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Si(AngouriMath.Entity.Number.Complex)

Method (no overloads)

Summary

The sine integral, Si(z) = sum_(k >= 0) (-1)^k z^(2k + 1)/((2k + 1) (2k + 1)!), at the
working precision, for every complex z: entire, odd, real on the real line and
imaginary on the imaginary axis.

Remarks

With P digits of working precision, up to |z| = (P + 5) ln 10 the four
integrals are summed as their series, which cancel (|z| - |Im z|)/ln 10 digits for
Si and Ci, and (|z| - |Re z|)/ln 10 for Shi and Chi,
carried as guard digits. Beyond that they are the exponential integral's, whose
asymptotic series answers there: Shi(z) = (Ei(z) - Ei(-z))/2 - i pi/2 sgn(Im z),
Chi(z) = (Ei(z) + Ei(-z))/2 with i pi/2 sgn(Im z), or i pi on the
negative real axis, Si(z) = -i Shi(i z) and Ci(z) = Chi(i z) + ln z - ln(i z),
each branch term read off (ln z - ln(1/z))/2 in Ei(AngouriMath.Entity.Number.Complex).
On the axes where a value is real or imaginary, or has a known imaginary part, it is
given so, and not with a rounding residue: Ci(-x) is Ci(x) + i pi exactly.
https://github.com/asc-community/AngouriMath/issues/1501

AngouriMath

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AngouriMath

Si​(AngouriMath.​Entity.​Number.​Complex)

Si(AngouriMath.Entity.Number.Complex)