Erfc

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

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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:

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

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix Erfc function using MatrixFunction:

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 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 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:

LaplaceTransform:

MellinTransform:

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 piecewise‐constant 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 Birnbaum–Saunders 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:

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