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erfc

Complementary error function

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

Syntax

erfc(x)
erfc(x, n)

Description

erfc(x)=1erf(x)=2πxet2dt computes the complementary error function.

erfc(x,n)=xerfc(t,n1)dt with erfc(x, 0) = erfc(x) and erfc(x,1)=2πex2 returns the iterated integrals of the complementary error function. The calls erfc(x) and erfc(x, 0) are equivalent.

erfc is defined for all complex arguments x. For floating-point arguments, erfc returns floating-point results.

The implemented exact values are:

  • erfc(0) = 1, erfc(∞) = 0, erfc(-∞) = 2, erfc(i∞) = 1 - i∞, erfc(-i∞) = 1 + i∞

  • erfc(0,n)=12nΓ(n2+1), erfc(∞,n) = 0, erfc(-∞, n) = ∞

For all other arguments, the error function returns symbolic function calls.

If a numerical value of n is not an integer or if n < -1, the function call erfc(x, n) returns an error. The function also accepts symbolic values of n.

If n is a numerical value, you can use expand(erfc(x, n)) to apply the following rules. See Example 3.

  • The recurrence erfc(x,n)=erfc(x,n2)2nxerfc(x,n1)n

  • The reflection rule erfc(x,n)=(1)n+1erfc(x,n)+H(n,ix)in2n1n!, where H(n,ix) is the n-th degree Hermite polynomial at the point ix. See orthpoly::hermite.

For the function erfc with floating-point arguments of large absolute value, internal numerical underflow or overflow can happen. See Example 2. If a call to erfc causes underflow or overflow, this function returns:

  • The result truncated to 0.0 if x is a large positive real number

  • The result rounded to 2.0 if x is a large negative real number

  • RD_NAN if x is a large complex number and MuPAD® cannot approximate the function value

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.

Examples

Example 1

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

erfc(0), erfc(x + 1), erfc(-infinity), erfc(3/2), erfc(sqrt(2))

erfc(0, n), erfc(x + 1, -1), erfc(-infinity, 5)

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

float(erfc(3/2)), float(erfc(sqrt(2)))

Alternatively, use floating-points value as arguments:

erfc(-7.2), erfc(2.0 + 3.5*I), erfc(3.0, 4)

Example 2

For large complex arguments, the complementary error function can return :

erfc(38000.0 + 3801.0*I)

For large floating-point arguments with positive real parts, erfc can return values truncated to 0.0:

erfc(27281.1), erfc(27281.2)

Example 3

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

diff(erfc(x, 3), x, x)

limit(x/(1 + x)*(1 - erfc(x)), x = infinity)

expand(erfc(x, 3))

rewrite(erfc(x), erf),
rewrite(erfc(x), erfi)

series(erfc(x), x = infinity, 3)

Parameters

x

Arithmetical expression

n

Arithmetical expression representing an integer larger than or equal to -1.

Return Values

Arithmetical expression

Algorithms

erf, erfc, and erfi are entire functions.

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