InverseErf

InverseErf[s]

gives the inverse error function obtained as the solution for z in .

Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• Explicit numerical values are given only for real values of s between and .
• InverseErf[z0,s] gives the inverse of the generalized error function Erf[z0,z].
• For certain special arguments, InverseErf automatically evaluates to exact values.
• InverseErf can be evaluated to arbitrary numerical precision.
• InverseErf can be used with Interval and CenteredInterval objects. »
• InverseErf automatically threads over lists.

Examples

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Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin:

Series expansion at a singular point:

Scope(32)

Numerical Evaluation(5)

Evaluate numerically to high precision:

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

Evaluate InverseErf efficiently at high precision:

Evaluate numerically the inverse of the generalized error function:

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 InverseErf function using MatrixFunction:

Specific Values(2)

Exact results for specific arguments:

Find a real root of the equation :

Visualization(3)

Plot the InverseErf function:

Plot the inverse of the generalized error function for different values of :

Plot the inverse of the generalized error function for different values of :

Function Properties(9)

InverseErf is defined for all real values from the interval ():

InverseErf takes all real values:

InverseErf is an odd function:

InverseErf is an analytic function on its domain:

It is not analytic in general, as it has both singularities and discontinuities:

InverseErf is nondecreasing on its domain:

InverseErf is injective:

InverseErf is surjective:

InverseErf is neither non-negative nor non-positive:

InverseErf is neither convex nor concave on its domain:

Differentiation(2)

First derivative:

Higher derivatives:

Integration(3)

Indefinite integral of InverseErf:

Definite integral of InverseErf over its real domain:

Numerical approximation of the definite integral of InverseErf:

Series Expansions(2)

Taylor expansion for InverseErf:

Plot the first three approximations for InverseErf around :

Series expansion of the inverse of the generalized error function:

Function Identities and Simplifications(2)

Compose with the inverse error function:

Compose with the inverse generalized error function:

Function Representations(4)

Primary definition of the inverse error function:

Relation to the inverse of the generalized error function:

Relation to the inverse complementary error function:

Applications(3)

Generate Gaussian-distributed random numbers:

The number of standard deviations for a 99% confidence interval in the Gaussian distribution:

Plot InverseErf:

Properties & Relations(5)

Solve a transcendental equation:

Numerically find a root of a transcendental equation:

Obtain InverseErf as the solution of a differential equation:

InverseErf is a numeric function:

In TraditionalForm, is automatically interpreted as an inverse error function:

Possible Issues(1)

InverseErf evaluates numerically only for :

Neat Examples(1)

Riemann surface of InverseErf:

Wolfram Research (1996), InverseErf, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseErf.html (updated 2023).

Text

Wolfram Research (1996), InverseErf, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseErf.html (updated 2023).

CMS

Wolfram Language. 1996. "InverseErf." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/InverseErf.html.

APA

Wolfram Language. (1996). InverseErf. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseErf.html

BibTeX

@misc{reference.wolfram_2024_inverseerf, author="Wolfram Research", title="{InverseErf}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/InverseErf.html}", note=[Accessed: 14-September-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_inverseerf, organization={Wolfram Research}, title={InverseErf}, year={2023}, url={https://reference.wolfram.com/language/ref/InverseErf.html}, note=[Accessed: 14-September-2024 ]}