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%I #9 Jun 27 2023 15:08:09
%S 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,
%T 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,
%U 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
%N Decimal expansion of the upper limit x such that Integral_{u=0..Pi*x} u*cot(u) du = 0.
%C 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.
%H G. Freiman and H. Halberstam, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa49/aa4944.pdf">On a product of sines</a>, Acta Arithmetica 49 issue 4 (1987) 377-385.
%e x = 0.7912265710...
%p intu := proc(u) u*log(sin(u)) - int( log(sin(t)),t=Pi/2..u) ; evalf(%) ; end proc:
%p Digits := 80 : x := 0.79122 :
%p 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:
%t 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 *)
%K cons,nonn
%O 0,1
%A _R. J. Mathar_, Aug 01 2010
%E More terms from _Robert G. Wilson v_, Aug 03 2010
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