A175638 Decimal expansion of the upper limit x such that Integral_{u=0..Pi*x} u*cot(u) du = 0.

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

Offset

0,1

Comments

Because the integral from u=0 up to u=Pi/2 equals log(2)*Pi/2 = A086054/2, this is also the x such that Integral_{u=Pi/2..Pi*x} u*cot(u) du = -log(2)*Pi/2. By partial integration, Integral_{u} u*cot(u) du = u*log(sin(u)) - Integral_{u} log(sin(u)) du, used with a Newton method in the Maple implementation.

Links

Table of n, a(n) for n=0..104.

G. Freiman and H. Halberstam, On a product of sines, Acta Arithmetica 49 issue 4 (1987) 377-385.

Example

x = 0.7912265710...

Maple

intu := proc(u) u*log(sin(u)) - int( log(sin(t)), t=Pi/2..u) ; evalf(%) ; end proc:
Digits := 80 : x := 0.79122 :
for it from 1 to 10 do x0 := intu(evalf(Pi*x))+Pi*log(2)/2 ; xnew := x-evalf(x0)/Pi^2/x/cot(Pi*x) ; x := evalf(xnew) ; print(x) ; end do:

Mathematica

First@ RealDigits@ Re[ FindRoot[ Integrate[ u*Cot[u], {u, 0, x*Pi}], {x, 0.7}, WorkingPrecision -> 2^7][[1, 2]]] (* Robert G. Wilson v, Aug 03 2010 *)

Crossrefs

Sequence in context: A199392 A120670 A357330 * A091900 A222135 A086318

Adjacent sequences: A175635 A175636 A175637 * A175639 A175640 A175641

Keyword

cons,nonn

Author

R. J. Mathar, Aug 01 2010

Extensions

More terms from Robert G. Wilson v, Aug 03 2010

Status

approved