Difference between revisions of "Table of superfunctions"

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−
For a given function $T$, called [[transfer function]], the holomorphic solution $F$ of [[Transfer equation]]
 +
For a given function \(T\), called [[transfer function]], the holomorphic solution \(F\) of [[Transfer equation]]
−
: $ \!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ F(z+1)=T(F(z)) $
 +
: \( \!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ F(z+1)=T(F(z)) \)
−
is called [[superfunction]] with respect to $T$.
+
is called [[superfunction]] with respect to \(T\).
   
−
The inverse function, id est, $G=F^{-1}$ is called [[Abel function]] with respect to $T$; it satisfies the [[Abel equation]]
 +
The inverse function, id est, \(G=F^{-1}\) is called [[Abel function]] with respect to \(T\); it satisfies the [[Abel equation]]
−
: $ \!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ G(T(z))=G(z)+1 $
 +
: \( \!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ G(T(z))=G(z)+1 \)
   
−
In any pair of holomorphic functions $F$, $G\!=\!F^{-1}$,
+
In any pair of holomorphic functions \(F\), \(G\!=\!F^{-1}\),
−
function $F$ can be declared as [[superfunction]],
+
function \(F\) can be declared as [[superfunction]],
−
function $G$ can be declared as [[Abel function]], and then the corresponding [[transfer function]]
 +
function \(G\) can be declared as [[Abel function]], and then the corresponding [[transfer function]]
 
can be expressed as follows:
  
can be expressed as follows:
−
: $ \!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ T(z)=F(1+G(z)) $
 +
: \( \!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ T(z)=F(1+G(z)) \)
   
 
The table of examples of superfunctions can be constructed in a way, similar to the construction of a [[table of indefinite integrals]]. (Some special function is differentiated; if the result fits in one line of the table and cannot be easy expressed through other integrands already presented in table, then the result is qualified as new "integrand" and its integral is added to the table.) For superfunctions, any pair, function and its inverse function, are declared as [[superfunction]] and the [[Abel function]]; If the [[Transfer function]] by (3) can be simplified to fit the cell in the table, it is added to the
 
The table of examples of superfunctions can be constructed in a way, similar to the construction of a [[table of indefinite integrals]]. (Some special function is differentiated; if the result fits in one line of the table and cannot be easy expressed through other integrands already presented in table, then the result is qualified as new "integrand" and its integral is added to the table.) For superfunctions, any pair, function and its inverse function, are declared as [[superfunction]] and the [[Abel function]]; If the [[Transfer function]] by (3) can be simplified to fit the cell in the table, it is added to the
Line 23: Line 23:
 
|-
  
|-
 
! |
 
! |
−
! scope="col" | $T(z)~$
 +
! scope="col" | \(T(z)~\)
−
! scope="col" | $F(z)~$
 +
! scope="col" | \(F(z)~\)
−
! scope="col" | $G(z)~$
 +
! scope="col" | \(G(z)~\)
 
! scope="col" | Comments, keywords, refs
  
! scope="col" | Comments, keywords, refs
 
|-
  
|-
 
! | 00
  
! | 00
−
| $~ c$
+
| \(~ c\)
−
| $~ c$
 +
| \(~ c\)
−
| $~$ does not exist
 +
| \(~\) does not exist
 
| [[constant]]
  
| [[constant]]
 
|-
  
|-
 
! | 01
  
! | 01
−
| $~z+1$
+
| \(~z+1\)
−
| $~z+b$
 +
| \(~z+b\)
−
| $~z-b$
 +
| \(~z-b\)
 
| [[Unity increment]], [[addition]], [[substraction]]
  
| [[Unity increment]], [[addition]], [[substraction]]
 
|-
  
|-
 
! | 02
  
! | 02
−
| $~z+b$
+
| \(~z+b\)
−
| $~z\,b$
 +
| \(~z\,b\)
−
| $~z/b$
 +
| \(~z/b\)
−
| $b\! \ne \! 0$, [[addition]], [[multiplication]], [[division]]
 +
| \(b\! \ne \! 0\), [[addition]], [[multiplication]], [[division]]
 
|-
  
|-
 
! | 03
  
! | 03
−
| $~z\, b$
+
| \(~z\, b\)
−
| $~ b^z$
 +
| \(~ b^z\)
−
| $~\log_b(z)$
 +
| \(~\log_b(z)\)
−
| $b\! \ne \! 0$, $b\ne 1 ~$ [[multiplication]], [[exponential|exp]], [[logarithm|log]]
 +
| \(b\! \ne \! 0\), \(b\ne 1 ~\) ; [[multiplication]], [[exponential|exp]], [[logarithm|log]]
 
|-
  
|-
 
! | 03a
  
! | 03a
−
| $~a+b\, z$
+
| \(~a+b\, z\)
−
| $~ A \frac{1-b^z}{1-b}$
 +
| \(\displaystyle ~ a \frac{1-b^z}{1-b}\)
−
| $~\log_b\Big(1+\frac{b-1}{A}z \big)$
 +
| \(\displaystyle ~\log_b\Big(1+\frac{b-1}{a}z \big)\)
−
| $b\! \ne \! 0$, $b\ne 1 ~$ [[logarithm|log]]
 +
| \(a \ne 0 ~\), \(~b\! \ne \! 0\), \(b\ne 1 ~\) ; [[logarithm|log]]
 
|-
  
|-
 
! | 04
  
! | 04
−
| $~b^z$
+
| \(~b^z\)
−
| $~\mathrm{tet}_b(z)$
 +
| \(~\mathrm{tet}_b(z)\)
−
| $~\mathrm{ate}_b(z)$
 +
| \(~\mathrm{ate}_b(z)\)
−
| $b\! > \! 1$, [[exponential|exp]], [[tetration]], [[arctetration]]
 +
| \(b\! > \! 1\), [[exponential|exp]], [[tetration]], [[arctetration]]
 
|-
  
|-
 
! | 05
  
! | 05
−
| $~z^b$
+
| \(~z^b\)
−
| $~\exp(b^z)$
 +
| \(~\exp(b^z)\)
−
| $~\log_b(\ln(z))$
 +
| \(~\log_b(\ln(z))\)
−
| $b\ne \pm 1$ , [[Power function]], [[Logarithm|log]], [[exp]]
+
| \(b\ne \pm 1\) , [[Power function]], [[Logarithm|log]], [[exp]]
  ⚫
|-
  ⚫
! | 050
  +
| \(~c z^{1+r}\)
  +
| \(~ c^{-1/r} \exp\big( (1\!+\!r)^z\, c^{1/r}\big)\)
  +
| \(~ \frac{ \ln\big(c^{-1/r}\, \ln(c^{1/r}\,z\,) \big)}{\ln(1+r)}\)
  +
| \(c\!>\!0\), \(r\!>\!0\) , [[Power function]]
 
|-
  
|-
 
! | 05a
  
! | 05a
−
| $\displaystyle \frac{-a^2}{z}$
+
| \(\displaystyle \frac{-a^2}{z}\)
−
| $~\displaystyle a\, \tan\left(\frac{2}{\pi} z\right)$
 +
| \(~\displaystyle a\, \tan\left(\frac{\pi}{2} z\right)\)
−
| $~\displaystyle \frac{2}{\pi} \arctan(z/a)$
 +
| \(~\displaystyle \frac{2}{\pi} \arctan(z/a)\)
−
| [[Iterate of linear fraction]]
  
⚫
|-
  
⚫
! | 05b
  
⚫
| $\displaystyle \frac{-a^2}{z}$
 
⚫
| $~\displaystyle a\, \tan\Big(\frac{2}{\pi} z\Big)$
  
−
| $~\displaystyle \frac{2}{\pi} \arctan(z/a)$
  
 
| [[Iterate of linear fraction]]
  
| [[Iterate of linear fraction]]
 
|-
  
|-
 
! | 05c
  
! | 05c
−
| $\displaystyle \frac{z}{c+z}$
+
| \(\displaystyle \frac{z}{c+z}\)
−
| $~\displaystyle \frac{1-c}{1-c^z}$
 +
| \(~\displaystyle \frac{1-c}{1-c^z}\)
−
| $~\displaystyle \log_c \Big(1-\frac{1-c}{z}\Big) $
 +
| \(~\displaystyle \log_c \Big(1-\frac{1-c}{z}\Big) \)
−
| [[Iterate of linear fraction]]
 +
| \(c\ne 0\), \(c\ne 1\); [[Iterate of linear fraction]]
  +
|-
  +
! | 05d
  ⚫
| \(\displaystyle \frac{z}{1+z}\)
  +
| \(~1/z\)
  +
| \(~1/z \)
  ⚫
| \(\displaystyle T^n(z)=\frac{z}{1+n z}\)
 
|-
  
|-
 
! | 06
  
! | 06
−
| $~\log_b(z)$
+
| \(~\log_b(z)\)
−
| $~\mathrm{tet}_b(-z)$
 +
| \(~\mathrm{tet}_b(-z)\)
−
| $~-\mathrm{ate}(z)$
 +
| \(~-\mathrm{ate}_b(z)\)
 
| [[ArcTetration]]
 
| [[ArcTetration]]
 
|-
  
|-
 
! | 07
  
! | 07
−
| $~ \ln(b+\mathrm{e}^z)$
+
| \(~ \ln(b+\mathrm{e}^z)\)
−
| $~ \ln(bz)$
 +
| \(~ \ln(bz)\)
−
| $~\mathrm{e}^z/b$
 +
| \(~\mathrm{e}^z/b\)
 
| [[Logarithm]]
 
| [[Logarithm]]
 
|-
  
|-
 
! | 08
  
! | 08
−
| $~(a^b+z^b)^{1/b}$
+
| \(~(a^b+z^b)^{1/b}\)
−
| $~a z^{1/b} $
 +
| \(~a z^{1/b} \)
−
| $~(z/a)^b$
 +
| \(~(z/a)^b\)
 
| [[Exponential]]
 
| [[Exponential]]
 
|-
  
|-
 
! | 09
  
! | 09
−
| $~2 z^2-1 $
+
| \(~2 z^2-1 \)
−
| $~\cos(2^z)$
 +
| \(~\cos(2^z)\)
−
| $~\log_2(\arccos(z))$
 +
| \(~\log_2(\arccos(z))\)
 
| [[cosinus]], [[trigonometric functions]]
  
| [[cosinus]], [[trigonometric functions]]
 
|-
  
|-
 
! | 10
  
! | 10
−
| $~2 z^2-1$
+
| \(~2 z^2-1\)
−
| $~ \cosh(2^z)$
 +
| \(~ \cosh(2^z)\)
−
| $~\log_2(\mathrm{arccosh}(z))$
 +
| \(~\log_2(\mathrm{arccosh}(z))\)
 
| (compare to "09") [[Hyperbolic functions]]
 
| (compare to "09") [[Hyperbolic functions]]
 
|-
  
|-
 
! | 11
  
! | 11
−
| $~ 2 z/(1\!-\!z^2) $
+
| \(~ 2 z/(1\!-\!z^2) \)
−
| $~ \tan(2^z)$
 +
| \(~ \tan(2^z)\)
−
| $~ \log_2(\arctan(z))$
 +
| \(~ \log_2(\arctan(z))\)
 
| [[tangent]]
  
| [[tangent]]
 
|-
  
|-
 
! | 12
  
! | 12
−
| $~ 2z/(1\!+\!z^2)$
+
| \(~ 2z/(1\!+\!z^2)\)
−
| $~ \tanh(2^z)$
 +
| \(~ \tanh(2^z)\)
−
| $~ \log_2\big( 2 \ln\Big( \frac{z+1}{z-1} \Big) \Big)$
 +
| \(~ \log_2\big( 2 \ln\Big( \frac{z+1}{z-1} \Big) \Big)\)
 
| [[Exponential]]
 
| [[Exponential]]
 
|-
  
|-
 
! | 13
  
! | 13
−
| $~ z!$
+
| \(~ z!\)
−
| $~ \mathrm{SuperFactorial}(z)$
 +
| \(~ \mathrm{SuperFactorial}(z)\)
−
| $~ \mathrm{AbelFactorial}(z)$
 +
| \(~ \mathrm{AbelFactorial}(z)\)
 
| [[Factorial]], [[SuperFactorial]], [[AbelFactorial]] <ref name="fac">http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
  
| [[Factorial]], [[SuperFactorial]], [[AbelFactorial]] <ref name="fac">http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
 
</ref>
  
</ref>
 
|-
  
|-
 
! | 14
  
! | 14
−
| $~ u\, z\, (1\!−\!z)$
+
| \(~ u\, z\, (1\!−\!z)\)
−
| $~ \mathrm{LogisticSequence}(z)$
 +
| \(~ \mathrm{LogisticSequence}(z)\)
−
| $~ \mathrm{LogisticSequence}^{-1}(z)$
 +
| \(~ \mathrm{LogisticSequence}^{-1}(z)\)
−
| $u\!=\!\mathrm{const}$, [[Logistic sequence]], <ref name="logi">http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.</ref>
 +
| \(u\!=\!\mathrm{const}\), [[Logistic sequence]], <ref name="logi">http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.</ref>
 
|-
  
|-
 
! | 15
  
! | 15
−
<!-- | $~\mathrm{LambertW}(z \mathrm{e}^{z+t})$ !-->
 +
<!-- | \(~\mathrm{LambertW}(z \mathrm{e}^{z+t})\) !-->
−
| $~ \mathrm{Doya}(z)$
 +
| \(~ \mathrm{Doya}(z)\)
−
| $~ \mathrm{Tania}(z)$
 +
| \(~ \mathrm{Tania}(z)\)
−
| $~ (z+\ln(z)−1)$
 +
| \(~ (z+\ln(z)−1)\)
 
| [[Doya function|Doya]], [[Tania function|Tania]], [[LambertW]], [[WrightOmega]] <ref name="trala">
  
| [[Doya function|Doya]], [[Tania function|Tania]], [[LambertW]], [[WrightOmega]] <ref name="trala">
 
http://link.springer.com/article/10.1007/s10043-013-0058-6 (official version, registration is required)<br>
  
http://link.springer.com/article/10.1007/s10043-013-0058-6 (official version, registration is required)<br>
Line 155: Line 161:
 
|-
  
|-
 
! | 16
  
! | 16
−
| $~ \mathrm{Keller}(z)$
 +
| \(~ \mathrm{Keller}(z)\)
−
| $~ \mathrm{Shoka}(z)$
 +
| \(~ \mathrm{Shoka}(z)\)
−
| $~ \mathrm{ArcShoka}(z)$
 +
| \(~ \mathrm{ArcShoka}(z)\)
 
| [[Keller function|Keller]], [[Shoka function|Shoka]], [[ArcShoka]] <ref name="trala">
  
| [[Keller function|Keller]], [[Shoka function|Shoka]], [[ArcShoka]] <ref name="trala">
 
http://link.springer.com/article/10.1007/s10043-013-0058-6 (official version, registration is required)<br>
  
http://link.springer.com/article/10.1007/s10043-013-0058-6 (official version, registration is required)<br>
Line 165: Line 171:
 
|-
  
|-
 
! | 17
  
! | 17
−
<!--| $\frac{1}{4} \left(4 z-(3 z\!+\!2)
 +
<!--| \(\frac{1}{4} \left(4 z-(3 z\!+\!2)
 
\sin \left(\frac{\pi z}{2}\right)-(z\!+\!1) \cos
  
\sin \left(\frac{\pi z}{2}\right)-(z\!+\!1) \cos
 
\left(\frac{\pi z}{2}\right)-(2 z\!+\!1) \cos
  
\left(\frac{\pi z}{2}\right)-(2 z\!+\!1) \cos
−
(\pi z)+2\right)$!-->
 +
(\pi z)+2\right)\)!-->
−
| $ \begin{array}{c}
+
| \( \begin{array}{c}
 
n/2~ , \mathrm{ ~~if~~ } n/2 \in \mathbb N \\\!
  
n/2~ , \mathrm{ ~~if~~ } n/2 \in \mathbb N \\\!
 
\frac{3n\!+\!1)}{2} ,\mathrm{ ~if~ } \frac{n\!+\!1}{2} \in \mathbb N
  
\frac{3n\!+\!1)}{2} ,\mathrm{ ~if~ } \frac{n\!+\!1}{2} \in \mathbb N
−
\end{array}$
 +
\end{array}\)
−
| $~ \mathrm{SubCollatz}(z)$
 +
| \(~ \mathrm{SubCollatz}(z)\)
−
| $~ \mathrm{ArcSubCollatz}(z)$
 +
| \(~ \mathrm{ArcSubCollatz}(z)\)
 
| [[Holomorphic extension of the Collatz subsequence|Collatz subsequence]]
  
| [[Holomorphic extension of the Collatz subsequence|Collatz subsequence]]
 
|-
  
|-
 
! | 18
  
! | 18
−
| $~\displaystyle \frac{a^2\!+\!2az\!+\!bz}{b-z}$
 +
| \(~\displaystyle \frac{a^2\!+\!2az\!+\!bz}{b-z}\)
−
| $~\displaystyle \frac{az\!+\!b}{1\!-\!z}$
 +
| \(~\displaystyle \frac{az\!+\!b}{1\!-\!z}\)
−
| $~\displaystyle \frac{z\!-\!b}{z\!+\!a}$
 +
| \(~\displaystyle \frac{z\!-\!b}{z\!+\!a}\)
−
| $~\displaystyle T^n(z)=\frac{a^2 n + (b\!+\!a\!+na)z}{a+b-an-nz}$
 +
| \(~\displaystyle T^n(z)=\frac{a^2 n + (b\!+\!a\!+na)z}{a+b-an-nz}\)
 
|-
  
|-
 
! | 19
  
! | 19
−
| $~ \mathrm{zex}(z)\!=\! z \exp(z)$
 +
| \(~ \mathrm{zex}(z)\!=\! z \exp(z)\)
−
| $~ \mathrm{SuZex}(z)$
 +
| \(~ \mathrm{SuZex}(z)\)
−
| $~ \mathrm{AuZex}(z)$
 +
| \(~ \mathrm{AuZex}(z)\)
 
| [[Zex]], [[LambertW]], [[SuZex]], [[AuZex]]
  
| [[Zex]], [[LambertW]], [[SuZex]], [[AuZex]]
 
|-
  
|-
 
! | 20
  
! | 20
−
| $~ \mathrm{tra}(z)\!=\! z\!+\! \exp(z)$
 +
| \(~ \mathrm{tra}(z)\!=\! z\!+\! \exp(z)\)
−
| $~ \ln\Big(\mathrm{SuZex}(z)\Big)$
 +
|\(\mathrm{SuTra}(z)\!=\!\ln\!\big(\mathrm{SuZex}(z)\big)\! \)
−
| $~ \mathrm{AuZex}\Big(\exp(z)\big)$
 +
|\(\mathrm{AuTra}(z)\!=\!\mathrm{AuZex}\! \big(\exp(z)\big)\! \)
−
| [[Trappmann function]], [[SuZex]], [[AuZex]]
 +
| [[Trappmann function]], [[SuTra]], [[AuTra]] <ref>
  +
http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.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.6527-6541.
  +
</ref>
  +
|-
  +
! | 21
  +
| \(~ \sin(z)\)
  +
| \(~ \mathrm{SuSin}(z)\)
  +
| \(~ \mathrm{AuSin}(z)\)
  +
| [[sin]], [[SuSin]], [[AuSin]]
  +
|-
  +
! | 22
  +
| \(\!\mathrm{Nem}_q(z)\!=\!z\!+\!z^3\!+\!qz^4\!\)
  +
| \(~ \mathrm{SuNem}_q(z)\)
  +
| \(~ \mathrm{AuNem}_q(z)\)
  +
| [[Nemtsov function]], [[SuNem]], [[AuNem]]
  +
|-
  +
! |
  +
| \(t(z)=P(T( Q(z)))\)
  +
| \(~ f(z)=P(F(z))\)
  +
| \(~ g(z)=G(Q(z))\)
  +
| \(P(Q(z))\!=\!z\), \(T(z)=F(1\!+\!G(z))\)
 
|}
  
|}
   
 
==Extensions and uniqueness==
  
==Extensions and uniqueness==
−
The table above could be much longer. As it is mentioned, any pair of functions ($F$, $F^{-1}$) $=$ ($F$, $G$)
+
The table above could be much longer. As it is mentioned, any pair of functions (\(F\), \(F^{-1}\)) \(=\) (\(F\), \(G\))
−
can be interpreted as ([[superfunction]], [[Abel function]]) for the [[transfer function]] $T=z\mapsto F(1+F^{-1}(z))$.
 +
can be interpreted as ([[superfunction]], [[Abel function]]) for the [[transfer function]] \(T=z\mapsto F(1+F^{-1}(z))\).
   
−
Also, the functions from the table can be combined: for a holomorphic functions $P$ and $Q=P^{-1}$
+
Also, the functions from the table can be combined: for a holomorphic functions \(P\) and \(Q=P^{-1}\)
 
any line in the table (except the 0th) can be transformed as follows:
  
any line in the table (except the 0th) can be transformed as follows:
−
: $F(z) \longrightarrow P(F(z))~$, $~G(z) \longrightarrow G(Q(z))~$, $~T(z) \longrightarrow P(T(Q(z)))$
 +
: \(F(z) \longrightarrow P(F(z))~\), \(~G(z) \longrightarrow G(Q(z))~\), \(~T(z) \longrightarrow P(T(Q(z)))\)
   
 
In addition, the swap of the arguments is allowed:
 
In addition, the swap of the arguments is allowed:
−
: $F \longrightarrow G~$, $~G \longrightarrow F~$, $~T(z) \longrightarrow G(1\!+\!F(z))$
 +
: \(F \longrightarrow G~\), \(~G \longrightarrow F~\), \(~T(z) \longrightarrow G(1\!+\!F(z))\)
 
The replacement is assumed to be performed in parallel, not sequentially.
  
The replacement is assumed to be performed in parallel, not sequentially.
   
 
The general methods of construction of superfunctions allow to implement the efficient algorithms for solution of the [[transfer equation]] (1) and the [[Abel equation]] (2) even for those transfer functions, that cannot be simply expressed through the [[special function]]s with equation (3).
 
The general methods of construction of superfunctions allow to implement the efficient algorithms for solution of the [[transfer equation]] (1) and the [[Abel equation]] (2) even for those transfer functions, that cannot be simply expressed through the [[special function]]s with equation (3).
   
−
In general, superfunctions are not unique. The new superfunction can be obtained by the translation of the argument; for example, the superfunction in the line "9" of the Table can be obtained from than in line "8" by the imaginary displacement of the argument.
 +
In general, superfunctions are not unique. The new superfunction can be obtained by the translation of the argument; for example, the superfunction in the line "9" of the Table can be obtained from that in line "8" by the imaginary displacement of the argument.
 
Also, the different [[fixed point]]s of a [[transfer function]] can be used in the [[regular iteration]] method. Between the fixed points, the resulting superfunctions may agree with many decimal digits, but the deviation is easy to see for the complex values of the argument
  
Also, the different [[fixed point]]s of a [[transfer function]] can be used in the [[regular iteration]] method. Between the fixed points, the resulting superfunctions may agree with many decimal digits, but the deviation is easy to see for the complex values of the argument
 
<ref name="sqrt2">
  
<ref name="sqrt2">
Line 236: Line 264:
 
[[Shoka function]],
  
[[Shoka function]],
 
[[Superfunction]],
 
[[Superfunction]],
  +
[[Superfunctions]],
 
[[Transfer function]],
  
[[Transfer function]],
 
[[Transfer equation]],
 
[[Transfer equation]],
Line 244: Line 273:
 
[[Category:Abel function]]
  
[[Category:Abel function]]
 
[[Category:Articles in English]]
  
[[Category:Articles in English]]
  +
[[Category:Book]]
 
[[Category:English]]
  
[[Category:English]]
 
[[Category:Iteration]]
  
[[Category:Iteration]]

Latest revision as of 18:26, 30 July 2019

For a given function \(T\), called transfer function, the holomorphic solution \(F\) of Transfer equation

\( \!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ F(z+1)=T(F(z)) \)

is called superfunction with respect to \(T\).

The inverse function, id est, \(G=F^{-1}\) is called Abel function with respect to \(T\); it satisfies the Abel equation

\( \!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ G(T(z))=G(z)+1 \)

In any pair of holomorphic functions \(F\), \(G\!=\!F^{-1}\), function \(F\) can be declared as superfunction, function \(G\) can be declared as Abel function, and then the corresponding transfer function can be expressed as follows:

\( \!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ T(z)=F(1+G(z)) \)

The table of examples of superfunctions can be constructed in a way, similar to the construction of a table of indefinite integrals. (Some special function is differentiated; if the result fits in one line of the table and cannot be easy expressed through other integrands already presented in table, then the result is qualified as new "integrand" and its integral is added to the table.) For superfunctions, any pair, function and its inverse function, are declared as superfunction and the Abel function; If the Transfer function by (3) can be simplified to fit the cell in the table, it is added to the Table of superfunctions. The example of table of superfunctions appeared in 2010 in the Moscow University Physics bulletin [1].

Table

\(T(z)~\) \(F(z)~\) \(G(z)~\) Comments, keywords, refs
00 \(~ c\) \(~ c\) \(~\) does not exist constant
01 \(~z+1\) \(~z+b\) \(~z-b\) Unity increment, addition, substraction
02 \(~z+b\) \(~z\,b\) \(~z/b\) \(b\! \ne \! 0\), addition, multiplication, division
03 \(~z\, b\) \(~ b^z\) \(~\log_b(z)\) \(b\! \ne \! 0\), \(b\ne 1 ~\) ; multiplication, exp, log
03a \(~a+b\, z\) \(\displaystyle ~ a \frac{1-b^z}{1-b}\) \(\displaystyle ~\log_b\Big(1+\frac{b-1}{a}z \big)\) \(a \ne 0 ~\), \(~b\! \ne \! 0\), \(b\ne 1 ~\) ; log
04 \(~b^z\) \(~\mathrm{tet}_b(z)\) \(~\mathrm{ate}_b(z)\) \(b\! > \! 1\), exp, tetration, arctetration
05 \(~z^b\) \(~\exp(b^z)\) \(~\log_b(\ln(z))\) \(b\ne \pm 1\) , Power function, log, exp
050 \(~c z^{1+r}\) \(~ c^{-1/r} \exp\big( (1\!+\!r)^z\, c^{1/r}\big)\) \(~ \frac{ \ln\big(c^{-1/r}\, \ln(c^{1/r}\,z\,) \big)}{\ln(1+r)}\) \(c\!>\!0\), \(r\!>\!0\) , Power function
05a \(\displaystyle \frac{-a^2}{z}\) \(~\displaystyle a\, \tan\left(\frac{\pi}{2} z\right)\) \(~\displaystyle \frac{2}{\pi} \arctan(z/a)\) Iterate of linear fraction
05c \(\displaystyle \frac{z}{c+z}\) \(~\displaystyle \frac{1-c}{1-c^z}\) \(~\displaystyle \log_c \Big(1-\frac{1-c}{z}\Big) \) \(c\ne 0\), \(c\ne 1\); Iterate of linear fraction
05d \(\displaystyle \frac{z}{1+z}\) \(~1/z\) \(~1/z \) \(\displaystyle T^n(z)=\frac{z}{1+n z}\)
06 \(~\log_b(z)\) \(~\mathrm{tet}_b(-z)\) \(~-\mathrm{ate}_b(z)\) ArcTetration
07 \(~ \ln(b+\mathrm{e}^z)\) \(~ \ln(bz)\) \(~\mathrm{e}^z/b\) Logarithm
08 \(~(a^b+z^b)^{1/b}\) \(~a z^{1/b} \) \(~(z/a)^b\) Exponential
09 \(~2 z^2-1 \) \(~\cos(2^z)\) \(~\log_2(\arccos(z))\) cosinus, trigonometric functions
10 \(~2 z^2-1\) \(~ \cosh(2^z)\) \(~\log_2(\mathrm{arccosh}(z))\) (compare to "09") Hyperbolic functions
11 \(~ 2 z/(1\!-\!z^2) \) \(~ \tan(2^z)\) \(~ \log_2(\arctan(z))\) tangent
12 \(~ 2z/(1\!+\!z^2)\) \(~ \tanh(2^z)\) \(~ \log_2\big( 2 \ln\Big( \frac{z+1}{z-1} \Big) \Big)\) Exponential
13 \(~ z!\) \(~ \mathrm{SuperFactorial}(z)\) \(~ \mathrm{AbelFactorial}(z)\) Factorial, SuperFactorial, AbelFactorial [1]
14 \(~ u\, z\, (1\!−\!z)\) \(~ \mathrm{LogisticSequence}(z)\) \(~ \mathrm{LogisticSequence}^{-1}(z)\) \(u\!=\!\mathrm{const}\), Logistic sequence, [2]
15 \(~ \mathrm{Doya}(z)\) \(~ \mathrm{Tania}(z)\) \(~ (z+\ln(z)−1)\) Doya, Tania, LambertW, WrightOmega [3]
16 \(~ \mathrm{Keller}(z)\) \(~ \mathrm{Shoka}(z)\) \(~ \mathrm{ArcShoka}(z)\) Keller, Shoka, ArcShoka [3]
17 \( \begin{array}{c} n/2~ , \mathrm{ ~~if~~ } n/2 \in \mathbb N \\\! \frac{3n\!+\!1)}{2} ,\mathrm{ ~if~ } \frac{n\!+\!1}{2} \in \mathbb N \end{array}\) \(~ \mathrm{SubCollatz}(z)\) \(~ \mathrm{ArcSubCollatz}(z)\) Collatz subsequence
18 \(~\displaystyle \frac{a^2\!+\!2az\!+\!bz}{b-z}\) \(~\displaystyle \frac{az\!+\!b}{1\!-\!z}\) \(~\displaystyle \frac{z\!-\!b}{z\!+\!a}\) \(~\displaystyle T^n(z)=\frac{a^2 n + (b\!+\!a\!+na)z}{a+b-an-nz}\)
19 \(~ \mathrm{zex}(z)\!=\! z \exp(z)\) \(~ \mathrm{SuZex}(z)\) \(~ \mathrm{AuZex}(z)\) Zex, LambertW, SuZex, AuZex
20 \(~ \mathrm{tra}(z)\!=\! z\!+\! \exp(z)\) \(\mathrm{SuTra}(z)\!=\!\ln\!\big(\mathrm{SuZex}(z)\big)\! \) \(\mathrm{AuTra}(z)\!=\!\mathrm{AuZex}\! \big(\exp(z)\big)\! \) Trappmann function, SuTra, AuTra [4]
21 \(~ \sin(z)\) \(~ \mathrm{SuSin}(z)\) \(~ \mathrm{AuSin}(z)\) sin, SuSin, AuSin
22 \(\!\mathrm{Nem}_q(z)\!=\!z\!+\!z^3\!+\!qz^4\!\) \(~ \mathrm{SuNem}_q(z)\) \(~ \mathrm{AuNem}_q(z)\) Nemtsov function, SuNem, AuNem
\(t(z)=P(T( Q(z)))\) \(~ f(z)=P(F(z))\) \(~ g(z)=G(Q(z))\) \(P(Q(z))\!=\!z\), \(T(z)=F(1\!+\!G(z))\)

Extensions and uniqueness

The table above could be much longer. As it is mentioned, any pair of functions (\(F\), \(F^{-1}\)) \(=\) (\(F\), \(G\)) can be interpreted as (superfunction, Abel function) for the transfer function \(T=z\mapsto F(1+F^{-1}(z))\).

Also, the functions from the table can be combined: for a holomorphic functions \(P\) and \(Q=P^{-1}\) any line in the table (except the 0th) can be transformed as follows:

\(F(z) \longrightarrow P(F(z))~\), \(~G(z) \longrightarrow G(Q(z))~\), \(~T(z) \longrightarrow P(T(Q(z)))\)

In addition, the swap of the arguments is allowed:

\(F \longrightarrow G~\), \(~G \longrightarrow F~\), \(~T(z) \longrightarrow G(1\!+\!F(z))\)

The replacement is assumed to be performed in parallel, not sequentially.

The general methods of construction of superfunctions allow to implement the efficient algorithms for solution of the transfer equation (1) and the Abel equation (2) even for those transfer functions, that cannot be simply expressed through the special functions with equation (3).

In general, superfunctions are not unique. The new superfunction can be obtained by the translation of the argument; for example, the superfunction in the line "9" of the Table can be obtained from that in line "8" by the imaginary displacement of the argument. Also, the different fixed points of a transfer function can be used in the regular iteration method. Between the fixed points, the resulting superfunctions may agree with many decimal digits, but the deviation is easy to see for the complex values of the argument [5]. For uniqueness of the superfunction, the specification of its behavior in the complex plane is essential [6]. It seems, that the physically–meaningful superfunctions do not show the exponentical growth in the imaginary direction and can be constructed at the real fixed point with the regular iterations.

From the scientific point of view, it is difficult to say, what is "better" – to have the variety of superfunctions for each case of the life needs, or to have the unique superfunction, applying the strict requirements on its behavior at the complex values of the argument. But anyway, it is better to know about this variety, while it takes place, and to know about the uniqueness in the cases, when the superfunction and the Abel function are unigue [7].

Referenes

  1. ↑ 1.0 1.1 http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. Cite error: Invalid <ref> tag; name "fac" defined multiple times with different content
  2. ↑ http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
  3. ↑ 3.0 3.1 http://link.springer.com/article/10.1007/s10043-013-0058-6 (official version, registration is required)
    http://mizugadro.mydns.jp/PAPERS/2013or1.pdf (single column version with links for online reading)
    D.Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326.
  4. ↑ http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf
    http://mizugadro.mydns.jp/PAPERS/2013hikari.pdf D.Kouznetsov. Entire function with logarithmic asymptotic. Applied Mathematical Sciences, 2013, v.7, No.131, p.6527-6541.
  5. ↑ http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
  6. ↑ http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html D.Kouznetsov. Solutions of F(z+1)=exp(F(z)) in the complex z-plane. Mathematics of Computation, 78 p.1647-1670 (2009)
  7. ↑ http://www.springerlink.com/content/u7327836m2850246/ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)

Keywords: Abel Function, ArcTetration, Doya function, Iterate of linear fraction, Keller function, Shoka function, Superfunction, Superfunctions, Transfer function, Transfer equation, Tetration, Tania function,

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