erf

Error Function for Floating-Point and Symbolic Numbers

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

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

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

A =
    0.5205    0.9539    0.9545

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

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

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

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

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

ans =
[ 0.5204998778, 0.9538524394, 0.9544997361]

Error Function for Variables and Expressions

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

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

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

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

Error Function for Vectors and Matrices

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

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

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

ans =
[        0,  1]
[ erf(1/3), -1]
 
ans =
 erf(1)
 -Inf*1i

Special Values of Error Function

erf returns special values for particular parameters.

Compute the error function for x = 0, x = ∞, and x = –∞. Use sym to convert 0 and infinities to symbolic objects. The error function has special values for these parameters:

[erf(sym(0)), erf(sym(Inf)), erf(sym(-Inf))]

ans =
[ 0, 1, -1]

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

[erf(sym(i*Inf)), erf(sym(-i*Inf))]

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

Handling Expressions That Contain Error Function

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

Compute the first and second derivatives of the error function:

syms x
diff(erf(x), x)
diff(erf(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:

int(erf(x), x)
int(erf(log(x)), x)

ans =
exp(-x^2)/pi^(1/2) + x*erf(x)
 
ans =
x*erf(log(x)) - int((2*exp(-log(x)^2))/pi^(1/2), x)

Plot Error Function

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

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