Erfc
✖
Erfc
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 allBasic Examples (5)Summary of the most common use cases

https://wolfram.com/xid/0y8dob-gqdk4g

Plot over a subset of the reals:

https://wolfram.com/xid/0y8dob-g62iwn

Plot over a subset of the complexes:

https://wolfram.com/xid/0y8dob-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0y8dob-c8miyp

Series expansion at Infinity:

https://wolfram.com/xid/0y8dob-laddhh

Scope (40)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0y8dob-l274ju


https://wolfram.com/xid/0y8dob-cizwee

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

https://wolfram.com/xid/0y8dob-b3fv4a

Evaluate for complex arguments:

https://wolfram.com/xid/0y8dob-lqesq7

Evaluate Erf efficiently at high precision:

https://wolfram.com/xid/0y8dob-di5gcr


https://wolfram.com/xid/0y8dob-bq2c6r

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

https://wolfram.com/xid/0y8dob-nm3y5


https://wolfram.com/xid/0y8dob-lmyeh7

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0y8dob-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0y8dob-thgd2

Or compute the matrix Erfc function using MatrixFunction:

https://wolfram.com/xid/0y8dob-o5jpo

Specific Values (3)
Simple exact values are generated automatically:

https://wolfram.com/xid/0y8dob-e02kzi


https://wolfram.com/xid/0y8dob-hgfekk

Find the inflection point as the root of :

https://wolfram.com/xid/0y8dob-f2hrld


https://wolfram.com/xid/0y8dob-hihn50

Visualization (2)
Plot the Erfc function:

https://wolfram.com/xid/0y8dob-ecj8m7


https://wolfram.com/xid/0y8dob-e836u0


https://wolfram.com/xid/0y8dob-e7n1hs

Function Properties (9)
Erfc is defined for all real and complex values:

https://wolfram.com/xid/0y8dob-cl7ele


https://wolfram.com/xid/0y8dob-de3irc

Erfc takes all real values between 0 and 2:

https://wolfram.com/xid/0y8dob-evf2yr

Erfc has the mirror property :

https://wolfram.com/xid/0y8dob-heoddu

Erfc is an analytic function of x:

https://wolfram.com/xid/0y8dob-h5x4l2

It has no singularities or discontinuities:

https://wolfram.com/xid/0y8dob-mdtl3h


https://wolfram.com/xid/0y8dob-mn5jws

Erfc is nonincreasing:

https://wolfram.com/xid/0y8dob-nlz7s

Erfc is injective:

https://wolfram.com/xid/0y8dob-poz8g


https://wolfram.com/xid/0y8dob-ctca0g

Erfc is not surjective:

https://wolfram.com/xid/0y8dob-cxk3a6


https://wolfram.com/xid/0y8dob-frlnsr

Erfc is non-negative:

https://wolfram.com/xid/0y8dob-84dui

Erfc is neither convex nor concave:

https://wolfram.com/xid/0y8dob-8kku21

Differentiation (3)
Integration (3)
Indefinite integral of Erfc:

https://wolfram.com/xid/0y8dob-bponid

Definite integral Erfc:

https://wolfram.com/xid/0y8dob-b9jw7l


https://wolfram.com/xid/0y8dob-ed5h5p


https://wolfram.com/xid/0y8dob-ls0xa1


https://wolfram.com/xid/0y8dob-eowx9w

Series Expansions (4)
Taylor expansion for Erfc:

https://wolfram.com/xid/0y8dob-ewr1h8

Plot the first three approximations for Erfc around :

https://wolfram.com/xid/0y8dob-v2v18

General term in the series expansion of Erfc:

https://wolfram.com/xid/0y8dob-dznx2j

Asymptotic expansion of Erfc:

https://wolfram.com/xid/0y8dob-hb32tt

Erfc can be applied to a power series:

https://wolfram.com/xid/0y8dob-eneay1

Integral Transforms (3)
Compute the Fourier transform of Erfc using FourierTransform:

https://wolfram.com/xid/0y8dob-d71pqk


https://wolfram.com/xid/0y8dob-cb4me9


https://wolfram.com/xid/0y8dob-c26elq

Function Identities and Simplifications (3)
Use FunctionExpand to convert to other functions:

https://wolfram.com/xid/0y8dob-byom15


https://wolfram.com/xid/0y8dob-crn7k4

Integral definition of Erfc:

https://wolfram.com/xid/0y8dob-bh9t4q

Argument involving basic arithmetic operations:

https://wolfram.com/xid/0y8dob-h5mvqm


https://wolfram.com/xid/0y8dob-h8romk

Function Representations (4)

https://wolfram.com/xid/0y8dob-c333o7


https://wolfram.com/xid/0y8dob-lbst22

Erfc can be represented as a DifferentialRoot:

https://wolfram.com/xid/0y8dob-ecufp3

Erfc can be represented in terms of MeijerG:

https://wolfram.com/xid/0y8dob-joy7ok


https://wolfram.com/xid/0y8dob-ccfc26

TraditionalForm formatting:

https://wolfram.com/xid/0y8dob-i44j8y

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:

https://wolfram.com/xid/0y8dob-bmrcuj

The probability that a random value is greater than :

https://wolfram.com/xid/0y8dob-bkut9h


https://wolfram.com/xid/0y8dob-bwfdpi

The solution of the heat equation for a piecewise‐constant initial condition:

https://wolfram.com/xid/0y8dob-dgixnp
A check that the solution fulfills the heat equation:

https://wolfram.com/xid/0y8dob-e41rs

The plot of the solution for different times:

https://wolfram.com/xid/0y8dob-nwn7ib

Define the scaled complementary error function using HermiteH:

https://wolfram.com/xid/0y8dob-mr7mas


https://wolfram.com/xid/0y8dob-df5dsv

Interference pattern at the edge of a shadow:

https://wolfram.com/xid/0y8dob-mrd2h

The lifetime of a device follows a Birnbaum–Saunders distribution. Find the reliability of the device:

https://wolfram.com/xid/0y8dob-4m8ue6

The hazard function has the horizontal asymptote :

https://wolfram.com/xid/0y8dob-xz8iwk


https://wolfram.com/xid/0y8dob-zzzt7r

Find the reliability of two such devices in series:

https://wolfram.com/xid/0y8dob-xfhpfa

Find the reliability of two such devices in parallel:

https://wolfram.com/xid/0y8dob-7trpvs

Compare the reliability of both systems for and
:

https://wolfram.com/xid/0y8dob-qqswjr

Properties & Relations (3)Properties of the function, and connections to other functions
Use FunctionExpand to convert to other functions:

https://wolfram.com/xid/0y8dob-bnb4cw


https://wolfram.com/xid/0y8dob-g5ra2g

Compose with inverse functions:

https://wolfram.com/xid/0y8dob-d6rjln


https://wolfram.com/xid/0y8dob-ilf9o5

Solve a transcendental equation:

https://wolfram.com/xid/0y8dob-jpefd2


Possible Issues (3)Common pitfalls and unexpected behavior
For large arguments, intermediate values may underflow:

https://wolfram.com/xid/0y8dob-0jmyx


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

https://wolfram.com/xid/0y8dob-7jbus

Very large arguments can give unevaluated results:

https://wolfram.com/xid/0y8dob-bx6z96


Neat Examples (1)Surprising or curious use cases
A continued fraction whose partial numerators are consecutive integers:

https://wolfram.com/xid/0y8dob-ca284v

Its limit can be expressed in terms of Erfc:

https://wolfram.com/xid/0y8dob-czi5lw

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
]}
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
]}