2.2 Error Functions
25
With z ∈ C, both functions are defined in the complex domain and they are entire
functions. These functions fulfill the following symmetry relations:
erf(−z) = − erf(z)
(2.2a)
erfc(−z) = 2 − erfc(z)
(2.2b)
erf(¯ z) = erf(z),
(2.2c)
and thus erf is an odd function. Special values are
erf(0) = 0, erf(∞) = 1, erfc(0) = 1, and erfc(∞) = 0.
(2.3)
The derivatives of the error function hold
d n+1
dz n+1 erf(z) = (−1)
n 2
√
π
H n (z) exp(−z
2 ), n ∈ N,
(2.4)
where H n (z) are the Hermite polynomials.
The error functions are related to the incomplete gamma functions [4], Chap. 1.5,
via
erf(z) = prefactor ·
1
√
π
γ
1
2
, z
2
(2.5a)
erfc(z) = prefactor ·
1
√
π
Γ
1
2
, z
2
.
(2.5b)
These equations are in [4] without the prefactor and thus hold only if (z) > 0,
which is obvious due to the symmetry relations. For example, erf is an odd
function, whereas Eq. (2.5a) leads to an even function without prefactor. With
prefactor =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−1 : :(z) < 0
+1 : :(z) > 0
−1 : :(z) = 0 ∧ ∧(z) < 0
+1 : :(z) = 0 ∧ ∧(z) > 0
0 : :(z) = 0 ∧ ∧(z) = 0,
(2.6)
Equations (2.5a) and (2.5b) hold in the entire complex domain. Hence, the computation is based on the incomplete gamma functions. (See Chap. 1.5 for details.)
25
With z ∈ C, both functions are defined in the complex domain and they are entire
functions. These functions fulfill the following symmetry relations:
erf(−z) = − erf(z)
(2.2a)
erfc(−z) = 2 − erfc(z)
(2.2b)
erf(¯ z) = erf(z),
(2.2c)
and thus erf is an odd function. Special values are
erf(0) = 0, erf(∞) = 1, erfc(0) = 1, and erfc(∞) = 0.
(2.3)
The derivatives of the error function hold
d n+1
dz n+1 erf(z) = (−1)
n 2
√
π
H n (z) exp(−z
2 ), n ∈ N,
(2.4)
where H n (z) are the Hermite polynomials.
The error functions are related to the incomplete gamma functions [4], Chap. 1.5,
via
erf(z) = prefactor ·
1
√
π
γ
1
2
, z
2
(2.5a)
erfc(z) = prefactor ·
1
√
π
Γ
1
2
, z
2
.
(2.5b)
These equations are in [4] without the prefactor and thus hold only if (z) > 0,
which is obvious due to the symmetry relations. For example, erf is an odd
function, whereas Eq. (2.5a) leads to an even function without prefactor. With
prefactor =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−1 : :(z) < 0
+1 : :(z) > 0
−1 : :(z) = 0 ∧ ∧(z) < 0
+1 : :(z) = 0 ∧ ∧(z) > 0
0 : :(z) = 0 ∧ ∧(z) = 0,
(2.6)
Equations (2.5a) and (2.5b) hold in the entire complex domain. Hence, the computation is based on the incomplete gamma functions. (See Chap. 1.5 for details.)