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%I #53 May 15 2023 02:16:45
%S 4,4,9,3,4,0,9,4,5,7,9,0,9,0,6,4,1,7,5,3,0,7,8,8,0,9,2,7,2,8,0,3,2,2,
%T 0,8,2,2,1,5,5,8,3,8,7,2,2,9,0,0,4,0,8,0,2,8,9,5,8,2,3,9,6,1,9,2,6,9,
%U 5,0,3,1,4,5,9,7,1,0,4,0,9,8,7,2,9,0,5,7,8,0,9,4,5,5,8,7,9,6,9,1,5,2,1,7,6
%N Decimal expansion of smallest positive root of tan(x) = x.
%C Location (for x>0) of the first negative lobe of sinc(x) = sin(x)/x, where sinc(x) attains its absolute minimum of -0.217233628... The function sinc(x) is important in spectral theory (transient data truncation artifacts). - _Stanislav Sykora_, Mar 05 2012
%C Also the first root of the sinc(3,x) function, that is, the radial component of the 3D Fourier transform of 3-dimensional unit sphere. Also the first root of the spherical Bessel function of the 1st kind, j_1(x). - _Stanislav Sykora_, Nov 14 2013
%C Unique fixed point of the function arctan(x)+Pi, and this fixed point is attractive. - _Robert FERREOL_, May 09 2023
%D M. Abramowitz, I. A. Stegun, Editors, Handbook of Mathematical Functions, Dover Publications, 1965, Chapter 10.
%H G. C. Greubel, <a href="/A115365/b115365.txt">Table of n, a(n) for n = 1..5000</a>
%H Mohammad K. Azarian, <a href="http://www.ijpam.eu/contents/2008-46-1/3/3.pdf">On the Fixed Points of a Function and the Fixed Points of its Composite Functions</a>, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/27646421">Fixed Points of a Quadratic Polynomial, Problem 841</a>, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60.
%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/27646572">Solution to Fixed Points of a Quadratic Polynomial, Problem 841</a>, College Mathematics Journal Vol. 39, No. 1, January 2008, pp. 66-67.
%H Stanislav Sykora, <a href="http://dx.doi.org/10.3247/SL2Math07.002">K-Space Images of n-Dimensional Spheres and Generalized Sinc Functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tangent.html">Tangent</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TancFunction.html">Tanc Function</a>
%e 4.4934094579090641753...
%p Digits:=200; fsolve(x*cos(x)-sin(x),x,4..5);
%t RealDigits[FindRoot[Tan[x]==x,{x,4}, WorkingPrecision->128][[1,2]]][[1]] (* _Robert G. Wilson v_, Mar 05 2012; corrected by _Harvey P. Dale_, Mar 22 2012 *)
%t RealDigits[BesselJZero[3/2, 1], 10, 100][[1]] (* _Vladimir Reshetnikov_, May 13 2016 *)
%o (PARI) solve(x=4,4.5,tan(x)-x) \\ _Charles R Greathouse IV_, Jun 10 2012
%Y Cf. A102015 (continued fraction), A213053 (amplitude at x).
%Y Cf. A062546, A224196, A207528, A243108, A245333.
%K nonn,cons
%O 1,1
%A _Eric W. Weisstein_, Jan 21 2006
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