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10
Bessel Functions
Modified Bessel Functions
10.25
Definitions
10.27
Connection Formulas
§10.26
Graphics
ⓘ
Permalink:
http://dlmf.nist.gov/10.26
See also:
Annotations for
Ch.10
Contents
§10.26(i)
Real Order and Variable
§10.26(ii)
Real Order, Complex Variable
§10.26(iii)
Imaginary Order, Real Variable
§10.26(i)
Real Order and Variable
ⓘ
Keywords:
graphics
,
modified Bessel functions
Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/10.26.i
See also:
Annotations for
§10.26
and
Ch.10
Figure 10.26.1:
I
0
(
x
)
,
I
1
(
x
)
,
K
0
(
x
)
,
K
1
(
x
)
,
0
≤
x
≤
3
.
Magnify
ⓘ
Symbols:
I
ν
(
z
)
: modified Bessel function of the first kind
,
K
ν
(
z
)
: modified Bessel function of the second kind
and
x
: real variable
Permalink:
http://dlmf.nist.gov/10.26.F1
Encodings:
pdf
,
png
See also:
Annotations for
§10.26(i)
,
§10.26
and
Ch.10
Figure 10.26.2:
e
−
x
I
0
(
x
)
,
e
−
x
I
1
(
x
)
,
e
x
K
0
(
x
)
,
e
x
K
1
(
x
)
,
0
≤
x
≤
10
.
Magnify
ⓘ
Symbols:
e
: base of natural logarithm
,
I
ν
(
z
)
: modified Bessel function of the first kind
,
K
ν
(
z
)
: modified Bessel function of the second kind
and
x
: real variable
Permalink:
http://dlmf.nist.gov/10.26.F2
Encodings:
pdf
,
png
See also:
Annotations for
§10.26(i)
,
§10.26
and
Ch.10
Figure 10.26.3:
I
ν
(
x
)
,
0
≤
x
≤
5
,
0
≤
ν
≤
4
.
Magnify
3D
Help
ⓘ
Symbols:
I
ν
(
z
)
: modified Bessel function of the first kind
,
x
: real variable
and
ν
: complex parameter
Permalink:
http://dlmf.nist.gov/10.26.F3
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§10.26(i)
,
§10.26
and
Ch.10
Figure 10.26.4:
K
ν
(
x
)
,
0.1
≤
x
≤
5
,
0
≤
ν
≤
4
.
Magnify
3D
Help
ⓘ
Symbols:
K
ν
(
z
)
: modified Bessel function of the second kind
,
x
: real variable
and
ν
: complex parameter
Permalink:
http://dlmf.nist.gov/10.26.F4
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§10.26(i)
,
§10.26
and
Ch.10
Figure 10.26.5:
I
ν
′
(
x
)
,
0
≤
x
≤
5
,
0
≤
ν
≤
4
.
Magnify
3D
Help
ⓘ
Symbols:
I
ν
(
z
)
: modified Bessel function of the first kind
,
x
: real variable
and
ν
: complex parameter
Permalink:
http://dlmf.nist.gov/10.26.F5
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§10.26(i)
,
§10.26
and
Ch.10
Figure 10.26.6:
K
ν
′
(
x
)
,
0.3
≤
x
≤
5
,
0
≤
ν
≤
4
.
Magnify
3D
Help
ⓘ
Symbols:
K
ν
(
z
)
: modified Bessel function of the second kind
,
x
: real variable
and
ν
: complex parameter
Permalink:
http://dlmf.nist.gov/10.26.F6
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§10.26(i)
,
§10.26
and
Ch.10
§10.26(ii)
Real Order, Complex Variable
ⓘ
Permalink:
http://dlmf.nist.gov/10.26.ii
See also:
Annotations for
§10.26
and
Ch.10
Apply (
10.27.6
) and (
10.27.8
) to §
10.3(ii)
.
§10.26(iii)
Imaginary Order, Real Variable
ⓘ
Notes:
These graphs were produced at NIST.
Referenced by:
§10.45
Permalink:
http://dlmf.nist.gov/10.26.iii
See also:
Annotations for
§10.26
and
Ch.10
For the notation, see §
10.45
.
Figure 10.26.7:
I
~
1
/
2
(
x
)
,
K
~
1
/
2
(
x
)
,
0.01
≤
x
≤
3
.
Magnify
ⓘ
Symbols:
I
~
ν
(
x
)
: modified Bessel function of the first kind of imaginary order
,
K
~
ν
(
x
)
: modified Bessel function fo the second kind of imaginary order
and
x
: real variable
Keywords:
graphs
,
modified Bessel functions
,
of imaginary order
Permalink:
http://dlmf.nist.gov/10.26.F7
Encodings:
pdf
,
png
See also:
Annotations for
§10.26(iii)
,
§10.26
and
Ch.10
Figure 10.26.8:
I
~
1
(
x
)
,
K
~
1
(
x
)
,
0.01
≤
x
≤
3
.
Magnify
ⓘ
Symbols:
I
~
ν
(
x
)
: modified Bessel function of the first kind of imaginary order
,
K
~
ν
(
x
)
: modified Bessel function fo the second kind of imaginary order
and
x
: real variable
Keywords:
graphs
,
modified Bessel functions
,
of imaginary order
Permalink:
http://dlmf.nist.gov/10.26.F8
Encodings:
pdf
,
png
See also:
Annotations for
§10.26(iii)
,
§10.26
and
Ch.10
Figure 10.26.9:
I
~
5
(
x
)
,
K
~
5
(
x
)
,
0.01
≤
x
≤
3
.
Magnify
ⓘ
Symbols:
I
~
ν
(
x
)
: modified Bessel function of the first kind of imaginary order
,
K
~
ν
(
x
)
: modified Bessel function fo the second kind of imaginary order
and
x
: real variable
Keywords:
graphs
,
modified Bessel functions
,
of imaginary order
Permalink:
http://dlmf.nist.gov/10.26.F9
Encodings:
pdf
,
png
See also:
Annotations for
§10.26(iii)
,
§10.26
and
Ch.10
Figure 10.26.10:
K
~
5
(
x
)
,
0.01
≤
x
≤
3
.
Magnify
ⓘ
Symbols:
K
~
ν
(
x
)
: modified Bessel function fo the second kind of imaginary order
and
x
: real variable
Keywords:
graphs
,
modified Bessel functions
,
of imaginary order
Permalink:
http://dlmf.nist.gov/10.26.F10
Encodings:
pdf
,
png
See also:
Annotations for
§10.26(iii)
,
§10.26
and
Ch.10