Geoscience Reference
In-Depth Information
Appendix 5
The error function
The error function is defined as
x
2
√
π
e
−
y
2
d
y
erf(
x
)
=
(A5.1)
y
=
0
It is apparent that
erf(
−
x
)
=−
erf(
x
)
(A5.2)
and
(A5.3)
erf(0)
=
0
and
erf(
∞
)
=
1
(A5.4)
The complementary error function erfc(
x
)isdefined as
erfc(
x
)
=
1
−
erf(
x
)
∞
2
√
π
e
−
y
2
d
y
(A5.5)
=
x
The error function is shown in Fig. 7.5 and tabulated in Table A5.1.Aneasily programmable
approximation to the error function is
−
a
1
t
a
3
t
3
e
−
x
2
a
2
t
2
erf(
x
)
=
1
+
+
+
ε
(
x
)
(A5.6)
where
t
=
1
/
(1
+
0.470 47
x
),
a
1
=
0.348 024 2,
a
2
=−
0.095 879 8 and
a
3
=
0.747 855 6.
10
−
5
. (C. Hastings,
Approximations for
Digital Computers
, Princeton University Press, Princeton, 1955.)
In this text, the error function appears in solutions of the heat-conduction equation (see
Section 7.3.6). In more detailed thermal problems, the solutions may include repeated
integrations or derivatives of the error function. For example,
∞
The error in this approximation is
ε
(
x
)
≤
2.5
×
1
√
π
e
−
x
2
erfc(
y
)d
y
=
−
x
erfc(
x
)
x
and
d
d
x
(erf(
x
))
2
√
π
e
−
x
2
=
638
