A115365 Decimal expansion of smallest positive root of tan(x) = x.

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, 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, 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

Offset: 1

Author

Eric W. Weisstein, Jan 21 2006

Keywords

• nonn
• cons

Comments

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

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

Unique fixed point of the function arctan(x)+Pi, and this fixed point is attractive. - Robert FERREOL, May 09 2023

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

Examples

			4.4934094579090641753...
		

References

• M. Abramowitz, I. A. Stegun, Editors, Handbook of Mathematical Functions, Dover Publications, 1965, Chapter 10.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, 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.
• Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60.
• Mohammad K. Azarian, Solution to Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal Vol. 39, No. 1, January 2008, pp. 66-67.
• Stanislav Sykora, K-Space Images of n-Dimensional Spheres and Generalized Sinc Functions
• Eric Weisstein's World of Mathematics, Tangent
• Eric Weisstein's World of Mathematics, Tanc Function
• Index entries for transcendental numbers

Crossrefs

Cf. A102015 (continued fraction), A213053 (amplitude at x).

Cf. A062546, A224196, A207528, A243108, A245333.

Programs

Maple

Digits:=200; fsolve(x*cos(x)-sin(x),x,4..5);

Mathematica

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 *)
RealDigits[BesselJZero[3/2, 1], 10, 100][[1]] (* Vladimir Reshetnikov, May 13 2016 *)

PARI

solve(x=4,4.5,tan(x)-x) \\ Charles R Greathouse IV, Jun 10 2012

PARI

besseljzero(3/2,1) \\ Charles R Greathouse IV, Jan 23 2025