—   argument ($x\geq0$)

For real $x$, $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t$, $\operatorname{erfc}(x)=1-\operatorname{erf}(x)$, and $\operatorname{erfi}(x)=-i\operatorname{erf}(ix)$ [4].

Dawson's integral is $F(x)=e^{-x^2}\int_0^x e^{t^2}\,\mathrm{d}t=(\sqrt{\pi}/2)e^{-x^2}\operatorname{erfi}(x)$ [2].

The functions $\operatorname{erf}$, $\operatorname{erfi}$, $F$, $S$ and $C$ are odd, and $\operatorname{erfc}(-x)=2-\operatorname{erfc}(x)$.

Negative arguments are not listed separately. Values at negative arguments follow from (3).

At $x=0$ the value is exact, $\operatorname{erfi}(0)=0$, and is stored as the integer rather than as a hundred places of one.

The positive quadratic-coefficient integral $\int_0^1 e^{a t^2}\,dt=\sqrt\pi\operatorname{erfi}(\sqrt a)/(2\sqrt a)$ selects the square-root arguments. The coefficients $1\leq a\leq9$ form a short exploratory reference family; the finite cutoff is not intrinsic. The direct endpoints $1/2$ and $3/2$ complement the unit anchor in the subunit and crossover regimes. Negative arguments follow by oddness. These choices do not assert a published application for every coefficient. Definitions and the Dawson/error-function relation are in [8].

Sage

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

field = RealBallField(numberdb.bits(100, losing=64))
x = field(QQ(1))
x.erfi()

Python

import mpmath

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

Wikipedia: Error function

Wikipedia: Dawson function

Wikipedia: Fresnel integral

DLMF 7.2: Definitions

OEIS A099288: decimal expansion of the imaginary error function at 1

MathWorld: Probability Integral

DLMF 7.3: Graphics

DLMF 7.2 and 7.5: error functions and Dawson integral

Values of the error function $\operatorname{erf}(x)$ —   $\operatorname{erfi}(x)=-i\operatorname{erf}(ix)$

Values of the complementary error function $\operatorname{erfc}(x)$ —   another member of the error-function family

Values of Dawson's integral $F(x)$ —   $F(x)=(\sqrt{\pi}/2)e^{-x^2}\operatorname{erfi}(x)$

Values of the Fresnel sine integral $S(x)$ —   another member of the error-function family

Values of the Fresnel cosine integral $C(x)$ —   another member of the error-function family

Values of the exponential integral —   stores $\operatorname{Ei}(x)$ and $E_1(x)$, with exponential kernels $e^t/t$ and $e^{-t}/t$ rather than the Gaussian kernel $e^{-t^2}$

Values of the sine integral $\operatorname{Si}(x)$ —   stores $\operatorname{Si}(x)=\int_0^x\sin(t)/t\,\mathrm{d}t$, with linear phase in place of the Fresnel quadratic phase

Values of the cosine integral $\operatorname{Ci}(x)$ —   stores $\operatorname{Ci}(x)=\gamma+\log x+\int_0^x(\cos(t)-1)/t\,\mathrm{d}t$, with linear phase in place of the Fresnel quadratic phase

Values of the hyperbolic sine integral $\operatorname{Shi}(x)$ —   stores $\operatorname{Shi}(x)=\int_0^x\sinh(t)/t\,\mathrm{d}t$ at rational arguments

Values of the hyperbolic cosine integral $\operatorname{Chi}(x)$ —   stores $\operatorname{Chi}(x)=\gamma+\log x+\int_0^x(\cosh(t)-1)/t\,\mathrm{d}t$ at rational arguments

Values of the Gamma function at rational numbers —   contains $\Gamma(1/2)=\sqrt{\pi}$, the normalising constant in $\operatorname{erf}$ and $\operatorname{erfc}$

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it retains all earlier anchors and covers $x=\sqrt a$ for integers $1\leq a\leq9$, together with $x=1/2,3/2$ and the origin; the quadratic-coefficient family is a bounded exploratory reference, not a dense argument grid)

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.