A257822 -id:A257822 - cp's OEIS frontend

A257821 Decimal expansion of the unique real number a>0 such that the real part of li(-a) is zero.

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

Offset: 1

Stanislav Sykora, May 11 2015

As discussed in A257819, the real part of li(z) is a well behaved function for any real z, except for the singularity at z=+1. It has three roots: z=A070769 (the Soldner's constant), z=0, and z=-a. However, unlike in the other two cases, the imaginary part of li(-a) is not infinitesimal in the neighborhood of this root; it is described in A257822.

			2.4664082624126780751971033507759329502907808782774099823786...

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

Cf. A070769, A257819, A257822.

RealDigits[a/.FindRoot[Re[LogIntegral[-a]]==0,{a,2},WorkingPrecision->120]][[1]] (* Vaclav Kotesovec, May 11 2015 *)

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=-solve(x=-3,-1,real(li(x)))  \\ Better use excess realprecision

Satisfies real(li(-a)) = 0.

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

Original entry on oeis.org

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

Stanislav Sykora, May 11 2015

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).

			3.422733378777362789592375061797742805444394428668707820292256...

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

Cf. A000796, A257817, A257818, A257819, A257822.

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

RealDigits[Im[LogIntegral[-1]], 10, 120][[1]] (* Vaclav Kotesovec, May 11 2015 *)

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))

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

cp's OEIS Frontend

A257821 Decimal expansion of the unique real number a>0 such that the real part of li(-a) is zero.

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