A115365 - OEIS

%I #56 Jul 11 2024 06:36:34

%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

%C Further roots (intersections of y=x with other branches of y=tan(x)) are at x=7.725251... = A255272, x=10.9041216..., x=14.0661939..., x= 17.2207552.. etc. - _R. J. Mathar_, Jul 11 2024

%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