Environmental Engineering Reference
In-Depth Information
APPENDIX D
SPECIAL FUNCTIONS
D.1 ERROR FUNCTION
2
z
3
z
5
z
7
erf( )
z
=
z
−
+
−
+
…
(D.5)
3
2 5
!
3 7
!
π
The error function, erf(
z
), is defined by the relation
This relationship is particularly useful in estimating
erf(
z
) for small values of
z
. A closely related function
to the error function is the
complementary error func-
tion
, erfc(
z
), which is defined by
2
z
∫
2
−
x
(D.1)
erf(
z
=
e
dx
π
0
The error function is closely related to the cumula-
tive distribution function of a normal probability distri-
bution, and is defined for −∞ ≤
z
≤ ∞. The error function
is antisymmetric, such that
erfc
( )
z
= −
1
erf
( )
z
(D.6)
Values of the error function, erf(
z
), are tabulated
below.
erf
(
−
z
)
= −
erf
( )
z
(D.2)
TABLE D.1. Error Function
The constant before the integral sign in Equation
(D.1) is simply a normalizing constant, such that erf(
z
)
approaches 1 as
z
approaches infinity. For small values
of
z
, it is convenient to use the series expansion for
e
z
erf(
z
)
z
erf(
z
)
z
erf(
z
)
z
erf(
z
)
0.0
0.00000
0.8
0.74210
1.6
0.97635
2.4
0.99931
−
2
0.1
0.11246
0.9
0.79691
1.7
0.98379
2.5
0.99959
to obtain (Carslaw and Jaeger, 1959)
0.2
0.22270
1.0
0.84270
1.8
0.98909
2.6
0.99976
0.3
0.32863
1.1
0.88021
1.9
0.99279
2.7
0.99987
∞
∑
n
2
n
2
(
−
1
)
x
z
∫
0.4
0.42839
1.2
0.91031
2.0
0.99532
2.8
0.99992
erf( )
z
=
dx
(D.3)
n
!
0.5
0.52050
1.3
0.93401
2.1
0.99702
2.9
0.99996
π
0
n
=
0
0.6
0.60386
1.4
0.95229
2.2
0.99814
3.0
0.99998
0.7
0.67780
1.5
0.96611
2.3
0.99886
∞
1.00000
Since the series is uniformly convergent, it can be
integrated term by term to yield
∞
∑
n
2
n
+
1
2
(
−
1
2
)
z
D.2 BESSEL FUNCTIONS
erf( )
z
=
(D.4)
(
n
+
1
) !
n
π
n
=
0
D.2.1 Definition
which can also be written in the form (Hermance,
1999)
A second-order linear homogeneous differential equa-
tion of the form
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