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


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit numerical values are given only for real values of s between 0 and 2.
  • For certain special arguments, InverseErfc automatically evaluates to exact values.
  • InverseErfc can be evaluated to arbitrary numerical precision.
  • InverseErfc automatically threads over lists.
  • InverseErfc can be used with Interval and CenteredInterval objects. »


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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  (26)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

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

Evaluate InverseErfc efficiently at high precision:

InverseErfc threads elementwise over lists and arrays:

InverseErf can be used with Interval and CenteredInterval objects:

Specific Values  (4)

Exact results for specific arguments:

Find a real root of the equation :

Plot the InverseErfc function:

Plot the InverseErfc function reflected about :

Function Properties  (8)

InverseErfc is defined for all real values from the interval :

InverseErfc takes all real values:

InverseErfc is an analytic function on its domain:

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

InverseErfc is nonincreasing on its domain:

InverseErfc is injective:

InverseErfc is surjective:

InverseErfc is neither non-negative nor non-positive:

InverseErfc is neither convex nor concave:

Differentiation  (2)

First derivative:

Higher derivatives:

Integration  (3)

Indefinite integral of InverseErfc:

Definite integral of InverseErfc over its real domain:

Numerical approximation of the definite integral of InverseErfc:

Series Expansions  (2)

Series expansion for InverseErfc around :

Taylor expansion for InverseErfc around :

Plot the first three approximations for InverseErfc around :

Function Representations  (3)

Primary definition of the inverse error function:

Relation to the inverse complementary error function:

TraditionalForm formatting:

Applications  (1)

Generate Gaussian-distributed random numbers:

Properties & Relations  (4)

Solve a transcendental equation:

Numerically find a root of a transcendental equation:

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

InverseErfc is a numeric function:

Possible Issues  (1)

InverseErfc evaluates numerically only for :

Neat Examples  (1)

Riemann surface of InverseErfc:

Wolfram Research (1996), InverseErfc, Wolfram Language function, (updated 2023).


Wolfram Research (1996), InverseErfc, Wolfram Language function, (updated 2023).


Wolfram Language. 1996. "InverseErfc." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023.


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


@misc{reference.wolfram_2024_inverseerfc, author="Wolfram Research", title="{InverseErfc}", year="2023", howpublished="\url{}", note=[Accessed: 15-July-2024 ]}


@online{reference.wolfram_2024_inverseerfc, organization={Wolfram Research}, title={InverseErfc}, year={2023}, url={}, note=[Accessed: 15-July-2024 ]}