A335566 -id:A335566 - cp's OEIS frontend

A335563 Decimal expansion of the real part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

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

Offset: 0

Amiram Eldar, Jun 14 2020

			0.96226600633306068948508092593102537827547141928666...

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.

Cf. A003957, A335564 (the imaginary part), A335565, A335566.

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

A335564 Decimal expansion of the imaginary part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 14 2020

			1.11097438808469174490088735849365794466180028421014...

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.

Cf. A003957, A335563 (the real part), A335565, A335566.

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

A335565 Decimal expansion of the real part of the complex root of cos(x + iy) = x + iy with least x > 0 and y > 0.

Original entry on oeis.org

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

Amiram Eldar, Jun 14 2020

			5.86956037734761439408817256819284864079702760098009...

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.

Cf. A003957, A335563, A335564, A335566 (the imaginary part).

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]]

cp's OEIS Frontend

A335563 Decimal expansion of the real part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

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A335564 Decimal expansion of the imaginary part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

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A335565 Decimal expansion of the real part of the complex root of cos(x + iy) = x + iy with least x > 0 and y > 0.

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cp's OEIS Frontend

A335563 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.

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A335564 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.

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Mathematica

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.

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Author

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Programs

Mathematica

A335563 Decimal expansion of the real part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

A335564 Decimal expansion of the imaginary part of the complex root of cos(x + iy) = x - iy with least x > 0 and y > 0.

A335565 Decimal expansion of the real part of the complex root of cos(x + iy) = x + iy with least x > 0 and y > 0.