A226975 Decimal expansion I_1(1), the modified Bessel function of the first kind.

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

Offset: 0

Author

Horst-Holger Boltz, Jun 25 2013

Keywords

• nonn
• cons

Comments

This is also the derivative of the zeroth modified Bessel function at 1.

Examples

			0.56515910399248502720769602760986330732889962162109...
		

References

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

Links

• Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Crossrefs

Cf. A002454, A197036.

Programs

Mathematica

RealDigits[BesselI[1, 1], 10, 110][[1]]

PARI

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

SageMath

((1/2) * sum(1 / (4^x * factorial(x) * rising_factorial(2, x)), x, 0, oo)).n(360)
# Peter Luschny, Jan 29 2024

Formula

From Antonio Graciá Llorente, Jan 29 2024: (Start)

I_1(1) = (1/2) * Sum_{k>=0} (2*k)/(4^k*k!^2) = (1/2) * Sum_{k>=0} (2*k)/A002454(k).

Equals (1/2) * Sum_{k>=0} (4*k^2 + 4*k - 1) / (2*k)!!^2.

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

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

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