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• Factorial
  ===Definition through the integral=== Usually, the integral representation is used as definition. For <math>\Re(z)>-1</math>, define
  27 KB (3,925 words) - 18:26, 30 July 2019
• Tetration
  The substitution of into equation \(~ F(z\!+\!1)\!=\!\exp(F(z)/\mathrm e)~\) where \(x_0\) is solution of equation \(F(x_0)=1\).
  21 KB (3,175 words) - 23:37, 2 May 2021
• Superfunction
  ...cians of the same University were not so arrogant and used the symbol of [[integral]] and the [[Moebius surface]] at their logo, see second picture of the figu ...sponding [[Abel function]] <math>\mathcal{X}</math>, satisfying the [[Abel equation]]
  25 KB (3,622 words) - 08:35, 3 May 2021
• Natural pentation
  ...[[natural tetration]] tet; [[natural pentation]] satisfies the [[transfer equation]] In this equation, the [[tetration]] \(\mathrm{tet}\) appears as the [[transfer function]].
  7 KB (1,090 words) - 18:49, 30 July 2019
• SuperFactorial
  Superfactorial satisfies the [[transfer equation]] for integer values of the argument and with relatively simple integral
  18 KB (2,278 words) - 00:03, 29 February 2024
• Abel equation
  '''Abel equation''' is [[functional equation]] that relates some known function (considered as [[transfer function]]) \( ===Transfer equation===
  4 KB (598 words) - 18:26, 30 July 2019
• Proof:Noeter theorem
  can be presented in the form of the integral of divergence: ...at the assumption of the stationary action leads to the [[Lagrange–Euler equation]]
  9 KB (1,358 words) - 18:27, 30 July 2019
• Fourier transform
  '''Fourier transform''' is linear integral transform with the exponential [[kernel]]. If the integral converges, then, function \(B\) is called '''Fourier transform''' of functi
  11 KB (1,501 words) - 18:44, 30 July 2019
• Table of superfunctions
  ...called [[transfer function]], the holomorphic solution \(F\) of [[Transfer equation]] ...is called [[Abel function]] with respect to \(T\); it satisfies the [[Abel equation]]
  11 KB (1,565 words) - 18:26, 30 July 2019
• Natural tetration
  ...hrm {tet}\) or \(\mathrm {tet}_{\mathrm e}\); it satisfies the [[transfer equation]] ...tetration can be constructed for the holomorhix solution \( \varphi\) of equation
  14 KB (1,972 words) - 02:22, 27 June 2020
• Iterated integral
  '''Iterated integral''' of function \(f\) is function \(J^n f\) expressed with iteration of inte ...(and, in particular, fractional) values of \(n\), under condition that the integral converges. In this case, the expression \((x\!-\!t)^{n-1}\) is interpreted
  9 KB (1,321 words) - 18:26, 30 July 2019
• Tetration to Sheldon base
  ...ations about [[tetration]] \(\mathrm{tet}_s\), it satisfies the [[transfer equation]]: ...f the precise evaluation of \(\mathrm{tet}_s\) through the Cauchi integral equation <ref name="moc1">
  5 KB (707 words) - 21:33, 13 July 2020
• Iterated Cauchi
  '''Iterated Cauchi''' is algorithm of iterative solution of the [[Transfer equation]] ==The Cauchi integral==
  6 KB (987 words) - 10:20, 20 July 2020
• ArcSin
  \(f=\arcsin(z)\) is holomorphic solution \(f\) of equation ==Integral representaitons==
  9 KB (982 words) - 18:48, 30 July 2019
• Bessel transform
  The additional factor in the integral representation of the Bessel transform can be eliminated, multiplying each, W.N.Everitt, H.Kalf. The Bessel differential equation and the Hankeltransform.
  8 KB (1,183 words) - 10:21, 20 July 2020
• BesselJ0
  ...n]] of zero order''' and also \(J_0\), is [[entire function]], solution of equation ==Integral representation==
  6 KB (913 words) - 18:25, 30 July 2019
• Discrete Hankel transform
  ...ion in eigenfunctions of the Dirichlet problem for the Bessel differential equation. The final equation by GNU can be written as follows:
  7 KB (1,063 words) - 18:25, 30 July 2019
• BesselK0
  '''BesselK0''' or \(K_0\) is holomorphic function, solution \(f\) of equation The same equation (1) holds also for \(f(z)=\mathrm{BesselY0}(\mathrm i z)\), where [[Bessel
  3 KB (394 words) - 18:26, 30 July 2019
• BesselY0
  solution \(f=f(z)\) of the Bessel equation with integral representation
  3 KB (445 words) - 18:26, 30 July 2019
• Bessel function
  '''Bessel function''' referes to a solution \(f\) of the [[Bessel equation]] Due to singularity of the equation at \(z=0\), the regular solution should have specific behavior. This soluti
  13 KB (1,592 words) - 18:25, 30 July 2019

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