Difference between revisions of "Superfunction"
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| + | #redirect[[Superfunction]] |
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| − | '''Superfunction''' comes from iteration of another function. |
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| − | Roughly, for some function <math>T</math> (which is called [[Transfer function]]) and for some constant <math>t</math>, the superfunction $F$ could be defined with expression |
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| − | |||
| − | $\displaystyle {{F(z)} \atop \,} {= \atop \,} |
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| − | {T^z(t) \atop \,} {= \atop \,} |
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| − | {{\underbrace{T\Big(T\big(... T(t)...\big)\Big)}} \atop {z \mathrm{~evaluations~of~function~}T\! |
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| − | \!\!\!\!\!}}$ |
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| − | |||
| − | then <math>F</math> can be interpreted as superfunction of function <math>T</math>. |
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| − | Such definition is valid only for positive integer <math>z</math>. |
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| − | The most research and appllications around the superfunctions are related with various extensions of superfunction; and analysis of the existence, uniqueness and ways of the evaluation. |
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| − | <!-- In particular, :<math>S(1)=f(t)</math> !--> |
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| − | For simple function <math>T</math>, such as addition of a constant or multiplication by a constant, |
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| − | the superfunction can be expressed in terms of elementary function. |
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| − | <!-- |
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| − | Historically, first non-elementary superfunction considered was super-exponential or [[tetration]], that corresponds to |
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| − | !--> |
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| − | In particular, the [[Ackernann functions]] and [[tetration]] can be interpreted in terms of [[superfunction]]s. |
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| − | |||
| − | ==History== |
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| − | <!--[[Image:QFacQexp.jpg|right|600px|thumb|<math>\sqrt{!}</math> and <math>\sqrt{\exp}</math> in the complex plane]] |
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| − | [[Image:QFacQexp.jpg|right|100px]]!--> |
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| − | [[Image:Sqrt(factorial)LOGOintegralLOGO.jpg|200px|left|thumb|logos of Phys. and Math. depts of [[MSU]]]] |
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| − | [[File:QexpMapT400.jpg|200px|thumb| $u+\mathrm i v=\sqrt{\exp} (x+\mathrm i y)$]] |
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| − | [[File:QfacMapT500a.jpg|200px|thumb| $u+\mathrm i v=\sqrt{!}(x+\mathrm i y)$]] |
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| − | Analysis of superfunctions came from the application to the evaluation of fractional iterations of functions. |
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| − | [[Superfunction]]s and their inverse functions ([[Abel function]]s) allow evaluation of not only minus-first power of a function (inverse function), but also any real and even complex [[iteration]] of the function. Historically, the first function of such kind considered was <math>\sqrt{\exp}~</math> |
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| − | <ref name="kneser"> |
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| − | http://tori.ils.uec.ac.jp/PAPERS/Relle.pdf |
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| − | [[Helmuth Kneser]] |
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| − | Reelle analytische L¨osungen der Gleichung <math>\varphi(\varphi(x)) = e^x </math> und verwandter Funktionalgleichungen |
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| − | [[Journal fur die reine und angewandte Mathematik]] '''187''' (1950) 56-67 |
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| − | |||
| − | </ref>; then, function <math>\sqrt{!~}~</math> was used as logo of the Physics department of the [[Moscow State University]] |
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| − | <ref name="logo">Logo of the Physics Department of the Moscow State University. (In Russian); |
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| − | http://zhurnal.lib.ru/img/g/garik/dubinushka/index.shtml |
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| − | </ref><ref name="kandidov"> |
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| − | V.P.Kandidov. About the time and myself. (In Russian) |
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| − | http://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf: |
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| − | <blockquote>По итогам студенческого голосования победителями оказались значок с изображением |
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| − | рычага, поднимающего Землю, и нынешний с хорошо известной эмблемой в виде |
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| − | корня из факториала, вписанными в букву Ф. Этот значок, созданный студентом |
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| − | кафедры биофизики А.Сарвазяном, привлекал своей простотой и |
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| − | выразительностью. Тогда эмблема этого значка подверглась жесткой критике со |
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| − | стороны руководства факультета, поскольку она не имеет физического смысла, |
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| − | математически абсурдна и идеологически бессодержательна. |
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| − | </blockquote> |
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| − | </ref><ref name="naukai"> |
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| − | 250 anniversary of the Moscow State University. (In Russian) |
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| − | ПЕРВОМУ УНИВЕРСИТЕТУ СТРАНЫ - 250! |
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| − | http://nauka.relis.ru/11/0412/11412002.htm |
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| − | <blockquote> |
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| − | На значке физфака в букву "Ф" вписано стилизованное изображение корня из факториала (√!) - выражение, математического смысла не имеющее. |
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| − | </blockquote> |
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| − | </ref>. The [[complex map]]s of these functions are shown in figures at right. |
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| − | Mathematicians of the same University were not so arrogant and used the symbol of [[integral]] and the [[Moebius surface]] at their logo, see the figure at right. |
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| − | |||
| − | That time, researchers did not have computational facilities for evaluation of such functions, but |
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| − | the <math>\sqrt{\exp}</math> was more lucky than the <math>~\sqrt{!~}~~</math>; at least the existence of [[holomorphic function]] |
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| − | <math>\varphi</math> such that <math>\varphi(\varphi(z))=\exp(z)</math> has been demonstrated in 1950 by [[Helmuth Kneser]] <ref name="kneser"> |
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| − | http://tori.ils.uec.ac.jp/PAPERS/Relle.pdf |
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| − | H.Kneser. |
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| − | Reelle analytische Lösungen der Gleichung <math>\varphi(\varphi(x)) = e^x </math> und verwandter Funktionalgleichungen. |
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| − | Journal fur die reine und angewandte Mathematik |
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| − | '''187''' (1950) |
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| − | </ref>. Actually, for his proof, Kneser had constructed the [[superfunction]] of exp and corresponding [[Abel function]] <math>\mathcal{X}</math>, satisfying the [[Abel equation]] |
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| − | : <math>\mathcal{X}(\exp(z))=\mathcal{X}(z)+1</math> . |
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| − | The [[inverse function]], id est <math>F=\mathcal \chi^{-1}</math> is an [[entire function|entire]] super-exponential, although it is not real at the real axis; it cannot be interpreted as [[tetration]], because the condition <math>F(0)=1</math> cannot be realized for the entire super-exponential. The [[real function|real]] <math>\sqrt{\exp}</math> can be constructed with the [[tetration]] (which is also a superexponential), and the real <math>\sqrt{\rm Factorial}</math> can be constructed with the [[SuperFactorial]]. The plots of <math>\sqrt{\rm Factorial}~</math> and <math>\sqrt{\exp}~</math> in the compex plane are shown in the right hand side figure. |
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| − | |||
| − | ==Extensions== |
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| − | The recurrent formula of the preamble can be written as equations |
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| − | |||
| − | $F(z\!+\!1)=T(F(z)) ~ \forall z\in \mathbb{N} : z>0$ |
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| − | |||
| − | $F(1)=t$ |
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| − | |||
| − | Instead of the last equation, one could write |
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| − | |||
| − | $F(0)=t$ |
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| − | |||
| − | and extend the range of definition of superfunction <math>F</math> to the non-negative integers. Then, one may postulate |
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| − | |||
| − | $F(-1)=T^{-1}(t)$ |
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| − | |||
| − | and extend the range of validity to the negative integer values, at least while the inverse of the Transfer function is holomorphic. |
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| − | For example, |
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| − | |||
| − | $F(-2)=T^{-2}(t)=T^{-1}\Big(T^{-1}(t)\Big)$ |
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| − | |||
| − | and so on. However, the inverse function may happen to be not defined for some values of <math>t</math>. |
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| − | |||
| − | The [[tetration]] is considered as super-function of exponential for some real base <math>b</math>; in this case, |
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| − | |||
| − | $T=\exp_{b}$ |
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| − | |||
| − | then, at $t=1$, |
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| − | |||
| − | $F(-1)=\log_b(1)=0 ~ ~, ~ ~ \mathrm{but}$ |
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| − | |||
| − | $F(-2)=\log_b(0)~ \mathrm{~is~ not~ defined}$ |
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| − | |||
| − | For extension to non-integer values of the argument, superfunction should be defined in different way. |
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| − | |||
| − | ==Definition== |
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| − | For complex numbers <math>~p~</math> and <math>~q~</math>, such that <math>~p~</math> belongs to some connected domain <math>D\subseteq \mathbb{C}</math>,<br> |
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| − | <!-- <math>a \!\mapsto\! b</math> !--> |
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| − | superfunction (from <math>p \mapsto q</math>) of [[holomorphic function]] <math>~T~</math> on domain <math>C \in \mathbb C</math> is |
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| − | function <math> F </math>, [[holomorphic]] on domain <math>D</math>, such that |
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| − | |||
| − | $F(z\!+\!1)=T(F(z)) ~ \forall z\in D : z\!+\!1 \in D$ |
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| − | |||
| − | $F(p)=q$ |
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| − | |||
| − | ==Uniqueness== |
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| − | In general, the super-function is not unique. |
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| − | For a given base function <math>T</math>, from given <math>(p\mapsto q)</math> superfunciton <math>F</math>, another <math>(p \mapsto q)</math> superfunction <math>\tilde F</math> could be constructed as |
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| − | : <math>\tilde F(z)=F(z+\mu(z))</math> |
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| − | where <math>\mu</math> is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that <math> \mu(p)=0 </math>. |
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| − | |||
| − | The modified superfunction may have narrowed range of holomorphism. |
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| − | The narrower is the range of holomorphism, the wider is variety of superfunctions allowed <ref name="walker"> |
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| − | http://www.jstor.org/stable/2938713 |
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| − | P.Walker |
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| − | Infinitely differentiable generalized logarithmic and exponential functions |
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| − | [[Mathematics of computation]], '''196''' (1991), 723-733 |
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| − | </ref>. |
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| − | |||
| − | At large enough range of holomorphism, the super-function is expected to be unique, for each specific |
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| − | specific [[transfer function]] <math>T</math>. In particular, the <math>(C, 0\mapsto 1)</math> super-function of |
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| − | <math>\exp_b</math>, for <math>b>1</math>, is called [[tetration]] and is believed to be unique at least for |
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| − | <math>C= \{ z \in \mathbb{C} ~:~\Re(z)>-2 \}</math>; for the case <math> b>\exp(1/\mathrm{e})</math>, see <ref name="kouznetsov"> |
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| − | http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html |
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| − | (preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf ) |
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| − | D.Kouznetsov. |
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| − | Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. |
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| − | [[Mathematics of Computation]], |
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| − | '''78''' (2009) 1647-1670 |
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| − | <!--10.1090/S0025-5718-09-02188-7!--> |
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| − | </ref>. |
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| − | |||
| − | ==Examples== |
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| − | The short version of the [[table of superfunctions]] is suggested in |
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| − | <ref name="superfactorial"> |
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| − | http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf |
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| − | D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. [[Moscow University Physics Bulletin]], 2010, v.65, No.1, p.6-12. |
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| − | </ref>. A little bit more extended table is avilable at [[TORI]] |
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| − | <ref name="toritable"> |
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| − | http://tori.ils.uec.ac.jp/TORI/index.php/Table_of_superfunctions |
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| − | </ref>. |
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| − | Some superfunctions can be expressed with elementary functions, |
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| − | they are used without to mention that they are superfunctions. |
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| − | For example, for the transfer function "++", which means unity increment, |
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| − | the superfunction is just addition of a constant. |
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| − | |||
| − | ===Addition=== |
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| − | Chose a [[complex number]] <math>c</math> and define function |
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| − | <math>\mathrm{add}_c</math> with relation |
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| − | <math>\mathrm{add}_c(z)=c\!+\!z ~ \forall z \in \mathbb{C}</math> |
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| − | <!-- , where <math>c\in \mathbb{C}</math> is constant.!-->. |
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| − | Define function |
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| − | <math>\mathrm{mul_c}</math> with relation |
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| − | <math>\mathrm{mul_c}(z)=c\!\cdot\! z ~ \forall z \in \mathbb{C}</math>. |
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| − | |||
| − | Then, function <math>~\mathrm{mul_c}~</math> is '''superfunction''' (<math>~0</math> to <math>~ c~</math>) |
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| − | of function <math>~\mathrm{add_c}~</math> on <math>~\mathbb{C}~</math>. |
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| − | |||
| − | ===Multiplication=== |
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| − | Exponentiation <math>\exp_c</math> is superfunction (from 1 to <math>c</math>) of function <math>\mathrm{mul}_c </math>. |
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| − | |||
| − | ===Quadratic polynomials=== |
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| − | Let the transfer funciton $T$ be defined with $T(z)=2 z^2-1$. |
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| − | Then, |
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| − | $F(z)=\cos( \pi \cdot 2^z) $ is a |
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| − | $(\mathbb{C},~ 0\! \rightarrow\! 1)$ superfunction of $T$. |
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| − | |||
| − | Indeed, |
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| − | |||
| − | $F(z\!+\!1)=\cos(2 \pi \cdot 2^z)=2\cos(\pi \cdot 2^z)^2 -1 =T(F(z))$ |
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| − | |||
| − | and |
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| − | |||
| − | $F(0)=\cos(2\pi)=1$ |
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| − | |||
| − | In this case, the superfunction $F$ is periodic; its period |
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| − | |||
| − | $\tau=\frac{2\pi}{\ln(2)} \mathrm{i}\approx 9.0647202836543876194 \!~i $ |
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| − | |||
| − | and the superfunction approaches unity also in the negative direction of the real axis, |
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| − | |||
| − | $\lim_{x\rightarrow -\infty} F(x)=1$ |
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| − | |||
| − | The example above and the two examples below are suggested at |
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| − | <ref name="mueller">Mueller. Problems in Mathematics. http://www.math.tu-berlin.de/~mueller/projects.html </ref> |
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| − | |||
| − | In general, the transfer function $T$ has no need to be [[entire function]]. |
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| − | Here is the example with [[meromorphic function]] $T$. |
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| − | Let |
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| − | |||
| − | $T(z)=\frac{2z}{1-z^2} ~ \forall z\in D~$; $~ D=\mathbb{C} \backslash \{-1,1\}$ |
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| − | |||
| − | Then, function |
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| − | |||
| − | $F(z)=\tan(\pi 2^z)$ |
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| − | |||
| − | is $(C, 0\! \mapsto\! 0)$ superfunction of function $T$, where |
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| − | $C$ is the set of complex numbers except singularities of function $F$. |
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| − | For the proof, the trigonometric formula |
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| − | $\displaystyle |
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| − | \tan(2 \alpha)=\frac{2 \tan(\alpha)}{1-\tan(\alpha)^2}~~ |
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| − | \forall \alpha \in \mathbb{C} \backslash \{\alpha\in \mathbb{C} : \cos(\alpha)=0 || \sin(\alpha)=\pm \cos(\alpha) \} |
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| − | $ |
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| − | can be used at <math>\alpha=\pi 2^z </math>, that gives |
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| − | |||
| − | $\displaystyle |
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| − | T(F(z))=\frac{2 \tan(\pi 2^z)}{1-\tan(\pi 2^z)} = \tan(2 \pi 2^z)=F(z+1)$ |
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| − | <!-- However, such function <math>F(z)</math> allows the holomrphic extension to values, where <math>cos(\pi 2^z)=0</math>, |
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| − | setting it to zero in these points, but it has singularities, poles, at <math>2^z=\frac{1+2n}{4}</math> for integer <math>n</math>. |
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| − | !--> |
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| − | |||
| − | ===Algebraic function=== |
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| − | In the similar way one can consider the transfer function |
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| − | |||
| − | $T(z)=2z \sqrt{1-z^2}$ |
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| − | |||
| − | and |
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| − | |||
| − | $F(z)=\sin(\pi 2^z)$ |
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| − | |||
| − | which is $(C,~ 0\!\rightarrow \!0)$ superfunction of $H$ for |
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| − | $C= \{z\in \mathbb C : \Re( \cos(\pi 2^z))>0 \}$. |
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| − | |||
| − | ===Exponentiation=== |
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| − | Let |
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| − | <math>b>1</math>, |
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| − | <math>T(z)= \exp_b(z)</math>, |
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| − | <math> C= \{ z \in \mathbb{C} : \Re(z)>-2 \}</math>. |
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| − | Then, [[tetration]] <math> \mathrm{tet}_b </math> |
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| − | is a <math>(C,~ 0\! \rightarrow\! 1)</math> superfunction of <math>\exp_b</math>. |
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| − | |||
| − | ==Abel function== |
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| − | Inverse of superfunction $G=F^{-1}$ is called the [[Abel function]]; within some domain, it satisfies the Abel equation |
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| − | |||
| − | $T(G(z))=G(z)+1$ |
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| − | |||
| − | ==Applications of superfunctions and the Abel functions== |
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| − | |||
| − | Superfunctions, usially the [[tetration|superexponential]]s, are proposed as a fast-growing function for an |
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| − | upgrade of the [[floating point]] representation of numbers in computers. Such an upgrade would greatly extend the |
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| − | range of huge numbers which are still distinguishable from infinity. |
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| − | |||
| − | Other applications refer to the calculation of fractional iterates |
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| − | (or fractional power) of a function. Any holomorphic function can be declared as a "transfer function", then its superfunctions and |
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| − | corresponding Abel functions can be considered. |
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| − | ===Nest=== |
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| − | The $c$th iteration of some function $f$ can be expressed through the [[superfunction]] $F$ and the [[Abel function]] $G=F^{-1}$: |
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| − | :$ T^c(z)=F(c+G(z))$ |
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| − | In [[Mathematica]], there already exist the special name for the operation, that could evaluate such a expression. It is called [[Nest]]. |
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| − | <ref name="nest"> |
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| − | http://reference.wolfram.com/mathematica/ref/Nest.html |
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| − | </ref>. This function has 3 arguments. |
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| − | The first argument indicates the name of the function. |
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| − | The second argument indicates the initial value. |
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| − | The third (and last) argument indicates the number of iterations. |
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| − | Then, the iteration of function $T$ can be written as follows: |
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| − | :$ T^c(z)=\mathrm {Nest}[T,z,c]$ |
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| − | |||
| − | Unfortunately, in the version "Mathematica 8", the implementation of the [[Nest]] has serious restrictions: the number of iterations should allow the simplification to an integer constant. |
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| − | The intents to call function [[Nest]] with any other expression as the last argument cause the error messages. One may hope, in the future versions of Mathematica this bug will be corrected. |
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| − | |||
| − | The [[Table of superfunctions|table of known superfunctions]] and the corresponding [[Abel function]]s |
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| − | (similar to that suggested in <ref name="superfactorial"> |
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| − | http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf |
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| − | D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. [[Moscow University Physics Bulletin]], 2010, v.65, No.1, p.6-12. |
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| − | </ref>) |
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| − | could be loaded in Mathematica in a manned similar to that the [[table of integrals]] is loaded. This would allow the correct implementation of [[Nest]] for the case of non–integer number of iterations. |
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| − | |||
| − | ===Transition from a function to its inverse function=== |
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| − | [[Image:Expc.jpg|200px|left|thumb|<math>\exp^c(x)</math> versus <math>x</math> for various <math>c</math>]] |
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| − | [[Image:Sqrt(exp)(z).jpg|600px|right|thumb|<math>\exp^c</math> in the complex plane for various <math>c</math>]] |
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| − | |||
| − | A [[superfunction]] <math>S</math> allows to calculate the fractional [[iteration]] <math>H^c</math> of some transfer function <math>H</math>. Once the superfunction <math>S</math> and the [[Abel function]] <math>A=S^{-1}</math> are established, |
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| − | the fractional iteration can be defined as |
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| − | <math>H^c(z)=S(c+A(z))</math>. Then, as <math>c</math> changes from 1 to <math>-1</math>, the holomorphic transition from function <math>H</math> to <math>H^{-1}</math> is relalised. The figure at left shows an example of transition from |
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| − | <math>\exp^{1}\!=\!\exp </math> to |
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| − | <math>\exp^{\!-1}\!=\!\ln </math>. |
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| − | Function <math>\exp^c</math> versus real argument is plotted for |
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| − | <math>c=2,1,0.9, 0.5, 0.1, -0.1,-0.5, -0.9, -1,-2</math>. The [[tetration]]al and ArcTetrational were used as superfunction |
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| − | <math>F</math> |
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| − | and Abel function <math>G</math> of the exponential. |
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| − | The figure at right shows these functions in the [[complex plane]]. |
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| − | At non-negative integer number of [[iteration]], the iterated exponential is [[entire function]]; at non-integer values, it has two [[branch points]], thich correspond to the [[fixed points]] <math>L</math> and |
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| − | <math>L^*</math> of natural logarithm. At <math>c\!\ge\! 0</math>, function <math>\exp^c(z)</math> remains [[holomorphic function|holomorphic]] at least in the strip <math>|\Im(z)|<\Im(L)\approx 1.3 </math> along the real axis. |
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| − | |||
| − | ===Nonlinear Optics=== |
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| − | In the investigation of the nonlinear response of optical materials, |
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| − | the sample is supposed to be optically thin, in such a way, |
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| − | that the intensity of the light does not change much as it goes through. |
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| − | Then one can consider, for example, the absorption as function of the intensity. |
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| − | However, at small variation of the intensity in the sample, |
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| − | the precision of measurement of the absorption as function of intensity is not good. |
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| − | The reconstruction of the superfunction from the Transfer Function allows to work with |
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| − | relatively thick samples, improving the precision of measurements. In particular, the |
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| − | Transfer Function of the similar sample, which is half thiner, |
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| − | could be interpreted as the square root (id est, half-iteration) of the Transfer Function of the initial sample. |
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| − | |||
| − | Similar example is suggested for a nonlinear optical fiber <ref name="kouznetsov"> |
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| − | http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html |
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| − | (preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf ) |
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| − | D.Kouznetsov. |
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| − | Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. |
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| − | [[Mathematics of Computation]], '''78''' (2009) 1647-1670. |
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| − | <!--doi=10.1090/S0025-5718-09-02188-7!--> |
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| − | </ref>. |
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| − | In particular, the [[Tania function]] can be evaluated using the [[Doya function]] and the [[regular iteration]] |
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| − | <ref name="2013or"> |
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| − | http://link.springer.com/article/10.1007/s10043-013-0058-6 |
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| − | Dmitrii Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326. |
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| − | Preprint: http://mizugadro.mydns.jp/PAPERS/2013orSuper.pdf |
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| − | </ref><!--<ref name="sinapo"> http://mizugadro.mydns.jp/PAPERS/2011singapo.pdf |
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| − | D.Kouznetsov. Transfer function of an amplifier and characterization of Materials. Singapore, 2011. (slideshow)</ref>!--> |
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| − | at the fixed point $0$, as $\mathrm{Doya}_t(0)\!=\!0$. The same method can be applied also to other [[transfer funciton]]s, even if they cannot be easy represented through the [[special function]]s. This may refer, for example, to the experimentally–measured transfer function, that has no need to coincide with the [[Doya function]]. |
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| − | |||
| − | ===Nonlinear Acoustics=== |
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| − | It may have sense to characterize the nonlinearities in the |
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| − | attenuation of shock waves in a homogeneous tube. This could find an application in some |
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| − | advanced muffler, using nonlinear acoustic effects to withdraw the energy of the sound waves |
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| − | without to disturb the flux of the gas. Again, the analysis of the nonlinear response, |
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| − | id est, the Transfer Function, |
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| − | may be boosted with the superfunction. |
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| − | |||
| − | ===Vaporization and condensation=== |
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| − | <!--For the separation of isotopes due to the different pressure of the saturated vapor for different components,!--> |
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| − | In analysis of condensation, the growth (or vaporization) of a small drop of liquid can be considered, |
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| − | as it diffuses down through a tube with some uniform concentration of vapor. |
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| − | In the first approximation, at fixed concentration of the vapor, |
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| − | the mass of the drop at the output end can be interpreted as the |
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| − | Transfer Function of the input mass. |
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| − | The square root of this Transfer Function will characterize the tube of half length. |
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| − | |||
| − | ===Snow avalanche=== |
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| − | The mass of a snowball, that rolls down from the hill, |
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| − | can be considered as a function of the path it already have passed. At fixed length of this path |
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| − | (that can be determined by the altitude of the hill) this mass can be considered also as a Transfer Function of the input mass. The mass of the snowball could be measured at the top of the hill and at thе bottom, giving the Transfer Function; then, the mass of the snowball as a function of the length it passed is superfunction. |
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| − | |||
| − | ===Operational element=== |
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| − | If one needs to build-up an operational element with some given transfer function <math>H</math>, |
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| − | and wants to realize it as a sequential connection of a couple of identical operational elements, then, each of these two elements should have transfer function |
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| − | <math> h=\sqrt{T}</math>. Such a function can be evaluated through the superfunction and the Abel function of the transfer function <math>T</math>. |
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| − | |||
| − | The operational element may have any origin: it can be realized as an electronic microchip, |
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| − | or a mechanical couple of curvilinear grains), or some asymmetric U-tube filled with different liquids, and so on. |
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| − | |||
| − | ==References== |
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| − | <references/> |
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| − | |||
| − | ==Keywords== |
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| − | [[Abel function]], |
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| − | [[Iteration]], |
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| − | [[Regular iteration]], |
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| − | [[SuperFactorial]], |
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| − | [[Transfer function]], |
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| − | [[Transfer equation]], |
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| − | [[Tetration]], |
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| − | [[Tania function]], |
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| − | [[Tania function]], |
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| − | [[Shoka function]], |
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| − | [[SuZex]], |
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| − | [[SuTra]] |
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| − | |||
| − | ===Additional links=== |
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| − | |||
| − | http://www.proofwiki.org/wiki/Definition:Superfunction |
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| − | |||
| − | http://en.citizendium.org/wiki/Superfunction |
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| − | |||
| − | [[Category:Articles in English]] |
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| − | [[Category:Iterate]] |
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| − | [[Category:Iteration]] |
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| − | [[Category:Mathematics]] |
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| − | [[Category:Superfunction]] |
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Revision as of 07:06, 1 December 2018
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