# 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:

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

..

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.