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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
$\mathrm{erfc}x$
and
$\mathrm{erfc}\left(10x\right)$
,
$-3\le x\le 3$
.
Magnify
ⓘ
Symbols:
$\mathrm{erfc}z$
: 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
$F\left(x\right)$
,
$-3.5\le x\le 3.5$
.
Magnify
ⓘ
Symbols:
$F\left(z\right)$
: 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
$C\left(x\right)$
and
$S\left(x\right)$
,
$0\le x\le 4$
.
Magnify
ⓘ
Symbols:
$C\left(z\right)$
: Fresnel integral
,
$S\left(z\right)$
: 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:
${|\mathcal{F}\left(x\right)|}^{2}$
,
$-8\le x\le 8$
.
Fresnel (
1818
)
introduced the integral
$\mathcal{F}\left(x\right)$
in his study of the interference pattern at the edge of a shadow. He observed that the intensity distribution is given by
${\left|\mathcal{F}\left(x\right)\right|}^{2}$
.
Magnify
ⓘ
Symbols:
$\mathcal{F}\left(z\right)$
: 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:
$|\mathrm{erf}\left(x+\mathrm{i}y\right)|$
,
$-3\le x\le 3$
,
$-3\le y\le 3$
. Compare §
7.13(i)
.
Magnify
3D
Help
ⓘ
Symbols:
$\mathrm{erf}z$
: error function
,
$\mathrm{i}$
: 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:
$|\mathrm{erfc}\left(x+\mathrm{i}y\right)|$
,
$-3\le x\le 3$
,
$-3\le y\le 3$
. Compare §§
7.12(i)
and
7.13(ii)
.
Magnify
3D
Help
ⓘ
Symbols:
$\mathrm{erfc}z$
: complementary error function
,
$\mathrm{i}$
: 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