Inverse of the complementary error function

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.




inverfc(x) = inverf(1 - x) computes the inverse of the complementary error function. This function is defined for all complex arguments x.

The inverse function inverfc is singular at the points x = 0 and x = 2.

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 complementary error function inverfc(x) returns floating-point values for real values x from the interval [0, 2]. The implemented exact values are: inverfc(0) = ∞, inverfc(1) = 0, inverfc(2) = -∞. 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 complementary error function with exact and symbolic arguments:

inverfc(0), inverfc(1), inverfc(2), inverfc(15), inverfc(x/5), inverfc(1/5), inverfc(sqrt(2))

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

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

Alternatively, use floating-points value as arguments:

inverfc(0.2), inverfc(sqrt(2.0))

For floating-point arguments x from the interval [0, 2], inverfc returns floating-point values:

inverfc(0.5), inverfc(1.25)

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

inverfc(-1.25), inverfc(2.5)

Example 2

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

diff(inverfc(x), x)

float(ln(3 + inverfc(sqrt(PI))))

limit(1/inverfc(x), x = 1, Right);
limit(1/inverfc(x), x = 1, Left)

rewrite(inverfc(x), inverf)

series(inverfc(x), x = 1)

Return Values

Arithmetical expression

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