A212186 Decimal expansion of the integral over exp(x)/sqrt(1-x^2) dx between 0 and 1.

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

Offset: 1

Author

R. J. Mathar, Feb 13 2013

Keywords

• cons
• easy
• nonn

Comments

This appears as the first integral in an attempt to expand exp(x) in a Chebyshev series between 0 and 1. Other integrals of the higher order terms in that expansion are generally bootstrapped from the lower order terms.

If we substitute x=cos(y), this is the integral over exp(cos(y)) dy from y=0 to y=Pi/2, which matches (apart from the upper limit) eq. 3.915.4 of the Gradsteyn-Ryzhik tables. - R. J. Mathar, Feb 15 2013

Examples

			3.104379017855555098181769863187794767228...
		

Links

• A. Karageorghis, A note on the Chebyshev coefficients of the general order derivative of an infinitely differentiable function, J. Comp. Appl. Math 21 (1988) 129-132.

Programs

Mathematica

RealDigits[ Pi*(BesselI[0, 1] + StruveL[0, 1])/2, 10, 107] // First (* Jean-François Alcover, Feb 21 2013 *)
RealDigits[Integrate[Exp[x]/Sqrt[1-x^2],{x,0,1}],10,120][[1]] (* Harvey P. Dale, Jul 05 2025 *)

Formula

Equals Pi*(A197036+A197037)/2 .