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NIST
7
Error Functions, Dawson’s and Fresnel Integrals
Properties
7.2
Definitions
7.4
Symmetry
§7.3
Graphics
ⓘ
Permalink:
http://dlmf.nist.gov/7.3
See also:
Annotations for
Ch.7
Contents
§7.3(i)
Real Variable
§7.3(ii)
Complex Variable
§7.3(i)
Real Variable
ⓘ
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/7.3.i
See also:
Annotations for
§7.3
and
Ch.7
Figure 7.3.1:
Complementary error functions
erfc
x
and
erfc
(
10
x
)
,
-
3
≤
x
≤
3
.
Magnify
ⓘ
Symbols:
erfc
z
: complementary error function
and
x
: real variable
Keywords:
error functions
,
graphics
Permalink:
http://dlmf.nist.gov/7.3.F1
Encodings:
pdf
,
png
See also:
Annotations for
§7.3(i)
,
§7.3
and
Ch.7
Figure 7.3.2:
Dawson’s integral
F
(
x
)
,
-
3.5
≤
x
≤
3.5
.
Magnify
ⓘ
Symbols:
F
(
z
)
: Dawson’s integral
and
x
: real variable
Keywords:
Dawson’s integral
,
graphics
Permalink:
http://dlmf.nist.gov/7.3.F2
Encodings:
pdf
,
png
See also:
Annotations for
§7.3(i)
,
§7.3
and
Ch.7
Figure 7.3.3:
Fresnel integrals
C
(
x
)
and
S
(
x
)
,
0
≤
x
≤
4
.
Magnify
ⓘ
Symbols:
C
(
z
)
: Fresnel integral
,
S
(
z
)
: Fresnel integral
and
x
: real variable
Keywords:
Fresnel integrals
,
Fresnel integrals and Cornu’s spiral
,
diffraction of light
,
graphics
Permalink:
http://dlmf.nist.gov/7.3.F3
Encodings:
pdf
,
png
See also:
Annotations for
§7.3(i)
,
§7.3
and
Ch.7
Figure 7.3.4:
|
ℱ
(
x
)
|
2
,
-
8
≤
x
≤
8
.
Fresnel (
1818
)
introduced the integral
ℱ
(
x
)
in his study of the interference pattern at the edge of a shadow. He observed that the intensity distribution is given by
|
ℱ
(
x
)
|
2
.
Magnify
ⓘ
Symbols:
ℱ
(
z
)
: Fresnel integral
and
x
: real variable
Keywords:
Fresnel integrals
,
applications
,
interference patterns
Referenced by:
§7.5
,
Sidebar 7.SB1
Permalink:
http://dlmf.nist.gov/7.3.F4
Encodings:
pdf
,
png
See also:
Annotations for
§7.3(i)
,
§7.3
and
Ch.7
§7.3(ii)
Complex Variable
ⓘ
Notes:
These surfaces were produced at NIST.
Permalink:
http://dlmf.nist.gov/7.3.ii
See also:
Annotations for
§7.3
and
Ch.7
Figure 7.3.5:
|
erf
(
x
+
i
y
)
|
,
-
3
≤
x
≤
3
,
-
3
≤
y
≤
3
. Compare §
7.13(i)
.
Magnify
3D
Help
ⓘ
Symbols:
erf
z
: error function
,
i
: imaginary unit
and
x
: real variable
Keywords:
error functions
,
graphics
Referenced by:
§7.13(i)
Permalink:
http://dlmf.nist.gov/7.3.F5
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§7.3(ii)
,
§7.3
and
Ch.7
Figure 7.3.6:
|
erfc
(
x
+
i
y
)
|
,
-
3
≤
x
≤
3
,
-
3
≤
y
≤
3
. Compare §§
7.12(i)
and
7.13(ii)
.
Magnify
3D
Help
ⓘ
Symbols:
erfc
z
: complementary error function
,
i
: imaginary unit
and
x
: real variable
Keywords:
error functions
,
graphics
Referenced by:
§7.13(ii)
Permalink:
http://dlmf.nist.gov/7.3.F6
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§7.3(ii)
,
§7.3
and
Ch.7