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Erfi

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Erfi[z]
    gives the imaginary error function erfi(z) = -I Erf[I z]
    = (2/Sqrt[Pi]) Integral_0^z e^(t^2) dt.
Erfi[0] = 0, Erfi[Infinity] = Infinity, Erfi[I Infinity] = I. An entire
function, odd in z. Real and complex inputs evaluate numerically at
machine or arbitrary (MPFR) precision; D[Erfi[z], z] = (2/Sqrt[Pi]) E^(z^2).
Listable.

Examples

No verified examples yet for this function.

Implementation notes

  • Exact special values: Erfi[0] = 0, Erfi[Infinity] = Infinity,

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Erfi[0]
Out[1]= 0

In[1]:= Erfi[-x]
Out[1]= -Erfi[x]

In[1]:= N[Erfi[1], 30]
Out[1]= 1.650425758797542876025337729561

In[1]:= Series[Erfi[x], {x, 0, 7}]
Out[1]= 2/Sqrt[Pi] x + 2/3/Sqrt[Pi] x^3 + 1/5/Sqrt[Pi] x^5 + 1/21/Sqrt[Pi] x^7 + O[x]^8

In[2]:= D[Erfi[x], x]
Out[2]= (2 E^x^2)/Sqrt[Pi]

Notes

Erfi[z] = -I Erf[I z] = (2/Sqrt[Pi]) Integral_0^z e^(t^2) dt is the imaginary error function, an entire odd function with Erfi[0] = 0, Erfi[Infinity] = Infinity, and Erfi[I Infinity] = I. Compared with Erf, the sign of every Maclaurin coefficient is positive, reflecting the +t^2 in the integrand. Real and complex arguments evaluate numerically at machine or arbitrary (MPFR) precision, the derivative is (2/Sqrt[Pi]) E^(z^2), and Erfi is Listable.