cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A036793 Decimal expansion of (2/Pi)*Integral_{x=0..Pi} sin(x)/x dx.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Integral(sin(x)/x dx) = x - x^3/(3*3!) + x^5/(5*5!) - x^7/(7*7!) + ... . - Harry J. Smith, Apr 28 2009

Examples

			1.17897974447216727..., the constant in Gibbs phenomenon.

References

  • E. J. Borowski and J. M. Borwein, Dictionary of Mathematics, 3rd printing, Harper Collins, 1991, Gibbs phenomenon.
  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham Constant, p. 249.

Links

Crossrefs

Cf. A036791 (continued fraction), A061079 for Si( x ).

Programs

  • Mathematica
    RealDigits[ N[ (2/Pi)*SinIntegral[Pi], 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)
  • PARI
    { default(realprecision, 20080); y=0; x=Pi; m=x; x2=x*x; n=1; nf=1; s=1; while (x!=y, y=x; n++; nf*=n; n++; nf*=n; m*=x2; s=-s; x+=s*m/(n*nf)); x*=2/Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b036793.txt", n, " ", d)); } \\ Harry J. Smith, Apr 28 2009

Formula

A036792 divided by A019669. - R. J. Mathar, Mar 22 2011

A243267 Decimal expansion of 1/2+G/Pi, the highest limiting crest of a square wave Fourier series, where G is the Gibbs-Wilbraham constant.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 02 2014

Keywords

Examples

			1.08948987223608363511601442291245487...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham constant, p. 249.

Links

Crossrefs

Cf. A036792, A036793, A243268.

Programs

  • Mathematica
    G = SinIntegral[Pi]; RealDigits[1/2 + G/Pi, 10, 99] // First

A243268 Decimal expansion of 1/2-G/Pi, the lowest limiting trough of a square wave Fourier series, where G is the Gibbs-Wilbraham constant. [negated].

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 02 2014

Keywords

Examples

			-0.08948987223608363511601442291245487...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham constant, p. 249.

Links

Crossrefs

Cf. A036792, A036793, essentially the same as A243267.

Programs

  • Mathematica
    G = SinIntegral[Pi]; RealDigits[1/2 - G/Pi, 10, 97] // First

A036790 Continued fraction for Integral_{x=0..Pi} sin(x)/x.

Original entry on oeis.org

1, 1, 5, 1, 3, 15, 1, 5, 3, 2, 7, 2, 1, 62, 1, 3, 110, 1, 39, 1, 1, 1, 4, 3, 1, 2, 5, 2, 2, 28, 1, 1, 1, 2, 6, 2, 2, 1, 278, 3, 3, 1, 1, 4, 1, 4, 2, 5, 5, 2, 4, 1, 1, 1, 28, 1, 1, 203, 1, 89, 2, 2, 2, 1, 12, 1, 1, 3, 33, 1, 8, 3, 1, 2, 6, 1, 1, 3, 256, 1, 6, 5, 1, 6, 14, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Integral(sin(x)/x dx) = x - x^3/(3*3!) + x^5/(5*5!) - x^7/(7*7!) + ... - Harry J. Smith, May 01 2009

Examples

			1.851937051982466170361053370... = 1 + 1/(1 + 1/(5 + 1/(1 + 1/(3 + ...)))). - _Harry J. Smith_, May 01 2009

Links

Crossrefs

Cf. A036792 (decimal expansion).

Programs

  • PARI
    { allocatemem(932245000); default(realprecision, 21000); y=0; x=Pi; m=x; x2=x*x; n=1; nf=1; s=1; while (x!=y, y=x; n++; nf*=n; n++; nf*=n; m*=x2; s=-s; x+=s*m/(n*nf)); x=contfrac(x); for (n=1, 20000, write("b036790.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 01 2009

Extensions

Offset changed by Andrew Howroyd, Aug 03 2024

A245535 Decimal expansion of the analog of the Gibbs-Wilbraham constant for L_1 trigonometric polynomial approximation.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jul 25 2014

Keywords

Examples

			x0 = 1.376991769203938865765266614301624670814900061506257246...
g(x0) = 0.0657838882664480990565512180874704669499564803216...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham Constant, p. 249.

Links

Crossrefs

Cf. A036792, A036793, A243267, A243268.

Programs

  • Mathematica
    digits = 101; g[x_] := (PolyGamma[x/2] - PolyGamma[(x+1)/2] + 1/x)*Sin[Pi*x]/Pi; x0 = x /. FindRoot[g'[x] == 0, {x, 3/2}, WorkingPrecision -> digits+5]; RealDigits[g[x0], 10, digits] // First

Formula

Maximum g(x0) of the function g(x) = (psi(x/2) - psi((x+1)/2) + 1/x)*sin(Pi*x)/Pi, for x >= 1, where psi is the polygamma function.

A101512 Decimal expansion of the sine integral at 2*Pi.

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Jan 25 2005

Keywords

Comments

Si(Infinity)=Pi/2=~1.57079... (A019669).

Examples

			1.41815157613262845024578016229974942914245334929501627184272022969...

Crossrefs

Cf. A019669, A036792, A099281.

Programs

  • Mathematica
    RealDigits[ SinIntegral[2Pi], 10, 111] [[1]]

Formula

Equals Integral_{x=0..2*Pi} sin(x)/x dx.

A272335 Decimal expansion of a function approximation constant which is the analog of Gibbs's constant 2*G/Pi (A036793) for de la Vallée-Poussin sums.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 26 2016

Keywords

Examples

			1.14272812693068128481021845956657111930110150452947023957171253...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham Constant, p. 248.

Links

Crossrefs

Cf. A036792, A036793, A243267, A243268, A245535.

Programs

  • Mathematica
    (2/Pi)(2 SinIntegral[4 Pi/3] - SinIntegral[2 Pi/3]) // N[#, 101]& // RealDigits // First

Formula

Equals (2/Pi)*Integral_{t=0..2*Pi/3} (cos(t) - cos(2*t))/t^2 dt.
Equals (2/Pi)*(2*Si(4*Pi/3) - Si(2*Pi/3)), where Si is the Sine integral function.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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