A257820 Decimal expansion of the absolute value of the imaginary part of li(-1).

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

Offset: 1

Author

Stanislav Sykora, May 11 2015

Keywords

• nonn
• cons

Comments

The logarithmic integral function li(z) has a cut along the negative real axis which causes therein a discontinuity in the imaginary part of li(z). However, the absolute value of the imaginary part is continuous and its value is a well behaved function of any real argument, excepting z=+1. The above value corresponds to |imag(li(z))| at z=-1, the point where the corresponding real part (A257819) attains its maximum within the real interval (-infinity,+1).

Examples

			3.422733378777362789592375061797742805444394428668707820292256...
		

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. A000796, A257817, A257818, A257819, A257822.

Programs

Maple

evalf(Im(Li(-1)),120); # Vaclav Kotesovec, May 11 2015

Mathematica

RealDigits[Im[LogIntegral[-1]], 10, 120][[1]] (* Vaclav Kotesovec, May 11 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(-1))

Formula

Equals Pi*(1/2 + Sum[k=0..infinity]((-1)^k*Pi^(2*k)/(2*k+1)!/(2*k+1))).