A213053 -id:A213053 - cp's OEIS frontend

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

Eric W. Weisstein, Jan 21 2006

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

			4.4934094579090641753...

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

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

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

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

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

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 *)

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

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

A328227 Decimal expansion of positive solution to x^2 = 1 + (Pi + arccos(1/x))^2.

Original entry on oeis.org

4, 6, 0, 3, 3, 3, 8, 8, 4, 8, 7, 5, 1, 7, 0, 0, 3, 5, 2, 5, 5, 6, 5, 8, 2, 0, 2, 9, 1, 0, 3, 0, 1, 6, 5, 1, 3, 0, 6, 7, 3, 9, 7, 1, 3, 4, 1, 6, 0, 5, 3, 2, 3, 4, 6, 0, 3, 9, 4, 3, 0, 1, 1, 5, 4, 3, 8, 4, 5, 8, 7, 3, 1, 9, 6, 5, 9, 7, 0, 9, 9, 8, 7, 1, 6, 5, 4, 6, 9, 9, 7, 2, 2, 7, 2

Offset: 1

Jack Zhang, Oct 08 2019

We are in a rowboat on a circular lake, starting at the center. At the edge of the lake is a mean goblin. He can run k times as fast as we can row. This is the minimum value of k such that we will not be able to escape.
From Rian Hunter, Jun 16 2021: (Start)
For a spirograph defined by complex function z = p * e^(-i * b * t) + b * e^(i * t), this is the value of p as b->oo such that each petal is tangent to the next one.
If we consider the set of all right triangles such that their tangent value is equal to the opposite angle in radians, this value is equal to the negative secant of the right triangle from that set with the smallest nonzero opposite angle. (End)
The envelope of the t*x = sin(t*y) family of curves contains the set of y = (-1)^n*k_n*x straight lines (n > 0), where k_n is the solution of (n*Pi + arccos(1/k))^2 + 1 = k^2. This entry is k_1. See illustration, section Links. - Luc Rousseau, Mar 11 2022
Maximum negative value of x/sin(x). - Andrew Slattery, Jun 29 2022

			4.6033388487517003525565820291030165130673971341605323460394301154384587319659...

Rian Hunter, The Number Hiding Inside the Spirograph.
IBM Research, Ponder This Challenge - May 2001.
Luc Rousseau, A328227 viewed as the |slope| of an envelope.

Cf. A115365.

Equals 1/A213053.

NSolve[x^2==1+(Pi+ArcCos[1/x])^2,x,Reals,WorkingPrecision->100]

solve(x=4, 5, 1 + (Pi+acos(1/x))^2 - x^2) \\ Michel Marcus, Oct 08 2019

x=-sec(y), where decimal expansion of y is A115365.

Alternatively, x=sqrt(y^2+1).

A338670 Decimal expansion of the sum of the negative and positive local extreme values of the sinc function for x > 0 (negated).

Original entry on oeis.org

1, 4, 0, 8, 5, 9

Offset: 0

Bernard Schott, Apr 23 2021

The equation of the sinc function is y = sin(x)/x.
Equivalently, sum of f(x) = sinc(x) where x > 0 and f'(x) = 0. - David A. Corneth, May 01 2021
These extreme values are obtained when x_k > 0 is a solution to tan(x) = x (see Chronomath link), or equivalently to y = tanc(x) = tan(x)/x = 1. The corresponding k-th extreme value is y_k = sin(x_k)/x_k.
Every extremum y_k = (-1)^k/(k*Pi) + O(1/k^2), hence the series Sum_{k > 0} sin(x_k)/x_k is convergent.
However, this series is not absolutely convergent, just as (C_1)/2 diverges where C_1 is the corresponding du Bois-Reymond constant.

			-0.140859...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.3.18, pp. 285 and 303.

Serge Mehl, Comportement en zéro de sin(x)/x, ChronoMath.
Eric Weisstein's World of Mathematics, du Bois-Reymond constants.
Eric Weisstein's World of Mathematics, Sinc Function.
Eric Weisstein's World of Mathematics, Tanc Function.

Coordinates of the 1st extremum: A115365 (x_1), A213053 (y_1).

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

Equals Sum_{k >= 1} sinc(x_k) or Sum_{k >= 1} (-1)^k / sqrt(1+(x_k)^2), where x_k is the k-th positive root of x = tan(x).

More terms from Amiram Eldar, Apr 23 2021

Name clarified by N. J. A. Sloane, May 01 2021

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

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A328227 Decimal expansion of positive solution to x^2 = 1 + (Pi + arccos(1/x))^2.

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A338670 Decimal expansion of the sum of the negative and positive local extreme values of the sinc function for x > 0 (negated).

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