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7
Error Functions, Dawson’s and Fresnel Integrals
Properties
7.9
Continued Fractions
7.11
Relations to Other Functions
§7.10
Derivatives
ⓘ
Keywords:
Hermite polynomials
,
auxiliary functions for Fresnel integrals
,
derivatives
,
derivatives of the error function
,
error functions
,
relations to other functions
Notes:
These results may be verified by differentiation of the definitions given in §
7.2
.
Permalink:
http://dlmf.nist.gov/7.10
See also:
Annotations for
Ch.7
7.10.1
d
n
+
1
erf
z
d
z
n
+
1
=
(
−
1
)
n
2
π
H
n
(
z
)
e
−
z
2
,
n
=
0
,
1
,
2
,
…
.
ⓘ
Symbols:
H
n
(
x
)
: Hermite polynomial
,
π
: the ratio of the circumference of a circle to its diameter
,
d
f
d
x
: derivative of
f
with respect to
x
,
erf
z
: error function
,
e
: base of natural logarithm
,
z
: complex variable
and
n
: nonnegative integer
A&S Ref:
7.1.19
Permalink:
http://dlmf.nist.gov/7.10.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§7.10
and
Ch.7
For the Hermite polynomial
H
n
(
z
)
see §
18.3
.
7.10.2
w
′
(
z
)
=
−
2
z
w
(
z
)
+
(
2
i
/
π
)
,
ⓘ
Symbols:
π
: the ratio of the circumference of a circle to its diameter
,
w
(
z
)
: Faddeeva (or Faddeyeva) function
,
i
: imaginary unit
and
z
: complex variable
A&S Ref:
7.1.20
Permalink:
http://dlmf.nist.gov/7.10.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§7.10
and
Ch.7
7.10.3
w
(
n
+
2
)
(
z
)
+
2
z
w
(
n
+
1
)
(
z
)
+
2
(
n
+
1
)
w
(
n
)
(
z
)
=
0
,
n
=
0
,
1
,
2
,
…
.
ⓘ
Symbols:
w
(
z
)
: Faddeeva (or Faddeyeva) function
,
z
: complex variable
and
n
: nonnegative integer
A&S Ref:
7.1.20
Permalink:
http://dlmf.nist.gov/7.10.E3
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§7.10
and
Ch.7
7.10.4
d
f
(
z
)
d
z
=
−
π
z
g
(
z
)
,
d
g
(
z
)
d
z
=
π
z
f
(
z
)
−
1
.
ⓘ
Symbols:
f
(
z
)
: auxiliary function for Fresnel integrals
,
g
(
z
)
: auxiliary function for Fresnel integrals
,
π
: the ratio of the circumference of a circle to its diameter
,
d
f
d
x
: derivative of
f
with respect to
x
and
z
: complex variable
A&S Ref:
7.3.21
(in slightly different form)
Permalink:
http://dlmf.nist.gov/7.10.E4
Encodings:
TeX
,
TeX
,
pMML
,
pMML
,
png
,
png
See also:
Annotations for
§7.10
and
Ch.7