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- | 2 || 2 :<math> n! = n \cdot (n-1) \cdots 2 \cdot 1 , \,</math>27 KB (3,925 words) - 18:26, 30 July 2019
- ...\(b\!>\!1\), is holomorphic at least in \(\{ z \in \mathbb C : \Re(z)\!>\!-2\}\). http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>21 KB (3,175 words) - 23:37, 2 May 2021
- // for \(-2<x<2\) and \(1<b<5\) the \(x\),\(b\) coordinates shown with lines \(f=\)const.<b z_type old0(z_type d){ z_type q=sqrt(d); return -1.00186 KB (1,030 words) - 18:48, 30 July 2019
- { -7.5, 81.4, 6.2, 2}, { -39.0, 80.0, 6.2, 3},3 KB (564 words) - 18:33, 28 April 2023
- http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html ...H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.14 KB (2,275 words) - 18:25, 30 July 2019
- [[Image:Sqrt(factorial)LOGOintegralLOGO.jpg|200px]]<small> [[File:QexpMapT400.jpg|200px]]<br><small><center> \(u+\mathrm i v=\sqrt{\exp} (x+\mathrm i y)\)</center></small>25 KB (3,622 words) - 08:35, 3 May 2021
- 2. <i>Постепенное изменение генотипа из 48 хр ...ец. Заседание Президиума. 'Наука Урала' No. 2 (830), январь 2003.111 KB (2,581 words) - 16:54, 17 June 2020
- C3=2. /((1.-Q)*(1.-Q2) C5=2.*(7.+Q*(3.+Q*2.)) /((1.-Q)*(1.-Q2)*(1.-Q3)*(1.-Q43 KB (513 words) - 18:48, 30 July 2019
- ...le:SquareRootOfFactorial.png|400px|right|thumb| \(y\!=\! x!\) and \(y\!=\!\sqrt{!\,}(x)\) verus \(x\)]] [[Square root of factorial]] (half-iteration of [[Factorial]]), or \(\sqrt{!\,}\) is solution \(h\) of equation \(h(h(z))=z!\).13 KB (1,766 words) - 18:43, 30 July 2019
- : \(\mathrm{ArcFactorial}(2)=2\) ...aystyle \mathrm{ArcFactorial}\left( \frac{\sqrt{\pi}}{2}\right)\!=\frac{1}{2}\)3 KB (414 words) - 18:26, 30 July 2019
- n/2 , \mathrm{ ~~if~~ } n/2 \in \mathbb N \\ (3n\!+\!1)/2 ~ \mathrm{~~over-vice}5 KB (798 words) - 18:25, 30 July 2019
- http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html ...H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.20 KB (3,010 words) - 18:11, 11 June 2022
- ..." is [[superfunction]] of [[factorial]] constructed at its [[fixed point]] 2. The smallest integer larger than 2 (id est 3) is chosen as its value at zero, \(\mathrm{SuperFactorial}(0)=3~\18 KB (2,278 words) - 00:03, 29 February 2024
- : \(T^2(z)=T(T(z))\) ...H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1754 KB (547 words) - 23:16, 24 August 2020
- http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html<br> ...H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.11 KB (1,644 words) - 06:33, 20 July 2020
- :\(\sqrt{!\,}(z)=\mathrm{Nest}[\mathrm{Factorial},z,1/2]\) ...n function Nest is usable; for example, expression \(~\mathrm{Nest}[\sin,z,2]~\) returns \(~\sin(\sin(z))~\).3 KB (438 words) - 18:25, 30 July 2019
- ...an allows the nation to avoid big conflicts at least since the [[World War 2]]. Such a culture seems to be somehow related with the language, although t ...of the [[Physics Department of the Moscow State University]]), id est, \(\sqrt{!\,}\), !-->15 KB (2,106 words) - 13:37, 5 December 2020
- ::\( \displaystyle \lambda\left(x\right)=\frac{h}{\sqrt{2m\big(E-V(x)\big)}} \) ...suddenly switched-off. Then, the speed of atoms is determined as <math>~v=\sqrt{2gh}~</math>, where <math>~g~</math> is [[acceleration of free fall]], and16 KB (2,453 words) - 18:26, 30 July 2019
- ...'' of period <math>~L~</math>; this estimate is valid at <math>KL\!~\theta^2\ll 1</math>. See [[quantum reflection]] for the approximation (fit) of the <math>~\displaystyle r \approx \exp\!\left(-\sqrt{8\!~K\!~L}~\theta\right)~</math>, <!--where <math> ~K~</math> is wavenumber6 KB (906 words) - 07:04, 1 December 2018
- For base \( b\!=\!\mathrm e \!\approx\! 2.71 \), the natural ArcTetration is presented in figure at right with the [[ ...special function. [[Vladikavkaz Mathematical Journal]], 2010, v.12, issue 2, p.31-45.7 KB (1,091 words) - 23:03, 30 November 2019
- // for(m=-2;m<0;m+=2) {M(-4.6,m-.2) fprintf(o,"(%1d)s\n",m);} // for(m= 0;m<3;m+=2) {M(-4.4,m-.2) fprintf(o,"(%1d)s\n",m);}3 KB (529 words) - 14:32, 20 June 2013
- ...plify]] does not seem to handle well expressions with imaginary unity , I=\Sqrt[-1] . b = (-1 + Exp[(-2*I)*q - 2*s])*(-1 + Exp[(2*I)*q - 2*s])12 KB (1,901 words) - 18:43, 30 July 2019
- ...ссия, расследовавшая катастрофу [[Катынь-2]] (10 апреля 2010 года), перевела часы на 15 мин ...ноября 2009 года уведомлений о проведении 2 декабря того же года траурных митингов, п22 KB (406 words) - 00:38, 30 October 2020
- \frac{a\!+\!x}{\sqrt{1-(a\!+\!x)^2}}-\frac{a\!-\!x}{\sqrt{1-(a\!-\!x)^2}} :\( \!\!\!\!\! \!\!\!\!\! \!\!\!\!\! \!\!(2) ~ ~ ~ ~ \mathrm{Yulya}_a\!\Big( \mathrm{ArcYulya}_a(z) \Big) = z\)12 KB (1,754 words) - 18:25, 30 July 2019
- : \(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (2) ~ ~ ~ Tania function has two [[branch point]]s: \(~ -\!2\!\pm\! \mathrm i \pi~\). The position of the [[cut line]]s depends on the r27 KB (4,071 words) - 18:29, 16 July 2020
- : \( \displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \mathrm{Doya}(z)= \mathrm{Tania}\!\Big(1+\mathrm{ArcTania}(z)\Big)\) :\(t\!=\!2\), id est, \(~\mathrm{Doya}^2(x)=\mathrm{Doya}\big(\mathrm{Doya}(x)\big)\)19 KB (2,778 words) - 10:05, 1 May 2021
- )))))); DO(n,2) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; } // +x*(.2 +m*(-25./12 +m*(35./6. +m*(-5. +m)))) //reserve term for the testing3 KB (480 words) - 14:33, 20 June 2013
- : \(\!\!\!\!\!\!\!\!\!\!\!\!\!(1) ~ ~ ~ \displaystyle B(x)= \frac{1}{\sqrt{2\pi}} \int_{- \infty}^{\infty} \exp( - \mathrm{i} x y) A(y) ~ \mathrm{d} y \ ...!\!\!\!\!\!\!\!\!\!(2) ~ ~ ~ \displaystyle (\hat ふ A)(x)= \frac{1}{\sqrt{2\pi}} \int_{- \infty}^{\infty} \exp( - \mathrm{i} x y) A(y) ~ \mathrm{d} y11 KB (1,501 words) - 18:44, 30 July 2019
- // n should be 2^m ; o should be 1 or -1 ; q=N/2; p=2; for(m=1;p<N;m++) p*=2;1 KB (238 words) - 14:33, 20 June 2013
- B_k=\sum_{m=0}^{N-1} A_m \exp\Big(-\mathrm{i} \frac{2 \pi}{N}~ k~ m\Big)\) where \(N\) is natural number; usually, \(2^n\) for some natural \(n\);<br>6 KB (1,010 words) - 13:23, 24 December 2020
- : \( \!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ G(T(z))=G(z)+1 \) | \(\displaystyle \frac{-a^2}{z}\)11 KB (1,565 words) - 18:26, 30 July 2019
- [[Square root of exponential]] \(\varphi=\sqrt{\exp}=\exp^{1/2}\) is half-iteration of the [[exponential]], id est, such function that its Function \(\sqrt{\exp}\) should not be confused with5 KB (750 words) - 18:25, 30 July 2019
- ...T.jpg|400px|thumb|Fig.1. Iterates of \(T(z)=z^2~\): \(~y\!=\!T^n(x)\!=\!x^{2^n}~\) for various \(n\)]] [[File:FacIteT.jpg|400px|thumb|Fig.2. Iterates of [[Factorial]]: \(~y\!=\!\mathrm{Factorial~}^{~n}(x)~\) for va14 KB (2,203 words) - 06:36, 20 July 2020
- '''Fourier-2 transform''' is bidimensional [[Fourier transform]] \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! g(x,y)= \frac{1}{2\pi} \) \(\displaystyle\iint \mathrm dp ~\mathrm dq~ \exp(-i p x - i q y ) ~6 KB (954 words) - 18:27, 30 July 2019
- [[File:SquareRootOfFactorial.png|400px|right|thumb| \(y\!=\! x!\) и \(y\!=\!\sqrt{!\,}(x)\) как функции от \(x\)]] ...из факториала ([[Square root of factorial]]), то есть \(\sqrt{\,!\,}\) - голоморфная функкция \(f\) такая, что6 KB (312 words) - 18:33, 30 July 2019
- (id est, \(\sqrt{-\!1}~\) ) in [[Mathematica]] and the [[Identity function]], which is also : \(\!\!\!\!\!\!\!\!\! (2)\displaystyle ~ ~ ~ J^m J^n = J^{m+n}\)9 KB (1,321 words) - 18:26, 30 July 2019
- Символ \(\sqrt{\,!\,}\) установлен в качестве эмблемы Физфа7 KB (381 words) - 18:38, 30 July 2019
- z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t); if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z);2 KB (258 words) - 10:19, 20 July 2020
- ...anchpoint \(z=1/\mathrm e\); the second goes through the point \(z\!=\!\pi/2\), where the fixed points of logarithm are \(\pm \mathrm i\). ...f the cut, the function has real values; in addition, at \(z=\ln\big(\sqrt{2}\big)\), these values are integer4 KB (572 words) - 20:10, 11 August 2020
- :\( \displaystyle \arccos(z)=\frac{\pi}{2} - \arcsin(z)\) : \(\displaystyle \arccos[z]=\frac{\pi}{2}- \mathrm i ~ \mathrm{arccosh}(\mathrm i \, z)\)5 KB (754 words) - 18:47, 30 July 2019
- : \(\displaystyle \mathrm{Cip}(z) = \frac{1}{z} -\frac{z}{2}+\frac{z^3}{24}+\frac{z^5}{720}+...~\), \(~ |z|\!\ll\! 1\) ...\mathrm{Cip}'(z)=0\) has several solutions. One of them is \(z=f_0\approx 2.798386045783887\) .8 KB (1,211 words) - 18:25, 30 July 2019
- 1.01152306812684171, 1.51747364915328740, 2.26948897420495996, 3.00991738325939817, return s + log(2.*M_PI)/2. - z + (z+.5)*log(z);4 KB (487 words) - 07:00, 1 December 2018
- : \(\displaystyle \sin(z) = \frac{\exp(\mathrm i z)- \exp(-\mathrm i z)}{2~ \mathrm i}\) \( \arcsin(z)= -\mathrm i \ln\Big( \mathrm i z + \sqrt{1-z^2} \big)\)9 KB (982 words) - 18:48, 30 July 2019
- : \(\mathrm{Sazae} \approx ~ 2.798386045783887\) \frac{2(z\!-\!\mathrm{Tarao})}8 KB (1,137 words) - 18:27, 30 July 2019
- z_type acoscL(z_type z){ int n; z_type s,q; z*=-I; q=I*sqrt(1.50887956153832-z); z_type acoscB(z_type z){ z_type t=0.33650841691839534+z, u=sqrt(t), s; int n;1 KB (219 words) - 18:46, 30 July 2019
- if(Im(z)<0){if(Re(z)>=0){return I*log( z + sqrt(z*z-1.) );} else{return I*log( z - sqrt(z*z-1.) );}}3 KB (436 words) - 18:47, 30 July 2019
- ..._1 - \sqrt{ \frac{2}{\mathrm{Tarao}_1 } (\mathrm{Tarao}_1\!-\!x) } + \frac{2\, (\mathrm{Tarao}_1 \!-\! x )}{3~ \mathrm{Sazae}_1~ \mathrm{Tarao}_1}\) \sqrt{6 KB (896 words) - 18:26, 30 July 2019
- : \(\mathrm{ArcFactorial}(2)=2\) ...aystyle \mathrm{ArcFactorial}\left( \frac{\sqrt{\pi}}{2}\right)\!=\frac{1}{2}\)3 KB (376 words) - 18:26, 30 July 2019
- \(\mathrm{HankelKernel}(p,x)= \frac{1}{2 pi} \mathrm{BesselJ}_\nu(2 \pi px)\) \(\mathrm{BesselKernel}(p,x)= \mathrm{BesselJ}_\nu(2 \pi px)\)8 KB (1,183 words) - 10:21, 20 July 2020
- ...\!\!\!\! (2) ~ ~ ~ \mathrm i ~ \hbar ~ \dot \Psi = \frac{-\hbar^2 \nabla^2}{2m} \Psi + U(\vec x) ~ \Psi -\mathrm i ~ V(\vec x) ~ \Psi \) : \(\displaystyle \!\!\!\!\!\!\!\!\!\! (7) ~ ~ ~ \omega - (c+\mathrm i s)^2 = -\mathrm i \gamma\)15 KB (2,070 words) - 18:47, 30 July 2019
- : \( \!\!\!\!\!\!\!\!\!\! (2) \displaystyle ~ ~ ~ J_0(0)=1 ~, ~~ J_0'(0)=0\) \frac{ (-z^2/4)^n }6 KB (913 words) - 18:25, 30 July 2019
- : \( \mathrm{Sinc}(z)= 1-\frac{z^2}{6}+\frac{z^4}{120}-\frac{z^6}{5040}+ : \(\!\!\!\!\!\!\!\! \mathrm{ArcSinc}(1\!-\!t)= \sqrt{6 t} \left(4 KB (563 words) - 18:27, 30 July 2019
- f(t) = \sum_{m=1}^\infty (2 J_\nu(j_{\nu,m}t) / J_{\nu+1}(j_{\nu,m})^2) g_m. g_m = (2 / j_{\nu,M}^2)7 KB (1,063 words) - 18:25, 30 July 2019
- ...aystyle \!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \hat H = \frac{-\hbar^2}{2m} \nabla^2\) The substitution of (2),(3),(4) into (1) gives the "monochromatic" equation5 KB (743 words) - 18:47, 30 July 2019
- : \( \!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~ \displaystyle K_0(z) = \exp(-z)\sqrt{\frac{\pi}{2z}} ~ \Big( 1+ O(1/z)\Big)3 KB (394 words) - 18:26, 30 July 2019
- \displaystyle Y_0(x)=\frac{-2}{\pi} \int_0^\infty \cos(x \cosh(t)) \mathrm d t\) : \(Y_0(z)=Y_0(-z)+2~ \mathrm i~ J_0(-z)\)3 KB (445 words) - 18:26, 30 July 2019
- f''(z)+f'(z)/z + (z^2\!-\!1)f(z) = 0\) : \(f(0) = 0~\) and \(~f'(0)=1/2\)3 KB (439 words) - 18:26, 30 July 2019
- - 9./4.)*u + 2.)* c; return (C+S)/sqrt(2.*M_PI*z);}2 KB (190 words) - 18:47, 30 July 2019
- return s*z/2.;} f=M_PI/4.+z; c=cos(f); s=sin(f); a=sqrt(2./M_PI/z); t=1./(z*z);2 KB (159 words) - 14:59, 20 June 2013
- \( \!\!\!\!\!\!\!\!\! (1) ~ ~ ~ f''(z)+f'(z)/z+(1-\nu/z^2)f(x) =0\) f(x) \approx x^\nu \left( \frac{2^{-\nu}}{\mathrm{Factorial}(\nu)}+ O(x^2) \right)\)13 KB (1,592 words) - 18:25, 30 July 2019
- +q*(-0.016073968025938425623 + 0.0099471839432434584856 *L //2 +q*(-7.693079900902931853e-10 + 2.9981625986338549158e-10*L //64 KB (370 words) - 18:46, 30 July 2019
- TeXForm[Expand[Series[(HankelH1[0, x]) (Pi I x/2)^(1/2), {x, Infinity, 5}]]] \(e^{i x} \left(1-\frac{i}{8 x}-\frac{9}{128 x^2}+\frac{754 KB (509 words) - 18:26, 30 July 2019
- ...\!\!\!\!\!\!\!\! (1) \displaystyle ~ ~ ~ ふ_{\mathrm C}(x,y)=\sqrt{\frac{2}{\pi}} \cos(xy)\) : \(\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle \mathrm{CosFourier} f (x) = \int_0^\infty ~ ふ_{\mat6 KB (915 words) - 18:26, 30 July 2019
- ...displaystyle ふ_{m,n}=\frac{1}{\sqrt{N}} \exp\! \left( \frac{- \mathrm i ~2 ~\pi}{N}~m~n\right)\) Sometimes, the operator that deviate from \(\hat ふ\) with factor \(\sqrt{N}\) is considered; for such modified operator, the equation (4) does not h6 KB (1,032 words) - 18:48, 30 July 2019
- ...!\!\!\!\!(1) ~ ~ ~ \displaystyle \mathrm{CosFourier}(F)(x) ~=~ \sqrt{\frac{2}{\pi}}~ \int_0^\infty \! \cos(xy) ~F(y)~ \mathrm d y\) ...isplaystyle G_k = (\mathrm{DCTI}_N F)_k = \frac{1}{2} F_0 + \frac{(-1)^k}{2} F_{N} + \sum_{n=1}^{N-1} F_n \cos\! \left(\frac{\pi}{N} n k \right)~\) fo3 KB (482 words) - 18:26, 30 July 2019
- for(i=1;i<n;i+=2){ while (m >= 2 && j > m) { j -= m; m >>= 1; }4 KB (571 words) - 15:00, 20 June 2013
- <math>\displaystyle (\mathrm{DCTIV} ~f )_k = \sqrt{\frac{2}{N}} ~ ...~f_n~ \cos \left[\frac{\pi}{N} \left(n+\frac{1}{2}\right) \left(k+\frac{1}{2}\right) \right] \quad \quad k = 0, \dots, N-1.</math>3 KB (421 words) - 18:26, 30 July 2019
- ...isplaystyle G_k = (\mathrm{DCTI}_N F)_k = \frac{1}{2} F_0 + \frac{(-1)^k}{2} F_{N} + \sum_{n=1}^{N-1} F_n \cos\! \left(\frac{\pi}{N} n k \right)~\) fo : \( \!\!\!\!\!\!\!\!\!\!(2) ~ ~ ~ ~ (ふ_{\mathrm C1,N}~ F)_k=10 KB (1,447 words) - 18:27, 30 July 2019
- ..., the two cords that connect it with elementary body 1 and elementary body 2 are removed (cut); so, the 0th body remain free. However elementary body 2 and elementary body 3 are still connected with the ideal cord of length uni8 KB (1,036 words) - 18:25, 30 July 2019
- :<math>X_k = \frac{1}{2} (x_0 + (-1)^k x_{N}) For \(N=2^q\), the evaluation requires of order of \(Nq\) operations.10 KB (1,689 words) - 18:26, 30 July 2019
- _k = \sum_{n=0}^{N-1} ~ F_n~ \cos \left(\frac{\pi}{N} \left(n\!+\!\frac{1}{2}\right) k \right) ~ ~ ~\), \(~ ~ ~ k = 0, \dots, N\!-\!1\) For the simple and efficient implementation, \(N=2^q\) for some natural number \(q\). Note that the size of the arrays is for5 KB (682 words) - 18:27, 30 July 2019
- C3=2. /((1.-Q)*(1.-Q2) C5=2.*(7.+Q*(3.+Q*2.)) /((1.-Q)*(1.-Q2)*(1.-Q3)*(1.-Q43 KB (364 words) - 07:00, 1 December 2018
- ...ion of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. (Russian version: p.24-31) : \(\!\!\!\!\!\!\!\!\!\!\!(2) ~ ~ ~ ~ F(z\!+\!1)=T(F(z))\)6 KB (817 words) - 19:54, 5 August 2020
- The [[Shoko function]] is periodic; the period \(P=2 \pi \mathrm i\) is pure imaginary. The [[Shoko function]] has series of branchpoints at \(r+\pi(1\!+\!2 n) \mathrm i\) for integer values of \(n\);10 KB (1,507 words) - 18:25, 30 July 2019
- ...tion of the [[superfunction]] of the [[exponential]] to base \(b\!=\!\sqrt{2}\), constructed at the fixed point \(L\!=\!4\). 0.12022125769065893274e-1, 0.45849888965617461424e-2,1 KB (139 words) - 18:48, 30 July 2019
- ...tion of the [[Abel function]] of the [[exponential]] to base \(b\!=\!\sqrt{2}\), constructed at the fixed point \(L\!=\!4\). -0.587369764200886206e-2, 0.289686728710575713e-2,2 KB (163 words) - 18:47, 30 July 2019
- z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t); if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z);1 KB (209 words) - 15:01, 20 June 2013
- , 2.66666666666667 z_type LambertWe(z_type z){ int n,m=100; z_type t=1./M_E+z; t*=2*M_E; t=sqrt(t);5 KB (287 words) - 15:01, 20 June 2013
- (2) \(~ ~ ~ \log_s\Big(~ g\big( T(z)\big)~\Big) = 1 + \log_s\big(g(z)\big)~\) The substitution of (3) into (2) gives for function \(~G~\) the [[Abel equation]]8 KB (1,239 words) - 11:32, 20 July 2020
- (2) \(~ ~ ~ g\big(T(z)\big)= K \, g(z) \) http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html10 KB (1,627 words) - 18:26, 30 July 2019
- ...|thumb| \(u\!+\!\mathrm i v\!=\! \mathrm{ArcTra}(x\!+\!\mathrm i y)~\) by (2)]] (2) \(~ ~ ~ \mathrm{ArcTra}(z)=z-\mathrm{Tania}(z-1)\)10 KB (1,442 words) - 18:47, 30 July 2019
- z_type sutra(z_type z){ if( Re(z)<2. ) return sutra0(z); if( Re(z)<4. ) return tra(tra(sutra0(z-2.)));1 KB (197 words) - 15:03, 20 June 2013
- Simplify[2 Sin[x] Cos[x]] Sin[2 x]2 KB (259 words) - 18:45, 30 July 2019
- \( abcd^2 \) <math>f_1(x)=\frac{x+1}{\sqrt{1-x^2}}\label{ref1}\</math>5 KB (795 words) - 10:28, 26 March 2021
- [[File:Ack3a600.jpg|400px|thumb|Base \(b=\sqrt{2}\approx 1.41\)]] [[File:Ack4a600.jpg|400px|thumb|Binary tetration, \(b=2\)]]5 KB (761 words) - 12:00, 21 July 2020
- // Q has sense of parameter of the [[Nemtsov function]]; K=Q^2 C[2]=-0.625 + (-0.25 + K/2.)*K;16 KB (1,450 words) - 06:58, 1 December 2018
- ...|thumb|\(u\!+\!\mathrm i v=\mathrm{mori}\big(\sqrt{x\!+\!\mathrm i y}\big)^2=\mathrm{nori}(x\!+\!\mathrm i y)\)]] ...thumb| \(u\!+\!\mathrm i v=\mathrm{mori}\big(\sqrt{x\!+\!\mathrm i y}\big)^2\)]]!-->13 KB (1,759 words) - 18:45, 30 July 2019
- Some of numbers have own single-character names: 0,1,2,3,4,5,6,7,8,9. For example, : <math> 2=1++</math>5 KB (753 words) - 18:47, 30 July 2019
- \( P_c(z)=z^2+c\) \(p(z) = r z -r/2\),2 KB (229 words) - 18:44, 30 July 2019
- [[File:Fracit20t150.jpg|300px]]\( t(z)=\frac{z}{2+z} ~\); \(~y=t^n(x)\) versus \(x\) for various \(n\) (07) \(~ ~ ~ \displaystyle T(z)=P(t(Q(z)))= \frac{ABc - AB - A^2 ~+~ (A\!+\!B)z}{Bc-A+z}\)13 KB (2,088 words) - 06:43, 20 July 2020
- \((2) ~ ~ ~ \displaystyle A+B z= \lim_{M\rightarrow \infty} \frac{M A+ M B}{M+z} \frac{\frac{-A^2+A v-A w+u}{B^2}+\frac{v-A}{B} z}{\frac{A+w}{B}+z}\)5 KB (830 words) - 18:44, 30 July 2019
- https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3 [[File:978-620-2-67286-3-full.jpg|440px]]15 KB (2,166 words) - 20:33, 16 July 2023
- \(\mathrm{mori}(z)=\displaystyle \frac{ J_0 (L_1 z)}{1-z^2}\) where \(L_1\approx 2.4\) is first zero of the Bessel function, \(J_0(L_1)\!=\!0~\).3 KB (456 words) - 18:44, 30 July 2019
- return D + (153*m+2)/5 + 365*y + y/4 - y/100 + y/400 - 32045 - 2400000; } \rm Qu\! & 233.214 - 0.908933 x - 0.00361841 x^2 & 3.94609& 5.84960\\5 KB (433 words) - 18:47, 30 July 2019
- \((2) ~ ~ ~ \rho(x,t)=F(\mathrm e^x,t)~ \mathrm e^x\) From equation (2), the distribution \(F\) can be expressed through the new function \(\rho\)5 KB (865 words) - 18:44, 30 July 2019
- ...\frac{1}{2}\mathrm{Lof}(z)-\mathrm{Lof}\Big(\frac{z}{2}\Big)-\frac{\ln(2)}{2} z \right)\) \frac{2^{-z/2}\,\sqrt{z!}}6 KB (883 words) - 18:44, 30 July 2019
- where \(\psi_n(z)=\frac{1}{\sqrt{N_n}}\)[[HermiteH]]\(_n(z)\, \exp(-z^2/2)\) is [[oscillator function]], {2^n \sqrt{\pi}}6 KB (770 words) - 18:44, 30 July 2019
- [[File:2014ruble15t.png|240px]]Fig.2. Fitting from 2014.12.14 ...sere loaded, see Fig.1; then, the data for Euro and then for Yen, see Fig.2. and Fig.3.18 KB (2,080 words) - 13:48, 1 February 2022
- {2.000000000000000, 0.3333333333333333, 0.1500000000000000, 0.0892857142857142 M=29; qq=.5-.5*z; q=sqrt(qq); s=c[M]*q; for(m=M-1;m>0;m--) {s+=c[m]; s*=qq;}4 KB (488 words) - 06:58, 1 December 2018
- 8.173825669330684e-9, -2.0624513960198102e-8, //11 -2.5497322696797093e-10,6.897078914225712e-11, //152 KB (354 words) - 06:58, 1 December 2018
- [[File:Arqnem20zt.jpg|400px|thumb|\(u\!+\!\mathrm i v=\mathrm{ArqNem}_{2}(x\!+\!\mathrm i y)\)]] \(q\!=\!2\).7 KB (1,319 words) - 18:46, 30 July 2019
- ...pg|200px|thumb|Thick green curve: \(y=\eta^x\); thin red curve: \(y=(\sqrt{2})^x\)]] These pictures look similar to those for the case \(b=\sqrt{2}\approx 1.414\), see article [[Base sqrt2]].4 KB (559 words) - 17:10, 10 August 2020
- ...apT.png|300px|thumb|[[Complex map|Map]] of [[exponent]] to base \(b=\sqrt{2}\); lines of constant \(u\) and lines of constant \(v\) show ...00.jpg|300px|thumb|[[Complex map|Map]] of [[Logarithm]] to base \(b=\sqrt{2}\); lines of constant \(u\) and lines of constant \(v\) show3 KB (557 words) - 18:46, 30 July 2019
- \(T_{n+1}(x)=2 x T_n(x)-T_{n-1}(x)\) ...)\, \sin(x\, y)\, \frac{\mathrm d x}{\sqrt{1\!-\!x^2}} = (-1)^n \frac{\pi}{2}\, J_{2n+1}(y)1 KB (186 words) - 18:48, 30 July 2019
- </ref> and \(~y=\sin^n(\pi/2)-\sin^n(x)~\) with \(~n\!=\!100~\) http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html<br>7 KB (1,031 words) - 03:16, 12 May 2021
- ...ansform, refers to the integral transform with kernel \(K(x,y)=\sqrt{\frac{2}{\pi}} \cos(xy)\); \(\displaystyle g(x)=\,\)[[CosFT]]\(f\,(x) \displaystyle =\sqrt{\frac{2}{\pi}} \int_0^\infty \cos(xy) \, f(y) \, \mathrm d y\)3 KB (468 words) - 18:47, 30 July 2019
- X[n_] = BesselJZero[0,n]/Sqrt[S]; W[n_] = Sqrt[2./S]/Abs[BesselJ[1, BesselJZero[0,n]]];8 KB (1,153 words) - 18:44, 30 July 2019
- [[ackermann]]\(_{2,x}(y)=x\, y\) \frac{\exp(\mathrm i x) + \exp(-\mathrm i x)}{2}\)3 KB (496 words) - 18:45, 30 July 2019
- Let \(~ T(z)=z+b z^2 + c z^3+..\) T[z_] = z + b z^2 + c z^3;11 KB (1,715 words) - 18:44, 30 July 2019
- \( F''+ 2 \mathrm i \dot F=0\) then, assuming some large positive \(M\), expression \(M^2 z/k\) has sense of the coordinate along the propagation of wave, and \(M x/8 KB (1,216 words) - 18:43, 30 July 2019
- ....jpg|300px|thumb| Normalised [[Hermite polynomial]]s, \(y=h_n(x)\) for \(n=2,3,4,5,6\)]] \(H_m(x)=(-1)^m \exp(-x^2) \partial_x^m \exp(-x^2)\)4 KB (628 words) - 18:47, 30 July 2019
- \(~ ~\mathrm{kori}(z)=\displaystyle \frac{J_0\big(L\, \sqrt{x}\big)}{1-z}\) where \(L=\,\) [[BesselJZero]]\([0,1]\approx 2.4048255576957727686\)14 KB (1,943 words) - 18:48, 30 July 2019
- \(\mathrm{kori}(z)=\displaystyle \frac{J_0\big( L_1 \sqrt{z} \big)}{1-z}\) where \(L_1\approx 2.4\) is first zero of the Bessel function \(J_0\). The corresponding limit f2 KB (328 words) - 10:27, 20 July 2020
- ...tional aproximation of function [[kori]]\((x)=\displaystyle \frac{J_0(L_1 \sqrt{x})}{1\!-\!x}\) ...implementation of integrals with function [[nori]]\((x)=\,\)[[kori]]\((x)^2\).4 KB (644 words) - 18:47, 30 July 2019
- 1 & 1-z & \frac{1}{2} \left(z^2-4 z+2\right) & \frac{1}{6} \left(-z^3+9 z^2-18 z+6\right) &5 KB (759 words) - 18:44, 30 July 2019
- ...ga}(x)=\displaystyle 2 \int_0^\infty \mathrm{mori}(p )^2 \exp(\mathrm i p^2 x) \, p \,\mathrm d p\) [[mori]]\((p)=\displaystyle \frac{J_0(L p)}{1-p^2}\)8 KB (1,256 words) - 18:44, 30 July 2019
- http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html<br> ...H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.2 KB (344 words) - 07:02, 1 December 2018
- DB L1=2.404825557695772768621631879326454643124; x=1./(z*z); t=-sqrt(2./M_PI/L1/z)/z/z;2 KB (188 words) - 07:03, 1 December 2018
- {1\!-\!x^2}~\), where \(J_0\) is [[Bessel function]] and \(L_1\approx 2.4\) is its first zero. {1\!-\!x^2}\)15 KB (2,303 words) - 18:47, 30 July 2019
- ...ga}(p)=2\int_0^\infty \frac{J_0(L_1 x)^2}{(1-x^2)^2} \, \exp(\mathrm i p x^2)\, x\, \mathrm dx\) where \(L_1\!=\)[[BesselJZero]][0,1]\(\approx\! 2.4\) is the first zero of the Bessel function; \(J_0(L_1)=0\).5 KB (750 words) - 10:00, 20 July 2020
- \(\mathrm{Nem}_q'(z)=1+3z^2+4qz^3\) \(z_0=\mathrm{NemBra}(0)=\mathrm i /\sqrt{3} \approx 0.6\, \mathrm i\)4 KB (618 words) - 18:46, 30 July 2019
- \(\mathrm{Nem}_q^{\prime}(z)=1+3\,z^2+4\, q\, z^3\) Let \(~ \rho=-1-8q^2+4\sqrt{q^2+4q^4}\)3 KB (400 words) - 18:48, 30 July 2019
- [[File:Nem100map.jpg|140px|thumb|Fig.2. \(u\!+\!\mathrm i v=\mathrm{Nem}_0(x\!+\!\mathrm i y)\)]] ...{Nem}_q\) is shown in figure 2 for \(q\!=\!0\) and in figure 3 for \(q\!=\!2\) with lines \(u\!=\!\mathrm{const}\) and14 KB (2,157 words) - 18:44, 30 July 2019
- For the set of orthogonal polynomials \(P_m\), \(m=0,1,2, ..\) {\int_a^b \rho(x)\, x\, P_m(x)^2\, \mathrm d x}6 KB (918 words) - 18:47, 30 July 2019
- \(F''(x)+(2n\!+\!1-x^2)\, F(x) = 0\) \(F_n(x)=h_n(x) \, \exp(-x^2/2)\) \(=N_n^{-1/2}\, H_n(x)\, \exp(-x^2/2)\)6 KB (846 words) - 18:47, 30 July 2019
- \(\displaystyle \rho=\sqrt{uv}\) \(\displaystyle z=\frac{u\!-\!v}{2}\)3 KB (470 words) - 18:43, 30 July 2019
- \frac{1}{r^2} \partial_r (r^2 R') - \frac{\ell(\ell\!+\!1)}{r^2} R8 KB (1,199 words) - 18:45, 30 July 2019
- Series[HankelH1[0, Sqrt[x] ]^2 Pi I Sqrt[x]/2, {x, Infinity, 2}] e^{2 i \sqrt{x}} \left(1-\frac{1}{4} i2 KB (325 words) - 18:44, 30 July 2019
- ...ansform, refers to the integral transform with kernel \(K(x,y)=\sqrt{\frac{2}{\pi}} \sin(xy)\); \(\displaystyle g(x)=\,\)[[SinFT]]\(f\,(x) \displaystyle =\sqrt{\frac{2}{\pi}} \int_0^\infty \sin(xy) \, f(y) \, \mathrm d y\)5 KB (807 words) - 18:44, 30 July 2019
- 2. Alternative representations of the function, relation of the function to o And kilogram be 2 lb.7 KB (991 words) - 18:48, 30 July 2019
- ...uggests routine F21E for evaluation of [[tetration]] to base \(b\!=\!\sqrt{2}\). //In order to evaluate \(\mathrm{tet}_{\sqrt{2}}(z)\), the routine should be called as F21E(z)1 KB (109 words) - 18:48, 30 July 2019
- ...Sqrt2f21l.cin]] is code for evaluation of [[arctetration]] to base \(\sqrt{2}\) of complex argument. DB TcL[23]={1., //coeff. of expansion of exp(-q(z+1.2 ...) by powers of (2-F).1 KB (145 words) - 18:47, 30 July 2019
- ...r evaluation of real–holomorphic superexponential to base \(b\!=\!\sqrt{2}\). //In order to evaluate \(\mathrm{tet}_{\sqrt{2}}(z)\), the routine should be called as F23E(z)2 KB (146 words) - 18:47, 30 July 2019
- ...or evaluation of real–holomorphic abelexponential to base \(b\!=\!\sqrt{2}\). //In order to evaluate \(\mathrm{tet}_{\sqrt{2}}(z)\), the routine should be called as F23L(z)2 KB (168 words) - 18:47, 30 July 2019
- ...Sqrt2f43e.cin]] is routine for evaluation of superexponent to base \(\sqrt{2}\) that approach 4 at \(-\infty\) and has value 3 at zero. 0.12022125769065893274e-1, -0.45849888965617461424e-2,1 KB (131 words) - 10:44, 24 June 2020
- ...[Sqrt2f43e.cin]] is routine for evaluarion of abelexponent to base \(\sqrt{2}\) that approach 4 at \(-\infty\) and has value zero at 3. -0.587369764200886206e-2, 0.289686728710575713e-2,1 KB (124 words) - 18:46, 30 July 2019
- ...e infinitely growing superfunction of exponential, \(\mathrm{SuExp}_{\sqrt{2},5}\). 0.12022125769065893274e-1, 0.45849888965617461424e-2,1 KB (108 words) - 18:47, 30 July 2019
- ...at ecaluates the growing [[Abel function]] of the exponent to base \(\sqrt{2}\) -0.587369764200886206e-2, 0.289686728710575713e-2,1 KB (131 words) - 18:47, 30 July 2019
- \(q\!=\!1\) , green, and for \(q\!=\!2\) , blue]] [[File:Straro05at.jpg|300px|thumb|Fig.2. \(u\!+\!\mathrm i v=\mathrm{StraRo}_{0.5}(x\!+\!\mathrm i y\)]]4 KB (646 words) - 18:47, 30 July 2019
- [[File:Sunem2mdp.jpg|200px|thumb|\(u\!+\!\mathrm i v=\mathrm{SuNem}_{2}(x\!+\!\mathrm i y)\)]] ...Nem}_{q}(z) = {\displaystyle \frac{1}{\sqrt{-2z}}}\left(1+O\left(\frac{1}{\sqrt{-2z}} \right)\right)\)6 KB (967 words) - 18:44, 30 July 2019
- ...plot.jpg|300px|thumb| Graphics \(~y\!=\!\mathrm{SuSin}(x)~\) and \(~y\!=\!\sqrt{3/x}~\) ]] \((2)~ ~ ~\mathrm{SuSin}(1)=1\)15 KB (2,314 words) - 18:48, 30 July 2019
- // DB Q=2.; z_type t=1./sqrt(-2.*z);1 KB (228 words) - 07:06, 1 December 2018
- z_type nemF0(z_type z){ int m,n,k; z_type c[22],s; z_type L=log(-z); z_type x=sqrt(-.5/(NEMTSOVp*z)); c[2]=APQ[2][1]*L;2 KB (347 words) - 07:06, 1 December 2018
- \(y = \mathrm{Linear}(x) = 2.4 − 0.0048x\) \(y = \mathrm{Arc}(x) = 0.02\sqrt{(123−x)(471+x)}\)38 KB (636 words) - 18:37, 30 July 2019
- \(y=\mathrm{Linear}(x)=2.4 - 0.0048 x\) \(y=\mathrm{Arc}(x)=0.01\sqrt{(123-x)(471+x)} ~\)10 KB (89 words) - 18:36, 30 July 2019
- \(T^2(z)=T(T(z))\)<br> ...ачениях, \(\sin^2(z)=\sin(\sin(z))\), а вовсе не \(\sin(z)^2\).<br>19 KB (1,132 words) - 20:36, 16 July 2023
- ...plify]] does not seem to handle well expressions with imaginary unity , I=\Sqrt[-1] . b = (-1 + Exp[(-2*I)*q - 2*s])*(-1 + Exp[(2*I)*q - 2*s])2 KB (305 words) - 18:43, 30 July 2019
- ...ions. Bulletin (New Series) of the American Mathematical society, v.29, No.2 (1993) p.151-188.</ref><ref name="domsta"> ...l as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.15 KB (2,392 words) - 11:05, 20 July 2020
- \(y=\mathrm{Linear}(x)= 2.4-0.00048x\) \(y=\mathrm{Arc}(x)= c \sqrt{(a+x)(b-x)}\)39 KB (760 words) - 11:55, 22 April 2023
- D.Kouznetsov, H.Trappmann. Superfunctions and sqrt of Factorial. ...special function. [[Vladikavkaz Mathematical Journal]], 2010, v.12, issue 2, p.31-4512 KB (1,732 words) - 14:01, 12 August 2020
- ===2. Postulates=== ...e onarchica Italiana - U.M.I.// Nel corso del referendum istituzionale del 2 Giugno 1946 Napoli aveva dato l’83% dei voti alla Monarchia.60 KB (1,180 words) - 13:23, 24 July 2022
- http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br> ...H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.7 KB (1,082 words) - 07:03, 13 July 2020
- // superexponential to base \( \sqrt{2} \) 0.12022125769065893274e-1, 0.45849888965617461424e-2,1 KB (112 words) - 13:42, 7 July 2020
- x+1\over\sqrt{1-x^2}\label{ref2}457 bytes (68 words) - 20:12, 13 May 2021
- \( N = \log_{10}(100/0.5) = \log_{10}(200) \approx 2.301029995664 \approx 2 \) ===Exercise 2===10 KB (1,491 words) - 18:09, 11 June 2022
- \varphi(x)=\exp(-x^2/2)/\sqrt{2\pi} x^{2n} e^{-p x^2} \mathrm d x = \frac{(2n-1)!!}{2(2p)^n} \sqrt{\frac{\pi}{p}}2 KB (232 words) - 15:16, 3 April 2023
- ...2srav.png|300px}}<small><center> \( y_1=\exp_{\sqrt{2},2}^{~1/2}(x) \) , \( y_2=\exp_{\sqrt{2},4}^{~1/2}(x) \) ,<br> and the deviation \(D = y_1-y_2\)</center></small></div>5 KB (712 words) - 08:18, 9 May 2024
- echo 1/sqrt(8), "\n"; echo 2/(M_PI*sqrt(3)), "\n";2 KB (215 words) - 08:33, 9 May 2024
- ...345big.png|312px}}<small><center> \(y=\mathrm{Student}(n,x)\) for \(n\!= 1,2,3,4,5,\infty \) </center></small> ...a\left(\frac{\nu}{2}\right)} \left(1 + \frac{~ t^2}{\nu }\right)^{-(\nu+1)/2}</math>12 KB (1,727 words) - 03:21, 11 May 2024
- ...345big.png|360px}}<small><center> \(y=\mathrm{Student}(n,x)\) for \(n\!= 1,2,3,4,5,\infty \) </center></small> ...345big.png|360px}}<small><center> \(y=\mathrm{Ctudent}(n,x)\) for \(n\!= 1,2,3,4,5,\infty \) </center></small>5 KB (621 words) - 09:24, 10 May 2024
