—   argument ($x\geq0$)

For $x>0$, $\operatorname{Shi}(x)=\int_0^x\sinh(t)/t\,\mathrm{d}t$ and $\operatorname{Chi}(x)=\gamma+\log x+\int_0^x(\cosh(t)-1)/t\,\mathrm{d}t$ [4].

For $x>0$, $\operatorname{Ei}(x)=\operatorname{Shi}(x)+\operatorname{Chi}(x)$ [4].

The functions $\operatorname{Si}$ and $\operatorname{Shi}$ are odd [4]. With the principal branch of the logarithm, for $x>0$, $\operatorname{Ci}(-x)=\operatorname{Ci}(x)+i\pi$ and $\operatorname{Chi}(-x)=\operatorname{Chi}(x)+i\pi$ [5].

The small positive endpoints give a sparse reference from the subunit regime into exponential growth, not a denominator-bounded or decimal grid. The logarithmic endpoints are selected by the substitution $t=\log v$: the hyperbolic kernel then has rational endpoint data because $\sinh(\log v)=(v-v^{-1})/2$ and $\cosh(\log v)=(v+v^{-1})/2$. These are exploratory identification targets, not claims of separate published applications. The definitions and the real positive branch follow [4].

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

RBF = RealBallField(numberdb.bits(100, losing=64))
RBF(QQ(1)).Shi()

Python

import mpmath

mpmath.mp.dps = 100
mpmath.shi(1)

Wikipedia: Exponential integral

Wikipedia: Logarithmic integral function

Wikipedia: Trigonometric integral

DLMF 6.2: Definitions and Basic Properties

DLMF 6.4: Analytic Continuation

DLMF 8.19: Generalized Exponential Integral

OEIS A099283: decimal expansion of $\operatorname{Shi}(1)$

Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ —   the companion hyperbolic integral; their sum is $\operatorname{Ei}$ (2)

Values of the exponential integral —   $\operatorname{Ei}=\operatorname{Shi}+\operatorname{Chi}$

Values of the sine integral $\operatorname{Si}(x)$ —   the same integral with $\sin$ in place of $\sinh$

Wilbraham-Gibbs constant —   stores $\operatorname{Si}(\pi)$ and its normalisations

Stieltjes constants —   contains $\gamma_0$, the Euler-Mascheroni constant appearing in $\operatorname{Ei}$, $\operatorname{Ci}$ and $\operatorname{Chi}$

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it retains the exact origin and unit reference and includes $x=1/3,1/2,2,3,4,5,6$ and $\log 2,\log 3,\log 5$ as bounded exploratory hyperbolic-integral references)

How they were obtained:

All retained numeric strings, exact values and row annotations are unchanged. The standalone generate.py, balanced_values.py and selection.json reproduce the entire current selection. Each finite Arb enclosure at 733 bits must fit inside the interval claimed by the written last place; a second run uses 912 bits. New nonexact entries have 100 significant digits.

more

Every new entry has an independent mpmath quadrature, closed-form or inverse-function check at 150 and 200 decimal digits, with a nonzero known-integral control. Arguments are exact rationals or propagated balls from exact expressions; root arguments use strict sign-certified rational brackets. Lambert and saddle roots also pass endpoint sign tests on the written last-place interval. The measurements are in coverage-checks.json. Older attachments are retained for provenance; their previous enumerations are historical, and the active enumeration is selection.json.