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

Author

Robert G. Wilson v, Oct 09 2014

Keywords

• nonn
• cons

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

Examples

			0.86576948323965862428960184619184444137967919924876009961184822974244822945841...
The wedge angle in degrees:
49.6049374208547003776513077348112118247866748819092710723979907940346891648208... - _Stanislav Sykora_, May 31 2015
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Wikipedia, Gudermannian function.

Crossrefs

Cf. A248617, A257777, A258428.

Programs

Maple

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

Mathematica

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

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

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)