A248618 Decimal expansion of the solution when inverse Gudermannian(x) equals 1.

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

Offset

0,1

Comments

Inverse of A248617.

This is the angle of the unique wedge having its apex at the origin and kissing the exponential curve y=exp(x) on one side, and its inverse logarithmic function y=log(x) on the other side. - Stanislav Sykora, May 31 2015

Links

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Wikipedia, Gudermannian function.

Formula

Equals arcsin((exp(2)-1)/(exp(2)+1)). - Vaclav Kotesovec, Oct 11 2014

Equals atan(e)-atan(1/e) = A257777-A258428. - Stanislav Sykora, May 31 2015

From Amiram Eldar, Apr 07 2022: (Start)

Equals 2*arctan(tanh(1/2)).

Equals Integral_{x=0..1} sech(x) dx. (End)

Example

0.86576948323965862428960184619184444137967919924876009961184822974244822945841...
The wedge angle in degrees:
49.6049374208547003776513077348112118247866748819092710723979907940346891648208... - Stanislav Sykora, May 31 2015

Maple

evalf(arcsin((exp(2)-1)/(exp(2)+1)), 100) # Vaclav Kotesovec, Oct 11 2014

Mathematica

RealDigits[ Gudermannian[ 1], 10, 111][[1]]

Prog

(PARI) asin((exp(2)-1)/(exp(2)+1)) \\ Michel Marcus, Oct 11 2014
(PARI) atan(exp(1))-atan(1/exp(1)) \\ Stanislav Sykora, May 31 2015
(PARI) 2*atan(exp(1))-Pi/2 \\ Charles R Greathouse IV, Jun 02 2015

Crossrefs

Cf. A248617, A257777, A258428.

Sequence in context: A021540 A100199 A161883 * A197329 A360167 A046266

Adjacent sequences: A248615 A248616 A248617 * A248619 A248620 A248621

Keyword

nonn,cons

Author

Robert G. Wilson v, Oct 09 2014

Status

approved