# Erfi

Erfi[z]

gives the imaginary error function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• For certain special arguments, Erfi automatically evaluates to exact values.
• Erfi can be evaluated to arbitrary numerical precision.
• Erfi 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 Infinity:

## Scope(32)

### 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 Erfi efficiently at high precision:

### Specific Values(3)

Simple exact values are generated automatically:

Values at infinity:

Find the zero of Erfi:

### Visualization(2)

Plot the Erfi function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(4)

Erfi is defined for all real and complex values:

Erfi takes all real values:

Erfi is an odd function:

Erfi has the mirror property :

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the derivative:

### Integration(3)

Indefinite integral of Erfi:

Definite integral of an odd integrand over an interval centered at the origin is 0:

More integrals:

### Series Expansions(4)

Taylor expansion for Erfi:

Plot the first three approximations for Erfi around :

General term in the series expansion of Erfi:

Asymptotic expansion of Erfi:

Erfi can be applied to a power series:

### Function Identities and Simplifications(3)

Integral definition of Erfi:

Erfi of an inverse function:

Argument involving basic arithmetic operations:

### Function Representations(5)

Relationship of Erfi to Erf:

Series representation of Erfi:

Erfi can be represented as a DifferentialRoot:

Erfi can be represented in terms of MeijerG:

## Applications(3)

Solve a differential equation:

An isothermal solution of the forcefree Vlasov equation:

Integrating over the particle velocities gives the marginal distribution for the particle density:

A solution of the timedependent Schrödinger equation for the sudden opening of a shutter:

Verify correctness:

This plots the timedependent solution:

## Properties & Relations(1)

The imaginary error function for large imaginary-part arguments can be very close to :

## Possible Issues(1)

For large arguments, intermediate values may overflow:  Use DawsonF: