Difference between revisions of "Zooming equation"
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[[Zooming equation]] is tentative name for the equation 
[[Zooming equation]] is tentative name for the equation 

−  (1) 
+  (1) \(~ ~ ~ T\big(f(z)\big)= f( K\, z)~\) 
−  where 
+  where \(~T~\) is some given function such that \(T(0)\!=\!0\) and \(~K~\) is constant, that depends on \(T\). 
+  Usually, it is assumed that \(K\) is positive real number, id est, \(K>0\), and \(T\) is real–holomorphic, 

+  at least in some vicinity of zero, and \(~K\!=\!T'(0)~\). Function \(~f~\) is requested to builtup. 

−  The physical sense of function 
+  The physical sense of function \(~T~\) is transfer function; it describes the variation of value of function \(~f~\) at the scaling, zooming of its argument with factor \(~K~\). 
−  The tentative name for the solution 
+  The tentative name for the solution \(~f~\) of the [[zooming equation]] is [[zooming function]]. 
−  ==Schroeder 
+  ==Schroeder equation== 
−  The solution 
+  The solution \(~f\) of the zooming equation is related to the solution \(~g~\) of the [[Schroeder equation]] 
−  (2) 
+  (2) \(~ ~ ~ g\big(T(z)\big)= K \, g(z) \) 
−  it is assumed that 
+  it is assumed that \(~f=g^{1}\) and \(~g=f^{1}\), id est, in wide ranges of values of \(z\), the relations below hold: 
−  (3) 
+  (3) \(~ ~ ~ f(g(z))=z\) 
−  (4) 
+  (4) \(~ ~ ~ g(f(z))=z\) 
==[[Regular iteration]]== 
==[[Regular iteration]]== 

−  +  Regular iteration is procedure that leads to noninteger iterates of a transfer function, that are regular in some vicinity of its fixed point. Term [[Regular iteration]] had been suggested in 1958 by G. Seekers 

+  <ref> 

−  the [[zooming function]] $~f~$ and the [[Schroeder function]] $~g~$ can be used to construct the non–integer [[iterate]] $~T^r~$, that is regular in vicinity of zero, id est, the [[regular iterate]] 

+  http://link.springer.com/article/10.1007%2FBF02559539 

+  G. Szekeres. Regular iteration of real and complex functions. 

+  Acta Mathematica, September 1958, Volume 100, Issue 3, pp 203258. 

+  </ref>. 

+  
+  Assuming that zero is [[fixed point]] of the transfer function \(~T~\) is holomorphioc (regular) in vicinity of zero, 

+  the [[zooming function]] \(~f~\) and the [[Schroeder function]] \(~g\!=\!f^{1}~\) can be used to construct the non–integer [[iterate]] \(~T^n~\), that is regular in vicinity of zero, id est, the [[regular iterate]] 

−  (5) 
+  (5) \(\displaystyle ~ ~ ~ T^n(z)=f\Big(k^n\, g(z)\Big)\) 
−  Such a rule can be extended to the cases of other fixed points 
+  Such a rule can be extended to the cases of other fixed points \(~L~\), performing the corresponding transform of the transfer function from \(~T~\) to \(~t~\), 
assuming that 
assuming that 

−  (6) 
+  (6) \(~ ~ ~ t(z)=T(z\!+\!L)L\) 
−  and treating 
+  and treating \(~t~\) as a new transfer function with fixed point zero. 
−  After to construct iterates of function 
+  After to construct iterates of function \(~t~\), regular at zero, the \(~r\)th iteration of function \(~T~\), regular at \(~L~\), can be expressed as 
−  (7) 
+  (7) \(~ ~ ~ T^n(z)=t^n(z\!\!L)+L\) 
−  If the transfer function 
+  If the transfer function \(~T~\) has more than one [[fixed point]]s, then, the noninteger iterate \(~T^n~\), regular at one of these fixed points, has no need to be regular at another [[fixed point]]; another fixed point may be [[branchpoint]] 
+  <ref name="q2"> 

http://www.ams.org/journals/mcom/201079271/S0025571810023422/home.html 
http://www.ams.org/journals/mcom/201079271/S0025571810023422/home.html 

http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf 
http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf 

D.Kouznetsov, H.Trappmann. Portrait of the four regular superexponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.17271756. 
D.Kouznetsov, H.Trappmann. Portrait of the four regular superexponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.17271756. 

</ref>. The noninteger iterates of various functions can be constructed also with the [[superfunction]] and [[Abel function]], considering the [[transfer equation]] instead of the [[zooming equation]] 
</ref>. The noninteger iterates of various functions can be constructed also with the [[superfunction]] and [[Abel function]], considering the [[transfer equation]] instead of the [[zooming equation]] 

−  <ref> 
+  <ref name="logi"> 
http://tori.ils.uec.ac.jp/PAPERS/2010logistie.pdf 
http://tori.ils.uec.ac.jp/PAPERS/2010logistie.pdf 

http://www.springerlink.com/content/u712vtp4122544x4 
http://www.springerlink.com/content/u712vtp4122544x4 

D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.9198. (Russian version: p.2431) 
D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.9198. (Russian version: p.2431) 

DOI 10.3103/S0027134910020049 
DOI 10.3103/S0027134910020049 

−  </ref><ref> 
+  </ref><ref name="superfae"> 
http://tori.ils.uec.ac.jp/PAPERS/2010superfae.pdf 
http://tori.ils.uec.ac.jp/PAPERS/2010superfae.pdf 

http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 

D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.612. (Russian version: p.814) 
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.612. (Russian version: p.814) 

</ref>. 
</ref>. 

+  
+  ==Inverse problem== 

+  
+  In principle, any holomorphic functions \(f\) and \(g\!=\!f^{1}\), such that \(f(0)=g(0)=0\), for any number \(K\), can be declared to be [[zooming function]] and [[Schroeder function]] 

+  for the [[transfer function]] \(T(z)\!=\! f(K\, g(z))\). 

+  Then, the iterates of such a transfer function is straightforward, \(T^n\!=\! f(K^n\, g(z))\). 

+  
+  The expressions above can be used to construct table of zooming functions and Abel functions. 

+  One take any pair of holomorphic functions \(f\) and \(g\!=\!f^{1}\), calculate \(T(z)\!=\! f(K\, g(z))\). 

+  If the simplification of this function fits the width of the column of the table, it can be declared as "transfer function, for with the zooming function and Schroeder function are constructed", and added to the table, together with corresponding \(f\) and \(g\). 

+  
+  ==Asymptotic expansion== 

+  
+  If function \(T\) is known, and \(T^{1}\) is fixed, but \(f\) and \(g\) are not known, these \(f\) and \(g\) can be constructed thrush their asymptotic expansions at small values of the argument. Assuming that \(T\) is regular at zero, expand is ax follows: 

+  
+  (11) \(~ ~ ~ T(z)=K z + a_2 z^2 + a_3 z^3 + ..\) 

+  
+  Assume, that \(K\ne 0\) and \(K\ne 1\). Then K is the zooming coefficient, and the asymptotic expansion can be searched in the following way: 

+  
+  (12) \(~ ~ ~ f(z)=K z + c_2 z^2 + c_3 z^3+ ..\) 

+  
+  Then the right hand side of the zooming equation expands as follows: 

+  
+  (13) \(~ ~ ~ f(K z)=K^2 z + c_2 K^2 z^2 + c_3 K^3 z^3+ ..\) 

+  
+  and the left hand side expands as follows: <! '''[[Warning! something is wrong in the formulas below! They need to be corrected!]]''' !> 

+  
+  (14) \(~ ~ ~ T(f(z))=K\, (K z + c_2 z^2 + c_3 z^3 + ..) + a_2 \, ( K z + c_2 z^2 + ..)^2+ a_3 \, ( K z + ..)^3+..\) 

+  
+  (15) \(~ ~ ~T(f(z))=K^2 z + c_2 K z^2 + c_3 K z^3 + .. + a_2 K^2 z^2 +2 a_2 K c_2 z^3 + .. + a_3 K^3 z^3 + ..\) 

+  
+  (16) \(~ ~ ~T(f(z))=K^2 z + (c_2 K+a_2 K^2) z^2 +(c_3 K + 2 a_2 c_2 K + a_3 K^3) z^3 + ..\) 

+  
+  Combarison of coefficients at equal powers of \(z\) in (13) and (16) gives set of equations for coefficients \(c\): 

+  
+  (17) \(~ ~ ~ c_2 K^2 = c_2 K+a_2 K^2\) 

+  
+  (18) \(~ ~ ~ c_3 K^3 = c_3 K + 2 a_2 c_2 K^3 + a_3 K^3\) 

+  
+  and so on. Some advanced programming language, Mathematica or Maple, are strongly recommended, if one needs to calculate many coefficients \(c\) of the asymptotic expansion (12). For various transfer functions, similar expansion for superfunctions (related with the zooming function) and equivalent iterates are considered in century 21 

+  <ref name="q2"> 

+  http://www.ams.org/journals/mcom/201079271/S0025571810023422/home.html 

+  http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf 

+  D.Kouznetsov, H.Trappmann. Portrait of the four regular superexponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.17271756. 

+  </ref><ref name="logi"> 

+  http://tori.ils.uec.ac.jp/PAPERS/2010logistie.pdf 

+  http://www.springerlink.com/content/u712vtp4122544x4 

+  D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.9198. (Russian version: p.2431) 

+  DOI 10.3103/S0027134910020049 

+  </ref><ref name="superfae"> 

+  http://tori.ils.uec.ac.jp/PAPERS/2010superfae.pdf 

+  http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 

+  D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.612. (Russian version: p.814) 

+  </ref>. 

+  
+  However, first coefficients can be calculated also manually. Equation (17) and (18) can be rewritten as follows: 

+  
+  (19) \(~ ~ ~ c_2 (K\!\!1)=a_2 K\) 

+  
+  (20) \(~ ~ ~ c_3 (K^2\!\!1)= (2a_2c_2a_3)K^2\) 

+  
+  With coefficients \(c\) and expansion (12), for \(K>1\), for some integer \(M>1\), the zooming function can be defined as follows 

+  
+  (21) \(~ ~ ~\displaystyle f_M(z)= K z + \sum_{m=1}^{M} c_m z^b\) 

+  
+  (22) \(~ ~ ~\displaystyle f(z)=\lim_{n\rightarrow \infty} T^M\! \big( f( K^{M} z)\big)\) 

+  
+  Expression under the limit becomes small, where the asymptotic expansion is valid. 

+  
+  ==Generalisation== 

+  
+  The regular iteration with the zooming equation above is useful for the transfer function \(T\), regular at its fixed point zero, 

+  while \(T'(0)=K>1\). This does not cover all the cases. 

+  
+  Expression (19) indicates that the expansion fails, if \(K\!=\! 1\), and this case is qualified as [[exotic iteration]]. 

+  For this "exotic" case, the [[superfunction]] and [[abel function]] can be used to calculate the non–integer iterate. 

+  <ref> 

+  http://www.ams.org/journals/mcom/000000000/S002557182012025907/S002557182012025907.pdf<br> 

+  http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf 

+  <! http://mizugadro.mydns.jp/PAPERS/2011e1e.pdf (preprint) 

+  https://bitbucket.org/bo198214/e1e/raw/64beab23fa73/main.pdf !> 

+  H.Trappmann, D.Kouznetsov. Computation of the Two Regular SuperExponentials to base exp(1/e). Mathematics of Computation. Math. Comp., v.81 (2012), p. 22072227. ISSN 10886842(e) ISSN 00255718(p) 

+  </ref><ref name="sin"> 

+  http://www.pphmj.com/references/8246.htm 

+  http://mizugadro.mydns.jp/PAPERS/2014susin.pdf 

+  D.Kouznetsov. Super sin. Far East Jourmal of Mathematical Science, v.85, No.2, 2014, pages 219238. 

+  </ref>. 

+  
+  It may happen, that the real–holomorphic transfer function has no real fixed points. 

+  One of such function is just \(T\!=\!\exp\). Problem of construction of realholomorphic iterates of exponent, in particular, its iterate half, had been reported by Helmuth Kneser in 1950, and in 2011, the solution through the [[Cauchi integral]] and the [[superfunction]] had been suggested. 

+  <ref name="knezer"> http://gdz.sub.unigoettingen.de/dms/load/img/?PPN=GDZPPN002175851&IDDOC=260718 

+  http://tori.ils.uec.ac.jp/PAPERS/Relle.pdf Helmuth Kneser Reelle analytische L¨osungen der Gleichung 

+  \(\varphi(\varphi(x))=e^x\) 

+  und verwandter Funktionalgleichungen Journal fur die reine und angewandte Mathematik 187 (1950) 5667 

+  </ref><ref name="analuxp"> 

+  http://www.ams.org/mcom/200978267/S0025571809021887/home.html <br> 

+  http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf 

+  D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex zplane. Mathematics of Computation 78 (2009), 16471670. 

+  </ref> 

+  
+  En fin, it may happen, that the realholomorphic transfer function has no fixed points at all (neither real, nor complex). 

+  One example of such function is [[Trappmann function]] \(\mathrm{tra}(z)\!=\!z\!+\!\exp(z)~\); in 2013, its noninteger iterates are reported. 

+  <ref> 

+  http://www.mhikari.com/ams/ams2013/ams1291322013/kouznetsovAMS1291322013.pdf <br> 

+  http://mizugadro.mydns.jp/PAPERS/2013hikari.pdf 

+  D.Kouznetsov. Entire function with logarithmic asymptotic. Applied Mathematical Sciences, 2013, v.7, No.131, p.65276541. 

+  </ref>. 

+  
+  The cases mentioned are considered also in the book [[Superfunctions]]. 

==References== 
==References== 
Latest revision as of 18:26, 30 July 2019
Zooming equation is tentative name for the equation
(1) \(~ ~ ~ T\big(f(z)\big)= f( K\, z)~\)
where \(~T~\) is some given function such that \(T(0)\!=\!0\) and \(~K~\) is constant, that depends on \(T\). Usually, it is assumed that \(K\) is positive real number, id est, \(K>0\), and \(T\) is real–holomorphic, at least in some vicinity of zero, and \(~K\!=\!T'(0)~\). Function \(~f~\) is requested to builtup.
The physical sense of function \(~T~\) is transfer function; it describes the variation of value of function \(~f~\) at the scaling, zooming of its argument with factor \(~K~\).
The tentative name for the solution \(~f~\) of the zooming equation is zooming function.
Contents
Schroeder equation
The solution \(~f\) of the zooming equation is related to the solution \(~g~\) of the Schroeder equation
(2) \(~ ~ ~ g\big(T(z)\big)= K \, g(z) \)
it is assumed that \(~f=g^{1}\) and \(~g=f^{1}\), id est, in wide ranges of values of \(z\), the relations below hold:
(3) \(~ ~ ~ f(g(z))=z\)
(4) \(~ ~ ~ g(f(z))=z\)
Regular iteration
Regular iteration is procedure that leads to noninteger iterates of a transfer function, that are regular in some vicinity of its fixed point. Term Regular iteration had been suggested in 1958 by G. Seekers ^{[1]}.
Assuming that zero is fixed point of the transfer function \(~T~\) is holomorphioc (regular) in vicinity of zero, the zooming function \(~f~\) and the Schroeder function \(~g\!=\!f^{1}~\) can be used to construct the non–integer iterate \(~T^n~\), that is regular in vicinity of zero, id est, the regular iterate
(5) \(\displaystyle ~ ~ ~ T^n(z)=f\Big(k^n\, g(z)\Big)\)
Such a rule can be extended to the cases of other fixed points \(~L~\), performing the corresponding transform of the transfer function from \(~T~\) to \(~t~\), assuming that
(6) \(~ ~ ~ t(z)=T(z\!+\!L)L\)
and treating \(~t~\) as a new transfer function with fixed point zero. After to construct iterates of function \(~t~\), regular at zero, the \(~r\)th iteration of function \(~T~\), regular at \(~L~\), can be expressed as
(7) \(~ ~ ~ T^n(z)=t^n(z\!\!L)+L\)
If the transfer function \(~T~\) has more than one fixed points, then, the noninteger iterate \(~T^n~\), regular at one of these fixed points, has no need to be regular at another fixed point; another fixed point may be branchpoint ^{[2]}. The noninteger iterates of various functions can be constructed also with the superfunction and Abel function, considering the transfer equation instead of the zooming equation ^{[3]}^{[4]}.
Inverse problem
In principle, any holomorphic functions \(f\) and \(g\!=\!f^{1}\), such that \(f(0)=g(0)=0\), for any number \(K\), can be declared to be zooming function and Schroeder function for the transfer function \(T(z)\!=\! f(K\, g(z))\). Then, the iterates of such a transfer function is straightforward, \(T^n\!=\! f(K^n\, g(z))\).
The expressions above can be used to construct table of zooming functions and Abel functions. One take any pair of holomorphic functions \(f\) and \(g\!=\!f^{1}\), calculate \(T(z)\!=\! f(K\, g(z))\). If the simplification of this function fits the width of the column of the table, it can be declared as "transfer function, for with the zooming function and Schroeder function are constructed", and added to the table, together with corresponding \(f\) and \(g\).
Asymptotic expansion
If function \(T\) is known, and \(T^{1}\) is fixed, but \(f\) and \(g\) are not known, these \(f\) and \(g\) can be constructed thrush their asymptotic expansions at small values of the argument. Assuming that \(T\) is regular at zero, expand is ax follows:
(11) \(~ ~ ~ T(z)=K z + a_2 z^2 + a_3 z^3 + ..\)
Assume, that \(K\ne 0\) and \(K\ne 1\). Then K is the zooming coefficient, and the asymptotic expansion can be searched in the following way:
(12) \(~ ~ ~ f(z)=K z + c_2 z^2 + c_3 z^3+ ..\)
Then the right hand side of the zooming equation expands as follows:
(13) \(~ ~ ~ f(K z)=K^2 z + c_2 K^2 z^2 + c_3 K^3 z^3+ ..\)
and the left hand side expands as follows:
(14) \(~ ~ ~ T(f(z))=K\, (K z + c_2 z^2 + c_3 z^3 + ..) + a_2 \, ( K z + c_2 z^2 + ..)^2+ a_3 \, ( K z + ..)^3+..\)
(15) \(~ ~ ~T(f(z))=K^2 z + c_2 K z^2 + c_3 K z^3 + .. + a_2 K^2 z^2 +2 a_2 K c_2 z^3 + .. + a_3 K^3 z^3 + ..\)
(16) \(~ ~ ~T(f(z))=K^2 z + (c_2 K+a_2 K^2) z^2 +(c_3 K + 2 a_2 c_2 K + a_3 K^3) z^3 + ..\)
Combarison of coefficients at equal powers of \(z\) in (13) and (16) gives set of equations for coefficients \(c\):
(17) \(~ ~ ~ c_2 K^2 = c_2 K+a_2 K^2\)
(18) \(~ ~ ~ c_3 K^3 = c_3 K + 2 a_2 c_2 K^3 + a_3 K^3\)
and so on. Some advanced programming language, Mathematica or Maple, are strongly recommended, if one needs to calculate many coefficients \(c\) of the asymptotic expansion (12). For various transfer functions, similar expansion for superfunctions (related with the zooming function) and equivalent iterates are considered in century 21 ^{[2]}^{[3]}^{[4]}.
However, first coefficients can be calculated also manually. Equation (17) and (18) can be rewritten as follows:
(19) \(~ ~ ~ c_2 (K\!\!1)=a_2 K\)
(20) \(~ ~ ~ c_3 (K^2\!\!1)= (2a_2c_2a_3)K^2\)
With coefficients \(c\) and expansion (12), for \(K>1\), for some integer \(M>1\), the zooming function can be defined as follows
(21) \(~ ~ ~\displaystyle f_M(z)= K z + \sum_{m=1}^{M} c_m z^b\)
(22) \(~ ~ ~\displaystyle f(z)=\lim_{n\rightarrow \infty} T^M\! \big( f( K^{M} z)\big)\)
Expression under the limit becomes small, where the asymptotic expansion is valid.
Generalisation
The regular iteration with the zooming equation above is useful for the transfer function \(T\), regular at its fixed point zero, while \(T'(0)=K>1\). This does not cover all the cases.
Expression (19) indicates that the expansion fails, if \(K\!=\! 1\), and this case is qualified as exotic iteration. For this "exotic" case, the superfunction and abel function can be used to calculate the non–integer iterate. ^{[5]}^{[6]}.
It may happen, that the real–holomorphic transfer function has no real fixed points. One of such function is just \(T\!=\!\exp\). Problem of construction of realholomorphic iterates of exponent, in particular, its iterate half, had been reported by Helmuth Kneser in 1950, and in 2011, the solution through the Cauchi integral and the superfunction had been suggested. ^{[7]}^{[8]}
En fin, it may happen, that the realholomorphic transfer function has no fixed points at all (neither real, nor complex). One example of such function is Trappmann function \(\mathrm{tra}(z)\!=\!z\!+\!\exp(z)~\); in 2013, its noninteger iterates are reported. ^{[9]}.
The cases mentioned are considered also in the book Superfunctions.
References
 ↑ http://link.springer.com/article/10.1007%2FBF02559539 G. Szekeres. Regular iteration of real and complex functions. Acta Mathematica, September 1958, Volume 100, Issue 3, pp 203258.
 ↑ ^{2.0} ^{2.1} http://www.ams.org/journals/mcom/201079271/S0025571810023422/home.html http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf D.Kouznetsov, H.Trappmann. Portrait of the four regular superexponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.17271756.
 ↑ ^{3.0} ^{3.1} http://tori.ils.uec.ac.jp/PAPERS/2010logistie.pdf http://www.springerlink.com/content/u712vtp4122544x4 D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.9198. (Russian version: p.2431) DOI 10.3103/S0027134910020049
 ↑ ^{4.0} ^{4.1} http://tori.ils.uec.ac.jp/PAPERS/2010superfae.pdf http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.612. (Russian version: p.814)
 ↑
http://www.ams.org/journals/mcom/000000000/S002557182012025907/S002557182012025907.pdf
http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular SuperExponentials to base exp(1/e). Mathematics of Computation. Math. Comp., v.81 (2012), p. 22072227. ISSN 10886842(e) ISSN 00255718(p)  ↑ http://www.pphmj.com/references/8246.htm http://mizugadro.mydns.jp/PAPERS/2014susin.pdf D.Kouznetsov. Super sin. Far East Jourmal of Mathematical Science, v.85, No.2, 2014, pages 219238.
 ↑ http://gdz.sub.unigoettingen.de/dms/load/img/?PPN=GDZPPN002175851&IDDOC=260718 http://tori.ils.uec.ac.jp/PAPERS/Relle.pdf Helmuth Kneser Reelle analytische L¨osungen der Gleichung \(\varphi(\varphi(x))=e^x\) und verwandter Funktionalgleichungen Journal fur die reine und angewandte Mathematik 187 (1950) 5667
 ↑
http://www.ams.org/mcom/200978267/S0025571809021887/home.html
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex zplane. Mathematics of Computation 78 (2009), 16471670.  ↑
http://www.mhikari.com/ams/ams2013/ams1291322013/kouznetsovAMS1291322013.pdf
http://mizugadro.mydns.jp/PAPERS/2013hikari.pdf D.Kouznetsov. Entire function with logarithmic asymptotic. Applied Mathematical Sciences, 2013, v.7, No.131, p.65276541.