# 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.

# 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(33)

### Numerical Evaluation(5)

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:

### 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(3)

Erfc is defined for all real and complex values:

Erfc takes all real values between 0 and 2:

Erfc has the mirror property :

### 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:

### 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:

## Applications(3)

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 via the HermiteH function:

## 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 neat continued fraction:

Its limit can be expressed through Erfc:

Introduced in 1991
(2.0)