Erfi

The imaginary error function $\operatorname{erfi}(z) = -i\operatorname{erf}(iz)$ .

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaError functionMathWorldErfiErfWikidataQ579262DLMF7.1.p47.2.E1FungrimErfi✓Fungrim entry01440f✓603a49✓Wolfram LanguageErfi!

Erfi(complex | signed_infinity) -> complex | signed_infinityas compute-engine declares it

SageMatherfi(complex | signed_infinity)mpmatherfi(complex | signed_infinity)Juliaenumeratio_ball(erfi, complex | signed_infinity)

Typed signatures

Type	Package	Applies
`(complex \| signed_infinity) -> complex \| signed_infinity`	`compute-engine`	always

Domain: Compute engine

Details

Defined as $\operatorname{erfi}(z) = -i\operatorname{erf}(iz)$ , real-valued for real $z$ (unlike $\operatorname{erf}(iz)$ itself, which is imaginary). See Erf.
$\operatorname{erfi}(0) = 0$ ; $\operatorname{erfi}(\pm\infty) = \pm\infty$ , unbounded, unlike $\operatorname{erf}$ 's $\pm 1$ saturation.
Odd: $\operatorname{erfi}(-z) = -\operatorname{erfi}(z)$ .
$\operatorname{erfi}(\widetilde\infty)$ , the undirected complex infinity, has no single value: it grows like $\exp(z^2)/z$ off the real axis while diverging along it, the same direction-dependence as Erf and Erfc there. compute-engine's model has one complex infinity and no directed one, so ours is a type error at boxing; Wolfram leaves the call unevaluated.
compute-engine requires an inexact (floating-point) argument to produce a numeric value under plain evaluation, the same convention as Erf.

Plot

ComplexPlot

real slice

1 more kept as data, checked against the oracles

Basic #

\mathrm{Erfi}(0)

Out

0

A floating-point argument evaluates directly to a high-precision decimal

\mathrm{Erfi}(0.5)

Out

0.614\,952\,094\,696\,510\,980\,84

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

\mathrm{Erfi}(\bigl\lbrack0, 1\bigr\rbrack)

Out

\bigl\lbrack0, \mathrm{Erfi}(1)\bigr\rbrack

Listable: threads over its limits at $\pm\infty$ , unbounded unlike Erfc's saturating 0/2

\mathrm{Erfi}(\bigl\lbrack∞, -∞\bigr\rbrack)

Out

\bigl\lbrack∞, -∞\bigr\rbrack

Possible issues #

Erfi(ComplexInfinity) is Indeterminate, inheriting Erf's direction-dependent growth off the real axis

Known value · DLMF 7.2.2

About

Examples	6
Checked against	Wolfram Language 5 agree1 differSageMath 4 agree2 differmpmath 5 agreeJulia 4 agree1 error
Crosswalk	Wikipedia MathWorld ×2 Wikidata DLMF ×2 Fungrim Fungrim entry ×2 Wolfram LanguageSageMathmpmathJulia

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