A257451 Decimal expansion of the location of the maximum of (1-cos(x))/x.

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

Offset: 1

Author

Stanislav Sykora, Apr 23 2015

Keywords

• nonn
• cons
• changed

Comments

Also, the first positive solution of x*sin(x)=(1-cos(x)).

The function hsinc(x) = (1-cos(x))/x is the Hilbert transform of sinc(x) = sin(x)/x. Both functions play a considerable role in various branches of physics, particularly in spectroscopy.

The value of hsinc(a) is in A257452.

The kissing points [x,y] of the two tangents with the smallest nonzero |x|, drawn from the apex [0,1] of the function y = cos(x) to itself, have the coordinates [+a,cos(a)] and [-a,cos(a)], respectively. The angle each of the tangents subtends with the Y axis is theta = atan(1/sin(a)). - Stanislav Sykora, Oct 17 2015

For a curve S in the xy-plane starting at the origin, pointing to the right, turning counterclockwise with constant curvature K, and with an arclength of 1, let Y denote the maximum y-value of any point in S. Then, this constant is equal to the value of K that maximizes Y. - Andrew Slattery, Sep 11 2021

The smallest positive fixed point of tan(x/2). - Michal Paulovic, Aug 27 2025

Examples

			2.3311223704144226136678359559171213382690776953861145751...
Added in support of the Oct 17 2015 comment:
cos(a) = -0.689157736645164443889295..., theta = atan(1/sin(a)) = 0.943742927149971739026594... rad = 54.072486671015691988683987... deg.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A257452.

Programs

Mathematica

RealDigits[x/.FindMaximum[(1-Cos[x])/x,x,WorkingPrecision->200] [[-1]]] [[1]] (* Harvey P. Dale, Mar 29 2022 *)

PARI

a = solve(x=1,3,x*sin(x)-1+cos(x))