gives the complementary error function .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Erfc[z] is given by .
  • For certain special arguments, Erfc automatically evaluates to exact values.
  • Erfc can be evaluated to arbitrary numerical precision.
  • Erfc automatically threads over lists.
  • Erfc can be used with Interval and CenteredInterval objects. »


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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (40)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate Erf efficiently at high precision:

Erfc threads elementwise over lists:

Erfc can be used with Interval and CenteredInterval objects:

Specific Values  (3)

Simple exact values are generated automatically:

Values at infinity:

Find the inflection point as the root of :

Visualization  (2)

Plot the Erfc function:

Plot the real part of :

Plot the imaginary part of :

Function Properties  (9)

Erfc is defined for all real and complex values:

Erfc takes all real values between 0 and 2:

Erfc has the mirror property erfc(TemplateBox[{z}, Conjugate])=TemplateBox[{{erfc, (, z, )}}, Conjugate]:

Erfc is an analytic function of x:

It has no singularities or discontinuities:

Erfc is nonincreasing:

Erfc is injective:

Erfc is not surjective:

Erfc is non-negative:

Erfc is neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the n^(th) derivative:

Integration  (3)

Indefinite integral of Erfc:

Definite integral Erfc:

More integrals:

Series Expansions  (4)

Taylor expansion for Erfc:

Plot the first three approximations for Erfc around :

General term in the series expansion of Erfc:

Asymptotic expansion of Erfc:

Erfc can be applied to a power series:

Integral Transforms  (3)

Compute the Fourier transform of Erfc using FourierTransform:



Function Identities and Simplifications  (3)

Use FunctionExpand to convert to other functions:

Integral definition of Erfc:

Argument involving basic arithmetic operations:

Function Representations  (4)

Relationship of Erfc to Erf:

Erfc can be represented as a DifferentialRoot:

Erfc can be represented in terms of MeijerG:

TraditionalForm formatting:

Applications  (5)

The CDF of NormalDistribution can be expressed in terms of the complementary error function:

The probability that a random value is greater than :

The solution of the heat equation for a piecewiseconstant initial condition:

A check that the solution fulfills the heat equation:

The plot of the solution for different times:

Define the scaled complementary error function using HermiteH:

Interference pattern at the edge of a shadow:

The lifetime of a device follows a BirnbaumSaunders distribution. Find the reliability of the device:

The hazard function has the horizontal asymptote :

Find the reliability of two such devices in series:

Find the reliability of two such devices in parallel:

Compare the reliability of both systems for and :

Properties & Relations  (3)

Use FunctionExpand to convert to other functions:

Compose with inverse functions:

Solve a transcendental equation:

Possible Issues  (3)

For large arguments, intermediate values may underflow:

The error function for large negative real-part arguments can be very close to 2:

Very large arguments can give unevaluated results:

Neat Examples  (1)

A continued fraction whose partial numerators are consecutive integers:

Its limit can be expressed in terms of Erfc:

Wolfram Research (1991), Erfc, Wolfram Language function, (updated 2022).


Wolfram Research (1991), Erfc, Wolfram Language function, (updated 2022).


Wolfram Language. 1991. "Erfc." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022.


Wolfram Language. (1991). Erfc. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_erfc, author="Wolfram Research", title="{Erfc}", year="2022", howpublished="\url{}", note=[Accessed: 14-July-2024 ]}


@online{reference.wolfram_2024_erfc, organization={Wolfram Research}, title={Erfc}, year={2022}, url={}, note=[Accessed: 14-July-2024 ]}