Kneser expansion

Kneser expansion is asymptotic representation of superexponential constructed at its fixed point.

While in this article, only the special natural superexponential is considered. In this case, the fixed point is

\( L= (-\mathrm{ProductLog}(-1))^*\approx\) \( 0.3181315\) \(+\) \(1.3372357 \,\mathrm i \)

The solution of \(F\) of the transfer equation

\( F(z+1)=\exp(F(z)) \)

is asymptotically expanded in the gollowing form:

\( \displaystyle F(z)=L+\exp(L z)+\sum_{n=2}^{M_0} a_{0,n} \exp(L n z) + O(\exp(L (M_0\!+ \!1)z)\) \(\displaystyle + b_1 \exp(2 \pi \mathrm i z) \left(1+\sum_{n=1}^{M_1} a_{1,n} \exp(L n z) + ..\right)\) \(\displaystyle + b_2 \exp(4 \pi \mathrm i z) \left(1+\sum_{n=1}^{M_2} a_{2,n} \exp(L n z) + ..\right) + .. \)

Coefficients \(a_{0,n}\) for \(n=1..12\) can be evaluated with the Mathematica code below M=12; f0 = L + e + Sum[a[n] e^n, {n,2,M}]; f1 = L + L e + Sum[a[n]L^n e^n, {n,2,M}]; s0 = ReplaceAll[Series[Exp[f0],{e,0,M}] - f1, Exp[L]->L]; co[2] = Extract[Solve[Coefficient[s0,e^2]==0, a[2]], 1]; A[2] = ReplaceAll[a[2], co[2]] s[3] = Simplify[ReplaceAll[s0, a[2] -> A[2]]];

For[m = 3, m <= M, Print[m]; co[m] = Extract[Solve[Coefficient[s[m], e^m] == 0, a[m]], 1]; A[m] = ReplaceAll[a[m], co[m]]; s[m+1] = Simplify[ReplaceAll[s[m], a[m]->A[m]]]; m++]

tableM = Table[{a[m], A[m]}, {m,2,M}]; Le = N[Conjugate[-ProductLog[-1]], 64] N[TableForm[Table[{a[m], ReplaceAll[A[m], L -> Le]}, {m, 2, M}]], 16]

Keywords
Iteration, Tetration, Transfer equation, Transfer function, Superexponential Superfunction

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