# Kneser expansion

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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]

## References

http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851 H.Kneser. Reelle analytische Lösungen der Gleichung φ(φ(x))=ex. Equationes Mathematicae, Journal fur die reine und angewandte Mathematik {\bf 187} 56–67 (1950)

http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.