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Error function
Plot of the error function
From Wikipedia, the free encyclopedia
In mathematics, the
error function
(also called the
Gauss error function
) is a special function
(nonelementary) which occurs in probability,
statistics, materials science, and partial differential
equations. It is defined as:
The
complementary error function
, denoted
erfc
,
is defined in terms of the error function:
The
complex error function
, denoted
w
(
x
), (also known as the Faddeeva function) is also defined in terms of the error
function:
Contents
1 Properties
1.1 Taylor series
1.2 Inverse function
2 Applications
2.1 Asymptotic expansion
2.2 Approximation with elementary functions
3 Related functions
3.1 Generalised error functions
3.2 Iterated integrals of the complementary error function
4 Implementation
5 Table of values
6 See also
7 References
8 External links
Properties
Error function  Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Error_function
1 of 8
2/2/2009 11:08 PM
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View Full Document Fig.2. Integrand exp(
−
z
2
) in the complex
z
plane.
Fig.3. erf(
z
) in the complex
z
plane.
The error function is odd:
Also, for any complex number
z
one has
where
z
*
is the complex conjugate of
z
.
The integrand
ƒ
= exp(
−
z
2
) and
ƒ
= erf(
z
) are shown in the complex
z
plane in figures 2 and 3. Level of Im(
ƒ
) = 0 is shown with a thick
green line. Negative integer values of Im(
ƒ
) are shown with thick red
lines. Positive integer values of
are shown with thick blue lines.
Intermediate levels of Im(
ƒ
) = constant are shown with thin green
lines. Intermediate levels of Re(
ƒ
) = constant are shown with thin red
lines for negative values and with thin blue lines for positive values.
At the real axis, the erf(
z
) approach unity at
z
→
+
∞
and
−
1 at
z
→
−∞
. At the imaginary axis, it tends to
.
Taylor series
The error function is an entire function; it has no singularities (except
that at infinity) and its Taylor expansion always converges.
The integral cannot be evaluated in closed form in terms of
elementary functions, but by expanding the integrand in a Taylor
series, one obtains the Taylor series for the error function as follows:
which holds for every complex number
z
. This result arises from the Taylor series expansion of
which is
and is then integrated term by term. The denominator terms are sequence A007680 in the OEIS.
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This note was uploaded on 11/02/2009 for the course APMA 2102 taught by Professor Keyes during the Spring '08 term at Columbia.
 Spring '08
 Keyes

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