follows instantly the multivalued-ity of tetration, so A must be multivalued. Also consider two singlevalued branch such that \( f_1(f_2^{-1}(z))=z \), hence that \( A[f_1]=A[f_2]=g \), so \( A^{-1}[g]=\{f_1,f_2\} \), is multivalued.
Also, it's easy to check the successor function \( P(z)=z+1 \) is a fixed point of \( A,A^{-1} \). So any technique using cauchy's integral to extend the definition of some sequence is non-self-consistent. Because if one uses \( a_j=\frac{1}{2\pi{i}}\int_C{\frac{\sum_{n=0}^\infty{a_nt^n}}{t^{j+1}}\mathrm{d}t} \) to generalize to all j, the result must be the same when using \( a_{-j}=\frac{1}{2\pi{i}}\int_C{\frac{\sum_{n=0}^\infty{a_{-n}t^n}}{t^{j+1}}\mathrm{d}t} \). So in the hyperoperator case, we have for all negative integer -n, the hyperoperator is the successor function following the successor function is the fixed point of the operator \( A^{-1} \), then it's not available to generalize hyperoperators in the usage of Cauchy's integral formula, otherwise any non-0th-1st-or-2nd hyperoperators should be considered multivalued.