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• Tetration
  and the asymptotic analysis with small parameter \(|1/z|\) determines ...exp(1/{\rm e})\), \(z\!\approx 0\), then it is expandable into the Taylor series
  21 KB (3,175 words) - 23:37, 2 May 2021
• Regular Iteration
  I.N.Baker. Permutable power series and regular iterate. (2006) ...fixpoint $a$ is called regular, iff $\phi$ is analytic at $a$ or has an [[asymptotic powerseries development]] at $a$.
  20 KB (3,010 words) - 18:11, 11 June 2022
• SuperFactorial
  ...p(k)\). The expansion of the left hand side of equation (1) into the power series with respect to \(\varepsilon\) and equalizing to zero the coefficients giv ...get the required precision of evaluation of \(F\). <!--with the asymptotic series.!-->
  18 KB (2,278 words) - 00:03, 29 February 2024
• Mathematica
  ...orial]], [[Abel Factorial]], etc.; of order of a dozen coefficients of the asymptotic expansion can be calculated analytically in the [[real time]]. (If manually As any other software, Mathematica has bugs. One of them is related to the asymptotic expansion of the [[Bessel function]] of non-trivial argument.
  12 KB (1,901 words) - 18:43, 30 July 2019
• Tania function
  At \(|z|\gg 1\), the truncated series (7) can be used for the evaluation of \(\mathrm{Tania}(z)\), while the argu For the evaluation, the asymptotic can be written also as follows:
  27 KB (4,071 words) - 18:29, 16 July 2020
• Doya function
  ...ction could be implemented as the appropriate combination of the truncated series. The substitution of the series to the Transfer Equation gives value of increment \(k=\exp(t)\) and the coe
  19 KB (2,778 words) - 10:05, 1 May 2021
• Iteration
  W.Bergweiler. Iteration of meromorphic functions. Bulletin (New Series) of the American Mathematical society, v.29, No.2 (1993) p.151-188. D.Kouznetsov. Entire function with logarithmic asymptotic. Applied Mathematical Sciences, 2013, v.7, No.131, p.6527-6541.
  14 KB (2,203 words) - 06:36, 20 July 2020
• Coshc
  ==Asymptotic expansions== the series converges in the whole compelex plane except zero.
  4 KB (509 words) - 18:26, 30 July 2019
• ArcCosc
  In the figure with the [[explicit plot]] of acosc, it is compared with its asymptotic denoted with name \(\mathrm{Left}_3\). expansion. The radius of convergence of this series, id est, \(- \mathrm{Tarao}\), is determined by the distance to the closest
  8 KB (1,137 words) - 18:27, 30 July 2019
• Acosc1
  ...|thumb| Acosc1(\(x\)) versus \(x\), red; [[ArcCosc]](\(x\)), blue and some asymptotic]] ...hm; however, the accurate handling of the branches is required to use this asymptotic for the evaluation.
  6 KB (896 words) - 18:26, 30 July 2019
• BesselJ0
  The series converges in the whole complex plane and, at the complex(double) arithmetic sL[x_] = Normal[Series[Log[HankelH1[0, x] Sqrt[Pi I x/2]], {x, Infinity, 16}]]
  6 KB (913 words) - 18:25, 30 July 2019
• BesselJ1
  \(J_1\) is [[entire function]], the series ...complex(double) arithmetics is available. For large values of \(|z|\), the asymptotic expansion can be used for the precise evaluation:
  3 KB (439 words) - 18:26, 30 July 2019
• Bessel function
  with the following asymptotic behaviour: Table of Integrals, Series, And Products.
  13 KB (1,592 words) - 18:25, 30 July 2019
• BesselH0
  ==Asymptotic expansions== TeXForm[Expand[Series[(HankelH1[0, x]) (Pi I x/2)^(1/2), {x, Infinity, 5}]]]
  4 KB (509 words) - 18:26, 30 July 2019
• Shoko function
  The [[Shoko function]] has series of branchpoints at \(r+\pi(1\!+\!2 n) \mathrm i\) for integer values of \(n and the series of cutlines \(r+\pi(1\!+\!2 n)\mathrm i + t\) for integer values of \(n\)
  10 KB (1,507 words) - 18:25, 30 July 2019
• SuZex
  The first two terms of the asymptotic expansion of [[SuZex]] can be used as the definition. ==Asymptotic expansion==
  7 KB (1,076 words) - 18:25, 30 July 2019
• SuZex approximation
  [[File:SuZexoMapJPG.jpg|600px|thumb|Fig.4. Map od the Asymptotic approximation \(Q_{20}~\) by equation (\(~\)); \(~u\!+\!\mathrm i v= Q_{20 The simple approximation of any function is, perhaps, the truncated Taylor series, which is, actually, a polynomial. The [[complex map]] of such a polynomial
  14 KB (2,037 words) - 18:25, 30 July 2019
• AuZex
  ...along the real axis. For this reason, namely this fixed point is chosen as asymptotic value of [[SuZex]] at minus infinity. Correspondently, its inverse function The series does not seem to converge; the more terms are taken into account, the narro
  6 KB (899 words) - 18:44, 30 July 2019
• Power function
  and the asymptotic analysis. For \(a_1\!=\!1\), these coefficients can be calculated with the q[m] = Coefficient[Series[f1[m, x] - T[f[m, x]], {x, 0, m}], x^m]
  15 KB (2,495 words) - 18:43, 30 July 2019
• SuTra
  D.Kouznetsov. Entire Function with Logarithmic Asymptotic. ...(z)|>\varepsilon\), \( |z| \rightarrow \infty\), [[SuTra]] has logarithmic asymptotic behavior
  9 KB (1,285 words) - 18:25, 30 July 2019

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