A197036 Decimal expansion of the Modified Bessel Function I of order 0 at 1.

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

Offset: 1

Author

R. J. Mathar, Oct 08 2011

Keywords

• cons
• easy
• nonn

Examples

			1.26606587775200833559824462521471753760767031135496...
		

References

• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 51, page 504.

Links

• M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions, Chapter 9.6.
• Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Crossrefs

Cf. A002454, A242282.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), this sequence (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

Programs

Maple

BesselI(0,1) ;evalf(%) ;

Mathematica

RealDigits[BesselJ[0, I], 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

PARI

besseli(0,1) \\ Charles R Greathouse IV, Feb 19 2014

Formula

I_0(1) = Sum_{k>=0} 1/(4^k*k!^2) = Sum_{k>=0} 1/A002454(k).

Equals (1/Pi)*Integral_{t=0..Pi} exp(cos(t)) dt.

Equals BesselJ(0,i). - Jianing Song, Sep 18 2021

From Amiram Eldar, Jul 09 2023: (Start)

Equals exp(-1) * Sum_{k>=0} binomial(2*k,k)/(2^k*k!).

Equals e * Sum_{k>=0} (-1/2)^k * binomial(2*k,k)/k!. (End)