

A257451


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


2



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
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OFFSET

1,1


COMMENTS

Also, the first positive solution of x*sin(x)=(1cos(x)).
The function hsinc(x) = (1cos(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 xyplane 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 yvalue of any point in S. Then, this constant is equal to the value of K that maximizes Y.  Andrew Slattery, Sep 11 2021


LINKS

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


EXAMPLE

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.


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A257452.
Sequence in context: A290003 A139460 A105244 * A209007 A145854 A097663
Adjacent sequences: A257448 A257449 A257450 * A257452 A257453 A257454


KEYWORD

nonn,cons


AUTHOR

Stanislav Sykora, Apr 23 2015


STATUS

approved



