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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
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
Notes:
These graphics were produced at NIST.
Keywords:
modified Bessel functions
Permalink:
http://dlmf.nist.gov/10.26.i
Figure 10.26.1:
${I}_{0}\left(x\right)$
,
${I}_{1}\left(x\right)$
,
${K}_{0}\left(x\right)$
,
${K}_{1}\left(x\right)$
,
$0\le x\le 3$
.
Symbols:
${I}_{\nu}\left(z\right)$
: modified Bessel function of the first kind
,
${K}_{\nu}\left(z\right)$
: modified Bessel function of the second kind
and
$x$
: real variable
Permalink:
http://dlmf.nist.gov/10.26.F1
Encodings:
pdf
,
png
Figure 10.26.2:
${\mathrm{e}}^{-x}{I}_{0}\left(x\right)$
,
${\mathrm{e}}^{-x}{I}_{1}\left(x\right)$
,
${\mathrm{e}}^{x}{K}_{0}\left(x\right)$
,
${\mathrm{e}}^{x}{K}_{1}\left(x\right)$
,
$0\le x\le 10$
.
Symbols:
$\mathrm{e}$
: base of exponential function
,
${I}_{\nu}\left(z\right)$
: modified Bessel function of the first kind
,
${K}_{\nu}\left(z\right)$
: modified Bessel function of the second kind
and
$x$
: real variable
Permalink:
http://dlmf.nist.gov/10.26.F2
Encodings:
pdf
,
png
Visualization Help
Figure 10.26.3:
${I}_{\nu}\left(x\right)$
,
$0\le x\le 5,0\le \nu \le 4$
.
Symbols:
${I}_{\nu}\left(z\right)$
: modified Bessel function of the first kind
,
$x$
: real variable
and
$\nu $
: complex parameter
Permalink:
http://dlmf.nist.gov/10.26.F3
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 10.26.4:
${K}_{\nu}\left(x\right)$
,
$0.1\le x\le 5,0\le \nu \le 4$
.
Symbols:
${K}_{\nu}\left(z\right)$
: modified Bessel function of the second kind
,
$x$
: real variable
and
$\nu $
: complex parameter
Permalink:
http://dlmf.nist.gov/10.26.F4
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 10.26.5:
${{I}_{\nu}}^{\prime}\left(x\right)$
,
$0\le x\le 5,0\le \nu \le 4$
.
Symbols:
${I}_{\nu}\left(z\right)$
: modified Bessel function of the first kind
,
$x$
: real variable
and
$\nu $
: complex parameter
Permalink:
http://dlmf.nist.gov/10.26.F5
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 10.26.6:
${{K}_{\nu}}^{\prime}\left(x\right)$
,
$0.3\le x\le 5,0\le \nu \le 4$
.
Symbols:
${K}_{\nu}\left(z\right)$
: modified Bessel function of the second kind
,
$x$
: real variable
and
$\nu $
: complex parameter
Permalink:
http://dlmf.nist.gov/10.26.F6
Encodings:
VRML
,
X3D
,
pdf
,
png
§10.26(ii)
Real Order, Complex Variable
Permalink:
http://dlmf.nist.gov/10.26.ii
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
For the notation, see §
10.45
.
Figure 10.26.7:
${\stackrel{~}{I}}_{1/2}\left(x\right),{\stackrel{~}{K}}_{1/2}\left(x\right),0.01\le x\le 3$
.
Symbols:
${\stackrel{~}{I}}_{\nu}\left(x\right)$
: modified Bessel function of the first kind of imaginary order
,
${\stackrel{~}{K}}_{\nu}\left(x\right)$
: modified Bessel function fo the second kind of imaginary order
and
$x$
: real variable
Keywords:
modified Bessel functions
Permalink:
http://dlmf.nist.gov/10.26.F7
Encodings:
pdf
,
png
Figure 10.26.8:
${\stackrel{~}{I}}_{1}\left(x\right),{\stackrel{~}{K}}_{1}\left(x\right),0.01\le x\le 3$
.
Symbols:
${\stackrel{~}{I}}_{\nu}\left(x\right)$
: modified Bessel function of the first kind of imaginary order
,
${\stackrel{~}{K}}_{\nu}\left(x\right)$
: modified Bessel function fo the second kind of imaginary order
and
$x$
: real variable
Keywords:
modified Bessel functions
Permalink:
http://dlmf.nist.gov/10.26.F8
Encodings:
pdf
,
png
Figure 10.26.9:
${\stackrel{~}{I}}_{5}\left(x\right),{\stackrel{~}{K}}_{5}\left(x\right),0.01\le x\le 3$
.
Symbols:
${\stackrel{~}{I}}_{\nu}\left(x\right)$
: modified Bessel function of the first kind of imaginary order
,
${\stackrel{~}{K}}_{\nu}\left(x\right)$
: modified Bessel function fo the second kind of imaginary order
and
$x$
: real variable
Keywords:
modified Bessel functions
Permalink:
http://dlmf.nist.gov/10.26.F9
Encodings:
pdf
,
png
Figure 10.26.10:
${\stackrel{~}{K}}_{5}\left(x\right),0.01\le x\le 3$
.
Symbols:
${\stackrel{~}{K}}_{\nu}\left(x\right)$
: modified Bessel function fo the second kind of imaginary order
and
$x$
: real variable
Keywords:
modified Bessel functions
Permalink:
http://dlmf.nist.gov/10.26.F10
Encodings:
pdf
,
png