WOLFRAM

Erfc[z]

gives the complementary error function .

Details

  • 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. »

Examples

open allclose all

Basic Examples  (5)Summary of the most common use cases

Evaluate numerically:

Out[1]=1

Plot over a subset of the reals:

Out[1]=1

Plot over a subset of the complexes:

Out[1]=1

Series expansion at the origin:

Out[1]=1

Series expansion at Infinity:

Out[1]=1

Scope  (40)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

Out[1]=1

Evaluate to high precision:

Out[1]=1

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

Out[2]=2

Evaluate for complex arguments:

Out[1]=1

Evaluate Erf efficiently at high precision:

Out[1]=1
Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Out[1]=1
Out[2]=2

Or compute average-case statistical intervals using Around:

Out[3]=3

Compute the elementwise values of an array:

Out[1]=1

Or compute the matrix Erfc function using MatrixFunction:

Out[2]=2

Specific Values  (3)

Simple exact values are generated automatically:

Out[1]=1

Values at infinity:

Out[1]=1

Find the inflection point as the root of :

Out[1]=1
Out[2]=2

Visualization  (2)

Plot the Erfc function:

Out[1]=1

Plot the real part of :

Out[1]=1

Plot the imaginary part of :

Out[2]=2

Function Properties  (9)

Erfc is defined for all real and complex values:

Out[1]=1
Out[2]=2

Erfc takes all real values between 0 and 2:

Out[1]=1

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

Out[1]=1

Erfc is an analytic function of x:

Out[1]=1

It has no singularities or discontinuities:

Out[2]=2
Out[3]=3

Erfc is nonincreasing:

Out[1]=1

Erfc is injective:

Out[1]=1
Out[2]=2

Erfc is not surjective:

Out[1]=1
Out[2]=2

Erfc is non-negative:

Out[1]=1

Erfc is neither convex nor concave:

Out[1]=1

Differentiation  (3)

First derivative:

Out[1]=1

Higher derivatives:

Out[1]=1
Out[2]=2

Formula for the n^(th) derivative:

Out[1]=1

Integration  (3)

Indefinite integral of Erfc:

Out[1]=1

Definite integral Erfc:

Out[1]=1

More integrals:

Out[1]=1
Out[2]=2
Out[3]=3

Series Expansions  (4)

Taylor expansion for Erfc:

Out[1]=1

Plot the first three approximations for Erfc around :

Out[6]=6

General term in the series expansion of Erfc:

Out[1]=1

Asymptotic expansion of Erfc:

Out[1]=1

Erfc can be applied to a power series:

Out[1]=1

Integral Transforms  (3)

Compute the Fourier transform of Erfc using FourierTransform:

Out[1]=1

LaplaceTransform:

Out[1]=1

MellinTransform:

Out[1]=1

Function Identities and Simplifications  (3)

Use FunctionExpand to convert to other functions:

Out[1]=1
Out[2]=2

Integral definition of Erfc:

Out[1]=1

Argument involving basic arithmetic operations:

Out[1]=1
Out[2]=2

Function Representations  (4)

Relationship of Erfc to Erf:

Out[1]=1
Out[2]=2

Erfc can be represented as a DifferentialRoot:

Out[1]=1

Erfc can be represented in terms of MeijerG:

Out[1]=1
Out[2]=2

TraditionalForm formatting:

Applications  (5)Sample problems that can be solved with this function

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

Out[1]=1

The probability that a random value is greater than :

Out[2]=2
Out[3]=3

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

A check that the solution fulfills the heat equation:

Out[2]=2

The plot of the solution for different times:

Out[3]=3

Define the scaled complementary error function using HermiteH:

Out[1]=1
Out[2]=2

Interference pattern at the edge of a shadow:

Out[1]=1

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

Out[1]=1

The hazard function has the horizontal asymptote :

Out[2]=2
Out[3]=3

Find the reliability of two such devices in series:

Out[4]=4

Find the reliability of two such devices in parallel:

Out[5]=5

Compare the reliability of both systems for and :

Out[6]=6

Properties & Relations  (3)Properties of the function, and connections to other functions

Use FunctionExpand to convert to other functions:

Out[1]=1
Out[2]=2

Compose with inverse functions:

Out[1]=1
Out[2]=2

Solve a transcendental equation:

Out[1]=1

Possible Issues  (3)Common pitfalls and unexpected behavior

For large arguments, intermediate values may underflow:

Out[1]=1

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

Out[1]=1

Very large arguments can give unevaluated results:

Out[1]=1

Neat Examples  (1)Surprising or curious use cases

A continued fraction whose partial numerators are consecutive integers:

Out[1]=1

Its limit can be expressed in terms of Erfc:

Out[2]=2
Wolfram Research (1991), Erfc, Wolfram Language function, https://reference.wolfram.com/language/ref/Erfc.html (updated 2022).
Wolfram Research (1991), Erfc, Wolfram Language function, https://reference.wolfram.com/language/ref/Erfc.html (updated 2022).

Text

Wolfram Research (1991), Erfc, Wolfram Language function, https://reference.wolfram.com/language/ref/Erfc.html (updated 2022).

Wolfram Research (1991), Erfc, Wolfram Language function, https://reference.wolfram.com/language/ref/Erfc.html (updated 2022).

CMS

Wolfram Language. 1991. "Erfc." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Erfc.html.

Wolfram Language. 1991. "Erfc." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Erfc.html.

APA

Wolfram Language. (1991). Erfc. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Erfc.html

Wolfram Language. (1991). Erfc. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Erfc.html

BibTeX

@misc{reference.wolfram_2025_erfc, author="Wolfram Research", title="{Erfc}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Erfc.html}", note=[Accessed: 29-March-2025 ]}

@misc{reference.wolfram_2025_erfc, author="Wolfram Research", title="{Erfc}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Erfc.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_erfc, organization={Wolfram Research}, title={Erfc}, year={2022}, url={https://reference.wolfram.com/language/ref/Erfc.html}, note=[Accessed: 29-March-2025 ]}

@online{reference.wolfram_2025_erfc, organization={Wolfram Research}, title={Erfc}, year={2022}, url={https://reference.wolfram.com/language/ref/Erfc.html}, note=[Accessed: 29-March-2025 ]}