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Erfc

Status: Stable

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

Description

Erfc[z]
    gives the complementary error function erfc(z) = 1 - erf(z).
Erfc[0] = 1, Erfc[Infinity] = 0, Erfc[-Infinity] = 2. An entire
function. Real inputs evaluate via libm/MPFR erfc (cancellation-free);
complex inputs via 1 - erf(z) at machine or arbitrary (MPFR) precision.
D[Erfc[z], z] = -(2/Sqrt[Pi]) E^(-z^2). Listable.

Examples

No verified examples yet for this function.

Implementation notes

  • Exact special values: Erfc[0] = 1, Erfc[Infinity] = 0,

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]:= Erfc[0]
Out[1]= 1

In[1]:= N[Erfc[2], 40]
Out[1]= 0.0046777349810472658379307436327470713891081

In[1]:= N[Erfc[1 + I], 25]
Out[1]= -0.31615128169794764488027107 - 0.19045346923783468628410886*I

In[1]:= Series[Erfc[x], {x, 0, 5}]
Out[1]= 1 + -2/Sqrt[Pi] x + 2/3/Sqrt[Pi] x^3 + -1/5/Sqrt[Pi] x^5 + O[x]^6

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

Notes

Erfc[z] = 1 - Erf[z] is the complementary error function, with Erfc[0] = 1, Erfc[Infinity] = 0, and Erfc[-Infinity] = 2. Real inputs evaluate through the cancellation-free erfc of libm / MPFR — important in the right tail, where the N[Erfc[2], 40] example keeps full precision instead of losing it to a 1 - Erf subtraction. Complex inputs route through 1 - Erf[z] at machine or arbitrary precision. The derivative is D[Erfc[z], z] = -(2/Sqrt[Pi]) E^(-z^2), and Erfc is Listable.