erfc

Complementary error function

Syntax

erfc(X)

erfc(K,X)

Description

erfc(X) represents the complementary error function of X, that is,erfc(X) = 1 - erf(X).

erfc(K,X) represents the iterated integral of the complementary error function of X, that is, erfc(K, X) = int(erfc(K - 1, y), y, X, inf).

Examples

Complementary Error Function for Floating-Point and Symbolic Numbers

Depending on its arguments, erfc can return floating-point or exact symbolic results.

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

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

A =
    0.4795    0.0461    0.0455

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

symA = [erfc(sym(1/2)), erfc(sym(1.41)), erfc(sqrt(sym(2)))]

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

Use vpa to approximate symbolic results with the required number of digits:

d = digits(10);
vpa(symA)
digits(d)

ans =
[ 0.4795001222, 0.04614756064, 0.0455002639]

Error Function for Variables and Expressions

For most symbolic variables and expressions, erfc returns unresolved symbolic calls.

Compute the complementary error function for x and sin(x) + x*exp(x):

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

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

Complementary Error Function for Vectors and Matrices

If the input argument is a vector or a matrix, erfc returns the complementary error function for each element of that vector or matrix.

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)
 1 + Inf*1i

Compute the iterated integral of the complementary error function for the elements of V and M, and the integer -1:

erfc(-1, M)
erfc(-1, V)

ans =
[             2/pi^(1/2), 0]
[ (2*exp(-1/9))/pi^(1/2), 0]
 
ans =
 (2*exp(-1))/pi^(1/2)
                  Inf

Special Values of Complementary Error Function

erfc returns special values for particular parameters.

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))]

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

Handling Expressions That Contain Complementary Error Function

Many functions, such as diff and int, can handle expressions containing erfc.

Compute the first and second derivatives of the complementary error function:

syms x
diff(erfc(x), x)
diff(erfc(x), x, 2)

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

Compute the integrals of these expressions:

syms x
int(erfc(-1, x), x)

ans =
erf(x)

int(erfc(x), x)

ans =
x*erfc(x) - exp(-x^2)/pi^(1/2)

int(erfc(2, x), x)

ans =
(x^3*erfc(x))/6 - exp(-x^2)/(6*pi^(1/2)) +...
(x*erfc(x))/4 - (x^2*exp(-x^2))/(6*pi^(1/2))

Plot Complementary Error Function

Plot the complementary error function on the interval from -5 to 5.

syms x
fplot(erfc(x),[-5 5])
grid on

Input Arguments

X — Input

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.

K — Input representing an integer larger than -2

number | symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

Input representing an integer larger than -2, specified as a number, symbolic number, variable, expression, or function. This arguments can also be a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions.

More About

Complementary Error Function

The following integral defines the complementary error function:

$$erfc(x)=\frac{2}{\sqrt{\pi }}{\displaystyle \underset{x}{\overset{\infty }{\int }}{e}^{-t^2}dt}=1-erf\left(x\right)$$

Here erf(x) is the error function.

Iterated Integral of Complementary Error Function

The following integral is the iterated integral of the complementary error function:

$$erfc\left(k,x\right)={\displaystyle \underset{x}{\overset{\infty }{\int }}erfc\left(k-1,y\right)dy}$$

Here, $$erfc\left(0,x\right)=erfc\left(x\right)$$.

Tips

• 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.

• For most symbolic (exact) numbers, erfc returns unresolved symbolic calls. You can approximate such results with floating-point numbers using vpa.

• At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, then erfc expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

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

[1] 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.