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Decimal expansion of the imaginary part of the complex root of cos(x + i*y) = x + i*y with least x > 0 and y > 0.
3

%I #8 Jun 14 2020 22:47:12

%S 2,5,4,4,8,8,5,7,6,6,8,8,5,7,0,9,3,2,6,5,5,5,9,7,0,4,2,5,6,7,3,0,9,9,

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

%U 3,8,5,7,2,0,0,9,8,1,3,1,8,0,4,5,3,1,7

%N Decimal expansion of the imaginary part of the complex root of cos(x + i*y) = x + i*y with least x > 0 and y > 0.

%H Henry E. Fettis, <a href="https://www.jstor.org/stable/2005324">Complex Roots of sin z = az, cos z = az, and cosh z = az</a>, Mathematics of Computation, Vol. 30, No. 135 (1976), pp. 541-545, Table 18, <a href="https://doi.org/10.1090/S0025-5718-1976-0418401-9">alternative link</a> (without the tables).

%H T. H. Miller, <a href="http://dx.doi.org/10.1017/S0013091500030868">On the numerical values of the roots of the equation cos x = x</a>, Proc. Edinburgh Math. Soc., Vol. 9 (1890), pp. 80-83.

%H T. Hugh Miller, <a href="https://doi.org/10.1017/S001309150003460X">On the imaginary roots of cos x = x</a>, Proc. Edinburgh Math. Soc., Vol. 21 (1902), pp. 160-162 (the last 3 pages of the pdf file).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DottieNumber.html">Dottie Number</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dottie_number">Dottie number</a>.

%e 2.54488576688570932655597042567309972354822580168081...

%t z = {x, y} /. FindRoot[{x == Cos[x]*Cosh[y], y == -Sin[x]*Sinh[y]}, {{x, 5}, {y, 2}}, WorkingPrecision -> 100]; RealDigits[z[[2]], 10, 90][[1]]

%Y Cf. A003957, A335563, A335564, A335565 (the real part).

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, Jun 14 2020