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A196823 Decimal expansion of the number c for which the curve y=1/(1+x^2) is tangent to the curve y=-c+cos(x), and 0

%I A196823 #17 Feb 16 2025 08:33:15
%S A196823 0,9,3,7,9,0,0,2,2,4,4,3,5,8,8,1,4,0,6,4,6,8,9,1,6,2,7,2,0,2,1,0,9,9,
%T A196823 8,6,7,0,9,0,1,2,8,8,0,7,8,5,3,3,2,8,7,2,7,1,6,2,8,5,9,7,3,8,8,1,3,4,
%U A196823 8,9,3,1,0,9,7,8,6,5,5,8,9,5,2,4,9,0,1,4,9,2,3,8,4,3,1,1,5,3,8,4
%N A196823 Decimal expansion of the number c for which the curve y=1/(1+x^2) is tangent to the curve y=-c+cos(x), and 0<x<2*Pi.
%H A196823 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WitchofAgnesi.html">Witch of Agnesi</a>
%e A196823 c=0.09379002244358814064689162720210998670901288078533287...
%t A196823 Plot[{1/(1 + x^2), -.094 + Cos[x]}, {x, 0, 1}]
%t A196823 t = x /. FindRoot[2 x == ((1 + x^2)^2) Sin[x], {x, .5, 1}, WorkingPrecision -> 100]
%t A196823 RealDigits[t]     (* A196822 *)
%t A196823 c = N[-Cos[t] + 1/(1 + t^2), 100]
%t A196823 RealDigits[-c]     (* A196823 *)
%t A196823 slope = N[-Sin[t], 100]
%t A196823 RealDigits[slope] (* A196824 *)
%Y A196823 Cf. A196822.
%K A196823 nonn,cons
%O A196823 0,2
%A A196823 _Clark Kimberling_, Oct 06 2011
%E A196823 0 prepended to get correct constant value by _Michel Marcus_, Feb 10 2015