This is where I'm hung up on further progress with this method: The HAM method requires the choice of an "auxiliary linear operator". This is a free parameter in the method equations.
The rule, it seems, for choosing this linear operator is to make it one which annihilates some (apparently, 3) initial terms of a "solution expression", which is a form in which to write the solution of the integral or differential equation in question. What we need is to find a form for the tetrational (or, perhaps, better, the solution of the Cauchy equations, which are given in an approximate form so as to approximate it) which looks like
\( \mathrm{tet}(z) = \sum_{n=0}^{\infty} a_n b_n(z) \)
where \( b_n \) are properly-chosen "basis functions". I have thought about the Kneser-mapping solution (i.e. regular iteration warped with theta mapping) as a possible set of basis functions, which gives a double summation over coefficients with terms \( b_{n,k}(z) = e^{(Ln + 2\pi i k)z} \) (this is for base \( e \). \( L \) is the fixed point of the logarithm) (note the sum over two indices instead of one), but the problem is this only covers half of the plane (as given, the upper half-plane), and the Cauchy equations require both halves of the plane.
Do you, perhaps, have any ideas as to how this could be done? The form of solution need not converge on the entire plane, only on and perhaps near the imaginary axis.
The rule, it seems, for choosing this linear operator is to make it one which annihilates some (apparently, 3) initial terms of a "solution expression", which is a form in which to write the solution of the integral or differential equation in question. What we need is to find a form for the tetrational (or, perhaps, better, the solution of the Cauchy equations, which are given in an approximate form so as to approximate it) which looks like
\( \mathrm{tet}(z) = \sum_{n=0}^{\infty} a_n b_n(z) \)
where \( b_n \) are properly-chosen "basis functions". I have thought about the Kneser-mapping solution (i.e. regular iteration warped with theta mapping) as a possible set of basis functions, which gives a double summation over coefficients with terms \( b_{n,k}(z) = e^{(Ln + 2\pi i k)z} \) (this is for base \( e \). \( L \) is the fixed point of the logarithm) (note the sum over two indices instead of one), but the problem is this only covers half of the plane (as given, the upper half-plane), and the Cauchy equations require both halves of the plane.
Do you, perhaps, have any ideas as to how this could be done? The form of solution need not converge on the entire plane, only on and perhaps near the imaginary axis.