Download Error Function and Complementary Error Function: Properties, Integrals, and Approximations and more Exercises Mathematical Physics in PDF only on Docsity!
Error and Complementary Error Functions
Reading Problems
- Background Outline
- Definitions
- Theory
- Gaussian function
- Error function
- Complementary Error function
- Relations and Selected Values of Error Functions
- Numerical Computation of Error Functions
- Rationale Approximations of Error Functions
- Assigned Problems
- References
Background
The error function and the complementary error function are important special functions which appear in the solutions of diffusion problems in heat, mass and momentum transfer, probability theory, the theory of errors and various branches of mathematical physics. It is interesting to note that there is a direct connection between the error function and the Gaussian function and the normalized Gaussian function that we know as the “bell curve”. The Gaussian function is given as
G(x) = Ae−x
(^2) /(2σ (^2) )
where σ is the standard deviation and A is a constant.
The Gaussian function can be normalized so that the accumulated area under the curve is unity, i.e. the integral from −∞ to +∞ equals 1. If we note that the definite integral
−∞
e−ax
2 dx =
π a
then the normalized Gaussian function takes the form
G(x) =
2 πσ
e−x^2 /(2σ^2 )
If we let
t^2 =
x^2 2 σ^2
and dt =
2 σ
dx
then the normalized Gaussian integrated between −x and +x can be written as
∫ (^) x
−x
G(x) dx =
π
∫ (^) x
−x
e−t^2 dt
or recognizing that the normalized Gaussian is symmetric about the y−axis, we can write
Definitions
1. Gaussian Function
The normalized Gaussian curve represents the probability distribution with standard distribution σ and mean μ relative to the average of a random distribution.
G(x) =
2 πσ
e−(x−μ)^2 /(2σ^2 )
This is the curve we typically refer to as the “bell curve” where the mean is zero and the standard distribution is unity.
2. Error Function
The error function equals twice the integral of a normalized Gaussian function between 0 and x/σ
y = erf x =
π
∫ (^) x
0
e−t
2 dt for x ≥ 0 , y [0, 1]
where
t =
x √ 2 σ
3. Complementary Error Function
The complementary error function equals one minus the error function
1 − y = erfc x = 1 − erf x =
π
x
e−t
2 dt for x ≥ 0 , y [0, 1]
4. Inverse Error Function
x = inerf y
inerf y exists for y in the range − 1 < y < 1 and is an odd function of y with a Maclaurin expansion of the form
inverf y =
∑^ ∞
n=
cn y^2 n−^1
5. Inverse Complementary Error Function
x = inerfc (1 − y)
x
NHxL
Figure 2.2: Plot of the Normalized Gaussian Function
Gaussian distributions have many convenient properties, so random variates with unknown distributions are often assumed to be Gaussian, especially in physics, astronomy and various aspects of engineering. Many common attributes such as test scores, height, etc., follow roughly Gaussian distributions, with few members at the high and low ends and many in the middle.
Computer Algebra Systems
Function Maple Mathematica
Probability Density Function statevalfpdf,dist PDF[dist, x ]
- frequency of occurrence at x
Cumulative Distribution Function statevalfcdf,dist CDF[dist, x ]
- integral of probability density function up to x dist = normald[μ, σ] dist = NormalDistribution[μ, σ] μ = 0 (mean) μ = 0 (mean) σ = 1 (std. dev.) σ = 1 (std. dev.)
Potential Applications
- Statistical Averaging:
Error Function
The error function is obtained by integrating the normalized Gaussian distribution.
erf x =
π
∫ (^) x
0
e−t
2 dt (1.4)
where the coefficient in front of the integral normalizes erf (∞) = 1. A plot of erf x over the range − 3 ≤ x ≤ 3 is shown as follows.
x
erf
Hx
L
Figure 2.3: Plot of the Error Function
The error function is defined for all values of x and is considered an odd function in x since erf x = −erf (−x).
The error function can be conveniently expressed in terms of other functions and series as follows:
erf x =
π
γ
, x^2
2 x √ π
M
, −x^2
2 x √ π
e−x
2 M
, x^2
π
∑^ ∞
n=
(−1)nx^2 n+ n!(2n + 1)
where γ(·) is the incomplete gamma function, M(·) is the confluent hypergeometric function of the first kind and the series solution is a Maclaurin series.
complementary Error Function
The complementary error function is defined as
erfc x = 1 − erf x
π
x
e−t
2 dt (1.8)
x
erf
Hx
L
Figure 2.4: Plot of the complementary Error Function
and similar to the error function, the complementary error function can be written in terms of the incomplete gamma functions as follows:
erfc x =
π
, x^2
As shown in Figure 2.5, the superposition of the error function and the complementary error function when the argument is greater than zero produces a constant value of unity.
Potential Applications
- Diffusion: In a similar manner to the transient conduction problem described for the error function, the complementary error function is used in the solution of the diffusion equation when the boundary conditions are constant surface heat flux, where qs = q 0
T (x, t) − Ti =
2 q 0 (αt/π)^1 /^2 k
exp
−x^2 4 αt
q 0 x k
erfc
x 2
αt
0 0.5 1 1.5 2 2.5 3 x
0.
0.
0.
0.
1
erf
Hx
L
Erf x + Erfc x
Erf x
Erfc x
Figure 2.5: Superposition of the Error and complementary Error Functions
and surface convection, where −k
∂T
∂x
x=
= h[T∞ − T (0, t)]
T (x, t) − Ti
T∞ − Ti
= erfc
x 2
αt
[
exp
hx k
h^2 αt k^2
)] [
erfc
x 2
αt
h
αt k
)]
Approximations
Power Series for Small x (x < 2)
Since
erf x =
π
∫ (^) x
0
e−t^2 dt =
π
∫ (^) x
0
∑^ ∞
n=
(−1)nt^2 n n!
dt (1.10)
and the series is uniformly convergent, it may be integrated term by term. Therefore
erf x =
π
∑^ ∞
n=
(−1)nx^2 n+ (2n + 1)n!
π
x 1 · 0!
x^3 3 · 1!
x^5 5 · 2!
x^7 7 · 3!
x^9 9 · 4!
Asymptotic Expansion for Large x (x > 2)
Since
erfc x =
π
x
e−t
2 dt =
π
x
t
e−t
2 t dt
we can integrate by parts by letting
u =
t
dv = e−t^2 d dt
du = −t−^2 dt v = −
e−t^2
therefore
x
t
e−t
2 t dt =
[
uv
]∞
x
x
v du =
[
2 t
e−t
2
]∞
x
x
e−t^2 t^2
dt
Thus
erfc x =
π
2 x
e−x
2 −
x
e−t^2 t^2
dt
Repeating the process n times yields
√ π 2
erfc x =
e−x
2
x
2 x^3
22 x^5
− · · · + (−1)n−^1
1 · 3 · · · (2n − 3) 2 n−^1 x^2 n−^1
+(−1)n^
1 · 3 · · · (2n − 1) 2 n
x
e−t^2 t^2 n^
dt (1.14)
Finally we can write
πxex
2 erfc x = 1 +
∑^ ∞
n=
(−1)n^
1 · 3 · 5 · · · (2n − 1) (2x^2 )n^
This series does not converge, since the ratio of the nth^ term to the (n−1)th^ does not remain less than unity as n increases. However, if we take n terms of the series, the remainder,
1 · 3 · · · (2n − 1) 2 n
x
e−t^2 t^2 n^
dt
is less than the nth^ term because
x
e−t^2 t^2 n^
dt < e−x^2 <
0
dt t^2 n
We can therefore stop at any term taking the sum of the terms up to this term as an approximation of the function. The error will be less in absolute value than the last term retained in the sum. Thus for large x, erfc x may be computed numerically from the asymptotic expansion.
√ πxex
2 erfc x = 1 +
∑^ ∞
n=
(−1)n^
1 · 3 · 5 · · · (2n − 1) (2x^2 )n
2 x^2
(2x^2 )^2
(2x^2 )^3
Repeated Integrals of the Complementary Error Function
inerfc x =
x
in−^1 erfc t dt n = 0, 1 , 2 ,... (1.26)
where
i−^1 erfc x =
π
e−x^2 (1.27)
i^0 erfc x = erfc x (1.28)
i^1 erfc x = ierfc x =
x
erfc t dt
π
exp(−x^2 ) − x erfc x (1.29)
i^2 erfc x =
x
i erfc t dt
[
(1 + 2x^2 ) erfc x −
π
x exp(−x^2 )
]
[erfc x − 2 x · ierfc x] (1.30)
The general recurrence formula is
2 nin^ erfc x = in−^2 erfc x − 2 xin−^1 erfc x (n = 1, 2 , 3 ,.. .) (1.31)
Therefore the value at x = 0 is
inerfc 0 −
2 n^ Γ
n 2
) (^) (n = − 1 , 0 , 1 , 2 , 3 ,.. .) (1.32)
It can be shown that y = in^ erfc x is the solution of the differential equation
d^2 y dx^2
dy dx
− 2 ny = 0 (1.33)
The general solution of
y′′^ + 2xy′^ − 2 ny = 0 − ∞ ≤ x ≤ ∞ (1.34)
is of the form
y = Ainerfc x + Binerfc (−x) (1.35)
Derivatives of Repeated Integrals of the Complementary Error
Function
d dx
[inerfc x] = (−1)n−^1 erfc x (n = 0, 1 , 2 , 3.. .) (1.36)
dn dxn
[
ex
2 erfc x
]
= (−1)n 2 nn!ex
2 inerfc x (n = 0, 1 , 2 , 3.. .) (1.37)
Numerical Computation of Error Functions
The power series form of the error function is not recommended for numerical computations when the argument approaches and exceeds the value x = 2 because the large alternat- ing terms may cause cancellation, and because the general term is awkward to compute recursively. The function can, however, be expressed as a confluent hypergeometric series.
erf x =
π
x e−x
2 M
, x^2
in which all terms are positive, and no cancellation can occur. If we write
erf x = b
∑^ ∞
n=
an 0 ≤ x ≤ 2 (1.52)
with
b =
2 x √ π
e−x
2 a 0 = 1 an =
x^2 (2n + 1)/ 2
an− 1 n ≥ 1
then erf x can be computed very accurately (e.g. with an absolute error less that 10 −^9 ). Numerical experiments show that this series can be used to compute erf x up to x = 5 to the required accuracy; however, the time required for the computation of erf x is much greater due to the large number of terms which have to be summed. For x ≥ 2 an alternate method that is considerably faster is recommended which is based upon the asymptotic expansion of the complementary error function.
erfc x =
π
x
e−t
2 dt
e−x^2 √ πx
2 Fo
x^2
x → ∞ (1.53)
which cannot be used to obtain arbitrarily accurate values for any x. An expression that converges for all x > 0 is obtained by converting the asymptotic expansion into a continued fraction
πex^2 erfc x =
x +
x +
x +
x +
x +
x +...
x > 0 (1.54)
which for convenience will be written as
erfc x =
e−x^2 √ π
x+
x+
x+
x+
x+
x > 0 (1.55)
It can be demonstrated experimentally that for x ≥ 2 the 16th approximant gives erfc x with an absolute error less that 10 −^9. Thus we can write
erfc x =
e−x^2 √ π
x+
x+
x+
x+
x
x ≥ 2 (1.56)
Using a fixed number of approximants has the advantage that the continued fraction can be evaluated rapidly beginning with the last term and working upward to the first term.
