Inverse of the error function

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.




inverf(x) computes the inverse of the error function. This function is defined for all complex arguments x.

The inverse function inverf is singular at the points x = -1 and x = 1.

The inverses of the error functions return floating-point values only for floating-point arguments that belong to a particular interval. Thus, the inverse of the error function inverf(x) returns floating-point values for real values x from the interval [-1, 1]. The implemented exact values are: inverf(-1) = -∞, inverf(0) = 0, inverf(1) = ∞. For all other arguments, the error functions return symbolic function calls.

MuPAD® can simplify expressions that contain error functions and their inverses. For real values x, the system applies the following simplification rules:

  • inverf(erf(x)) = inverf(1 - erfc(x)) = inverfc(1 - erf(x)) = inverfc(erfc(x)) = x

  • inverf(-erf(x)) = inverf(erfc(x) - 1) = inverfc(1 + erf(x)) = inverfc(2 - erfc(x)) = -x

For any value x, the system applies the following simplification rules:

  • inverf(-x) = -inverf(x)

  • inverfc(2 - x) = -inverfc(x)

  • erf(inverf(x)) = erfc(inverfc(x)) = x

  • erf(inverfc(x)) = erfc(inverf(x)) = 1 - x

Environment Interactions

When called with a floating-point argument, the functions are sensitive to the environment variable DIGITS, which determines the numerical working precision.


Example 1

You can call the inverse of the error function with exact and symbolic arguments:

inverf(-1), inverf(0), inverf(1), inverf(x + 1), inverf(1/5), inverf(1/sqrt(2))

To approximate exact results with floating-point numbers, use float:

float(inverf(1/5)), float(inverf(1/sqrt(2)))

Alternatively, use floating-points value as arguments:

inverf(0.2), inverf(1/sqrt(2.0))

For floating-point arguments x from the interval [-1, 1], inverf returns floating-point values:

inverf(-0.5), inverf(0.85)

For floating-point arguments outside of this interval, inverf returns symbolic function calls:

inverf(-5.3), inverf(10.0)

Example 2

diff, float, limit, rewrite, series, and other functions handle expressions involving the inverse of the error function:

diff(inverf(x), x)

float(ln(3 + inverf(1/sqrt(PI))))

limit(1/inverf(x), x = 0, Right);
limit(1/inverf(x), x = 0, Left)

rewrite(inverfc(x), inverf)

series(inverf(x), x = 0)

Return Values

Arithmetical expression

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