A335565 Decimal expansion of the real part of the complex root of cos(x + i*y) = x + i*y with least x > 0 and y > 0.
5, 8, 6, 9, 5, 6, 0, 3, 7, 7, 3, 4, 7, 6, 1, 4, 3, 9, 4, 0, 8, 8, 1, 7, 2, 5, 6, 8, 1, 9, 2, 8, 4, 8, 6, 4, 0, 7, 9, 7, 0, 2, 7, 6, 0, 0, 9, 8, 0, 0, 9, 6, 7, 2, 2, 4, 0, 8, 1, 2, 3, 5, 9, 7, 2, 4, 9, 9, 5, 0, 5, 2, 2, 1, 2, 9, 9, 2, 6, 9, 1, 3, 9, 9, 5, 2, 6
Offset: 1
Examples
5.86956037734761439408817256819284864079702760098009...
Links
- Henry E. Fettis, Complex Roots of sin z = az, cos z = az, and cosh z = az, Mathematics of Computation, Vol. 30, No. 135 (1976), pp. 541-545, Table 18, alternative link (without the tables).
- T. H. Miller, On the numerical values of the roots of the equation cos x = x, Proc. Edinburgh Math. Soc., Vol. 9 (1890), pp. 80-83.
- T. Hugh Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc., Vol. 21 (1902), pp. 160-162 (the last 3 pages of the pdf file).
- Eric Weisstein's World of Mathematics, Dottie Number.
- Wikipedia, Dottie number.
Programs
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Mathematica
z = {x, y} /. FindRoot[{x == Cos[x]*Cosh[y], y == -Sin[x]*Sinh[y]}, {{x, 5}, {y, 2}}, WorkingPrecision -> 100]; RealDigits[z[[1]], 10, 90][[1]]