A257818 Decimal expansion of the imaginary part of li(i), i being the imaginary unit.

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

Offset: 1

Author

Stanislav Sykora, May 10 2015

Keywords

• nonn
• cons

Comments

li(x) is the logarithmic integral function, extended to the whole complex plane. The corresponding real part is in A257817.

Examples

			2.941558494949385099300999980021326772089446035251921590122704439...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• Eric Weisstein's World of Mathematics, Logarithmic Integral
• Wikipedia, Logarithmic integral function

Crossrefs

Cf. A019669, A257817.

Programs

Maple

evalf(Im(Li(I)), 120); # Vaclav Kotesovec, May 10 2015
evalf(Pi/2*(1+Sum(((-1)^k*(Pi/2)^(2*k)/(2*k+1)!/(2*k+1)), k=0..infinity)), 120); # Vaclav Kotesovec, May 10 2015

Mathematica

RealDigits[Im[LogIntegral[I]], 10, 120][[1]] (* Vaclav Kotesovec, May 10 2015 *)

PARI

li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
  if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
  return(+Pi*I-eint1(-log(c))));}
  a=imag(li(I))

Formula

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