A224196 -id:A224196 - 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

A062546 Decimal expansion of the 2nd Du Bois-Reymond constant.

Original entry on oeis.org

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

Offset: 0

Jason Earls, Jun 26 2001

			0.194528049465325113...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 238.
Francois Le Lionnais, Les nombres remarquables, Paris, Hermann, 1983, p. 23.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Simon Plouffe, Du Bois-Reymond constant to 1024 digits.
Simon Plouffe, The Du Bois-Reymond constant to 1024 digits.
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A062545, A224196 (c3), A207528 (c4), A243108 (c5), A245333 (c6).

Cf. A073747 (has a similar continued fraction).

RealDigits[(E^2 - 7)/2, 10, 105][[1]] (* Jean-François Alcover, Jun 12 2018 *)

(exp(2)-7)/2 \\ Stefano Spezia, Dec 10 2024

Equals (exp(2)-7)/2.
Continued fraction: [0, 5, 7, 9, 11, 13, ...]. - Gerald McGarvey, Oct 15 2005
Equals exp(1)*cosh(1) - 4 = BesselI(1/2, 1)/BesselI(3/2, 1) - 3. - Terry D. Grant, Jul 20 2018

A207528 Decimal expansion of 4th du Bois-Reymond constant.

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Feb 24 2012

			0.0052407046777047711...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239.

Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A138730 (continued fraction expansion).

Cf. A062546 (c2), A224196 (c3), A243108 (c5), A245333 (c6).

RealDigits[(E^4 - 4E^2 - 25)/8, 10, 100][[1]]

2*Sum_{k>=1} (1 + (x_k)^2)^(-2) = (e^4 - 4*e^2 - 25)/8, where x_k is the k-th root of t = tan t and e = 2.71828... is the natural logarithm base.

A243108 Decimal expansion of the 5th du Bois-Reymond constant.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 19 2014

The k-th Du Bois Reymond constant c_k is asymptotic to 2/(1+r^2)^(k/2), where r = A115365 = 4.4934094579090641753... is smallest positive root of the equation tan(r) = r. - Vaclav Kotesovec, Aug 20 2014

			0.00105610210733480920562199158210781176744608061...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239.

Vaclav Kotesovec, Table of n, a(n) for n = 0..450
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A062546 (c2), A224196 (c3), A207528 (c4), A245333 (c6).

Cf. A115365, A079330, A088989.

A079330 Numerators of coefficients of odd powers of 1/q in the solution series for tan(x)/x=1.

Original entry on oeis.org

1, 2, 13, 146, 781, 16328, 6316012, 38759594, 9655714457, 50134571594, 25626917879, 638499558282328, 125381104727404588, 435948294065152496, 146414084312394268792, 1076603090723736731974978

Offset: 1

Eric W. Weisstein, Jan 03 2003

Series contributed by David W. Cantrell.

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 34, equation 34:7:4 at page 325.

Vaclav Kotesovec, Table of n, a(n) for n = 1..250
Vaclav Kotesovec, Asymptotic of the coefficients A079330 / A088989.
Eric Weisstein's World of Mathematics, Tanc Function.

Denominators are in A088989.

Cf. A224196.

Last/@Partition[CoefficientList[InverseSeries[Series[x+Cot[x], {x, 0, 50}], q], 1/q], 2]

a(n)/A088989(n) ~ c * (Pi/2)^(2*n) / n^(4/3), where c = GAMMA(1/3)/(2^(2/3)*3^(1/6)*Pi^(5/3)) = 0.208532... . - Vaclav Kotesovec, Aug 19 2014

A245333 Decimal expansion of the 6th du Bois-Reymond constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 18 2014

			0.000220674708176213169277990873376019646763182409385382227156501293654...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.12 Du Bois Reymond Constants, p. 238.

Eric Weisstein's World of Mathematics, du Bois-Reymond Constants

Cf. A062546 (c2), A224196 (c3), A207528 (c4), A243108 (c5).

c6 = (1/32)*(E^6 - 6*E^4 + 3*E^2 - 98); Join[{0, 0, 0}, RealDigits[c6, 10, 102] // First]

c_6 = (1/32)*(e^6 - 6*e^4 + 3*e^2 - 98).

A088989 Denominators of coefficients of odd powers of 1/q in the solution series for Tan[x]/x=1.

Original entry on oeis.org

1, 3, 15, 105, 315, 3465, 675675, 2027025, 241215975, 587577375, 138881925, 1581170716125, 140480936701875, 219150261254925, 32785574798851875, 106717045970262853125, 1424703093055003125, 33615869480632798734375

Offset: 1

David W. Cantrell (DWCantrell(AT)sigmaxi.net), Nov 01 2003

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 34, equation 34:7:4 at page 325.

Vaclav Kotesovec, Table of n, a(n) for n = 1..250
Vaclav Kotesovec, Asymptotic of the coefficients A079330 / A088989.
Eric Weisstein's World of Mathematics, Tanc Function.

Numerators are in A079330.

Cf. A224196.

Last/@Partition[CoefficientList[InverseSeries[Series[x+Cot[x], {x, 0, 50}], q], 1/q], 2]

A245532 Decimal expansion of b_3, a constant associated with the 3rd Du Bois Reymond constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 25 2014

			0.017327140547366991288083189869067399070958360636432145130492163368346...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.12 Du Bois Reymond Constants, pp. 238-239.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A224196.

Digits:=100: evalf((-1/4)*(exp(3)-3*exp(1)-12)); # Wesley Ivan Hurt, Jul 26 2014

b3 = (-1/4)*(E^3 - 3*E - 12); Join[{0}, RealDigits[b3, 10, 101] // First]

(12+3*exp(1)-exp(3))/4 \\ Charles R Greathouse IV, Jul 25 2014

b_3 = (-1/4)*(e^3 - 3*e - 12).

Equals 2*sum((-1)^(n+1)/(1+xi(n)^2)^(3/2), (n=1..infinity)), where xi(n) is the n-th positive solution to tan(x)=x.

A255272 Decimal expansion of the second smallest positive root of tan(x) = x.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 20 2015

This constant is quite close to 5*Pi/2 - 1/8 = 7.72898...

Searching for solutions x=k*Pi+Pi/2-e and small e, for k=1,2,3.... means via the approximation tan(x) = 1/e-e/3-e^3/45... that e is approximately 1/(k*Pi+Pi/2), so the constants x are close to k*Pi+Pi/2-1/(k*Pi+Pi/2). Here k=2 and the constant is close to 5*Pi/2-2/(5*Pi) = 7.7266576... - R. J. Mathar, Jul 11 2024

			7.72525183693770716419506893306298662637815930461...

Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A115365 (smallest positive root), A062546 (C_2 = 2nd du Bois-Reymond constant), A224196 (C_3), A207528 (C_4), A243108 (C_5), A245333 (C_6).

Cf. A079330, A088989.

xi[n_] := x /. FindRoot[Tan[x] == x, {x, n*Pi + Pi/2 - 1/(4*n)}, WorkingPrecision -> 102]; RealDigits[xi[2]] // First

solve(x=7,7.8,tan(x)-x) \\ Charles R Greathouse IV, Apr 20 2016

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

cp's OEIS Frontend

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

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A062546 Decimal expansion of the 2nd Du Bois-Reymond constant.

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A207528 Decimal expansion of 4th du Bois-Reymond constant.

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A243108 Decimal expansion of the 5th du Bois-Reymond constant.

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A079330 Numerators of coefficients of odd powers of 1/q in the solution series for tan(x)/x=1.

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A245333 Decimal expansion of the 6th du Bois-Reymond constant.

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A088989 Denominators of coefficients of odd powers of 1/q in the solution series for Tan[x]/x=1.

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A245532 Decimal expansion of b_3, a constant associated with the 3rd Du Bois Reymond constant.

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