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Ei

 Method with 2 overloads

• Ei​(AngouriMath.​Entity.​Number.​Complex)

  Summary

  The exponential integral, Ei(z) = gamma + (ln z - ln(1/z))/2 + sum_(k >= 1) z^k/(k k!),
  at the working precision, for every complex z; Ei(0) is -oo.
  

  Remarks

  (ln z - ln(1/z))/2 is ln z off the negative real axis and ln |z| on it, so Ei is real on the whole real line but 0, as SymPy's, mpmath's and
  Mathematica's is: on the positive axis it is the principal value of
  int_-oo^x e^t/t dt, on the negative axis -E1(-x), and just above or below
  the negative axis it is i pi more or less.
  With P digits of working precision, up to |z| = (P + 5) ln 10 it is summed
  as written. The terms peak near k = |z| at about e^|z|/|z|, while
  Ei(z) is about e^Re(z)/|z|, so the sum cancels (|z| - Re z)/ln 10 digits, which are carried as guard digits: none on the positive real axis, and twice
  |x|/ln 10 on the negative one.
  Beyond that it is e^z/z sum_k k!/z^k (Abramowitz and Stegun 5.1.51), cut at its
  smallest term, which is about e^(-|z|) of the sum and so below the precision
  there, and off the real axis the logarithm's branch adds i pi sgn(Im z).
  The result is rounded to DecimalPrecisionContext, whose
  exponent range bounds what can be represented: Ei(-500), about 1e-220,
  can come out as zero there.
  https://github.com/asc-community/AngouriMath/issues/1501
  

• Ei​(PeterO.​Numbers.​EDecimal,​PeterO.​Numbers.​EDecimal,​PeterO.​Numbers.​EContext,​System.​Int32)

  Summary

  Ei(re + i im) in a context of context's precision and
  extraDigits more, before any rounding back.