Calling erfc for a number that is not a symbolic object invokes the MATLAB erfc function. This function accepts real arguments only. If you want to compute the complementary error function for a complex number, use sym to convert that number to a symbolic object, and then call erfc for that symbolic object.

Input Arguments

`x`	Symbolic number, variable, or expression.
`A`	Vector or matrix of symbolic numbers, variables, or expressions.

Definitions

Complementary Error Function

The following integral defines the complementary error function:

Here erf(x) is the error function.

Examples

Compute the complementary error function for these numbers. Because these numbers are not symbolic objects, you get the floating-point results:

[erfc(1/2), erfc(1.41), erfc(sqrt(2))]

ans =
    0.4795    0.0461    0.0455

Compute the complementary error function for the numbers converted to symbolic objects. For most symbolic (exact) numbers, erfc returns unresolved symbolic calls:

[erfc(sym(1/2)), erfc(sym(1.41))]

ans =
[ erfc(1/2), erfc(141/100)]

Compute the complementary error function for x = 0, x = ∞, and x = –∞. The complementary error function has special values for these parameters:

[erfc(0), erfc(inf), erfc(-inf)]

ans =
     1     0     2

Compute the complementary error function for complex infinities. Use sym to convert complex infinities to symbolic objects:

[erfc(sym(i*inf)), erfc(sym(-i*inf))]

[ 1 - Inf*i, Inf*i + 1]

Compute the complementary error function for x and sin(x) + x*exp(x). For most symbolic variables and expressions, erfc returns unresolved symbolic calls:

syms x
f = sin(x) + x*exp(x);
erfc(x)
erfc(f)

ans =
erfc(x)
 
ans =
erfc(sin(x) + x*exp(x))

Now compute the derivatives of these expressions:

diff(erfc(x), x, 2)
diff(erfc(f), x)

ans =
(4*x*exp(-x^2))/pi^(1/2)
 
ans =
-(2*exp(-(sin(x) + x*exp(x))^2)*(cos(x) + exp(x) + x*exp(x)))/pi^(1/2)

Compute the complementary error function for elements of matrix M and vector V:

M = sym([0 inf; 1/3 -inf]);
V = sym([1; -i*inf]);
erfc(M)
erfc(V)

ans =
[         1, 0]
[ erfc(1/3), 2]
 
ans =
   erfc(1)
 Inf*i + 1

Algorithms

The toolbox can simplify expressions that contain error functions and their inverses. For real values x, the toolbox applies these simplification rules:

erfinv(erf(x)) = erfinv(1 - erfc(x)) = erfcinv(1 - erf(x)) = erfcinv(erfc(x)) = x
erfinv(-erf(x)) = erfinv(erfc(x) - 1) = erfcinv(1 + erf(x)) = erfcinv(2 - erfc(x)) = -x

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

erfcinv(x) = erfinv(1 - x)
erfinv(-x) = -erfinv(x)
erfcinv(2 - x) = -erfcinv(x)
erf(erfinv(x)) = erfc(erfcinv(x)) = x
erf(erfcinv(x)) = erfc(erfinv(x)) = 1 - x

References

Gautschi, W. "Error Function and Fresnel Integrals." Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.