A118322 Decimal expansion of perimeter of the closed portion of the bow curve.

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

Offset: 1

Author

Eric W. Weisstein, Apr 23 2006

Keywords

• nonn
• cons

Comments

Writing x=r*cos(phi), y=r*sin(phi), r=sin(phi)*(1-2*sin^2(phi))/cos^4(phi) in circular coordinates gives the arc length of one wing of Integral_{phi=0..Pi/4} sqrt((dx/dphi)^2 + (dy/dphi)^2) dphi = Integral_{s=0..1/sqrt(2)} sqrt(1-5*s^2+20*s^6) / (1-s^2)^3 ds. - R. J. Mathar, Mar 23 2010

Examples

			1.9215113651725125701...
		

Links

• Eric Weisstein's World of Mathematics, Bow

Programs

Maple

Digits := 120 : f := 2*sqrt(1-5*x^2+20*x^6)/(1-x^2)^3 ; Int(f,x=0..1/sqrt(2.0)) ; x := evalf(%) ; # R. J. Mathar, Mar 23 2010

Mathematica

f[x_] := 2*Sqrt[1-5*x^2+20*x^6]/(1-x^2)^3; First[ RealDigits[ NIntegrate[f[x], {x, 0, 1/Sqrt[2]}, WorkingPrecision -> 120], 10, 105]](* Jean-François Alcover, Jun 08 2012, after R. J. Mathar *)

Extensions

More digits from R. J. Mathar, Mar 23 2010