gives the imaginary error function .


  • 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.
  • Erfi can be used with Interval and CenteredInterval objects. »


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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 about the origin:

Series expansion at Infinity:

Scope  (39)

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 Erfi efficiently at high precision:

Erfi threads elementwise over lists:

Erfi can be used with Interval and CenteredInterval objects:

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  (10)

Erfi is defined for all real and complex values:

Erfi takes all real values:

Erfi is an odd function:

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

Erfi is an analytic function of x:

It has no singularities or discontinuities:

Erfi is nondecreasing:

Erfi is injective:

Erfi is surjective:

Erfi is neither non-negative nor non-positive:

Erfi is neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the n^(th) 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:

TraditionalForm formatting:

Applications  (4)

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:

Integrate along a line from the origin with direction , expressing with Erfi :

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:

Wolfram Research (1996), Erfi, Wolfram Language function, (updated 2022).


Wolfram Research (1996), Erfi, Wolfram Language function, (updated 2022).


Wolfram Language. 1996. "Erfi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022.


Wolfram Language. (1996). Erfi. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_erfi, author="Wolfram Research", title="{Erfi}", year="2022", howpublished="\url{}", note=[Accessed: 19-May-2024 ]}


@online{reference.wolfram_2024_erfi, organization={Wolfram Research}, title={Erfi}, year={2022}, url={}, note=[Accessed: 19-May-2024 ]}