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11
Struve and Related Functions
Struve and Modified Struve Functions
11.2
Definitions
11.4
Basic Properties
§11.3
Graphics
Permalink:
http://dlmf.nist.gov/11.3
Contents
§11.3(i)
Struve Functions
§11.3(ii)
Modified Struve Functions
§11.3(i)
Struve Functions
Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/11.3.i
Figure 11.3.1:
${\mathbf{H}}_{\nu}\left(x\right)$
for
$0\le x\le 12$
and
$\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$
.
Symbols:
${\mathbf{H}}_{\nu}\left(z\right)$
: Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F1
Encodings:
pdf
,
png
Figure 11.3.2:
${\mathbf{K}}_{\nu}\left(x\right)$
for
$$
and
$\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$
.
Symbols:
${\mathbf{K}}_{\nu}\left(z\right)$
: Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F2
Encodings:
pdf
,
png
Figure 11.3.3:
${\mathbf{H}}_{\nu}\left(x\right)$
for
$0\le x\le 12$
and
$\nu =-3,-2,-\frac{3}{2},-1,-\frac{1}{2}$
.
Symbols:
${\mathbf{H}}_{\nu}\left(z\right)$
: Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F3
Encodings:
pdf
,
png
Figure 11.3.4:
${\mathbf{K}}_{\nu}\left(x\right)$
for
$$
and
$\nu =-4,-3,-2,-1,0$
. If
$\nu =-\frac{1}{2},-\frac{3}{2}$
, …, then
${\mathbf{K}}_{\nu}\left(x\right)$
is identically zero.
Symbols:
${\mathbf{K}}_{\nu}\left(z\right)$
: Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F4
Encodings:
pdf
,
png
Visualization Help
Figure 11.3.5:
${\mathbf{H}}_{\nu}\left(x\right)$
for
$0\le x\le 8$
and
$-4\le \nu \le 4$
.
Symbols:
${\mathbf{H}}_{\nu}\left(z\right)$
: Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F5
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.6:
${\mathbf{K}}_{\nu}\left(x\right)$
for
$0\le x\le 8$
and
$-4\le \nu \le 4$
.
Symbols:
${\mathbf{K}}_{\nu}\left(z\right)$
: Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F6
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.7:
$\left|{\mathbf{H}}_{0}\left(x+\mathrm{i}y\right)\right|$
for
$-8\le x\le 8$
and
$-3\le y\le 3$
.
Symbols:
${\mathbf{H}}_{\nu}\left(z\right)$
: Struve function
and
$x$
: real variable
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F7
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.8:
$\left|{\mathbf{K}}_{0}\left(x+\mathrm{i}y\right)\right|$
(principal value) for
$-8\le x\le 8$
and
$-3\le y\le 3$
. There is a cut along the negative real axis.
Symbols:
${\mathbf{K}}_{\nu}\left(z\right)$
: Struve function
and
$x$
: real variable
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F8
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.9:
$|{\mathbf{H}}_{\frac{1}{2}}\left(x+\mathrm{i}y\right)|$
(principal value) for
$-8\le x\le 8$
and
$-3\le y\le 3$
. There is a cut along the negative real axis.
Symbols:
${\mathbf{H}}_{\nu}\left(z\right)$
: Struve function
and
$x$
: real variable
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F9
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.10:
$|{\mathbf{K}}_{\frac{1}{2}}\left(x+\mathrm{i}y\right)|$
(principal value) for
$-8\le x\le 8$
and
$-3\le y\le 3$
. There is a cut along the negative real axis.
Symbols:
${\mathbf{K}}_{\nu}\left(z\right)$
: Struve function
and
$x$
: real variable
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F10
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.11:
$\left|{\mathbf{H}}_{1}\left(x+\mathrm{i}y\right)\right|$
for
$-8.5\le x\le 8.5$
and
$-3\le y\le 3$
.
Symbols:
${\mathbf{H}}_{\nu}\left(z\right)$
: Struve function
and
$x$
: real variable
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F11
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.12:
$\left|{\mathbf{K}}_{1}\left(x+\mathrm{i}y\right)\right|$
(principal value) for
$-8\le x\le 8$
and
$-3\le y\le 3$
. There is a cut along the negative real axis.
Symbols:
${\mathbf{K}}_{\nu}\left(z\right)$
: Struve function
and
$x$
: real variable
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F12
Encodings:
VRML
,
X3D
,
pdf
,
png
§11.3(ii)
Modified Struve Functions
Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/11.3.ii
Figure 11.3.13:
${\mathbf{L}}_{\nu}\left(x\right)$
for
$$
and
$\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$
.
Symbols:
${\mathbf{L}}_{\nu}\left(z\right)$
: modified Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F13
Encodings:
pdf
,
png
Figure 11.3.14:
${\mathbf{M}}_{\nu}\left(x\right)$
for
$0\le x\le 16$
and
$\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$
.
Symbols:
${\mathbf{M}}_{\nu}\left(z\right)$
: modified Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F14
Encodings:
pdf
,
png
Figure 11.3.15:
${\mathbf{L}}_{\nu}\left(x\right)$
for
$$
and
$\nu =-3,-2,-\frac{3}{2},-1,-\frac{1}{2}$
.
Symbols:
${\mathbf{L}}_{\nu}\left(z\right)$
: modified Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F15
Encodings:
pdf
,
png
Figure 11.3.16:
${\mathbf{M}}_{\nu}\left(x\right)$
for
$$
and
$\nu =-3,-2,-\frac{3}{2},-1,-\frac{1}{2}$
.
Symbols:
${\mathbf{M}}_{\nu}\left(z\right)$
: modified Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F16
Encodings:
pdf
,
png
Visualization Help
Figure 11.3.17:
${\mathbf{L}}_{\nu}\left(x\right)$
for
$0\le x\le 5.6$
and
$-4\le \nu \le 4$
.
Symbols:
${\mathbf{L}}_{\nu}\left(z\right)$
: modified Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F17
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.18:
${\mathbf{M}}_{\nu}\left(x\right)$
for
$0\le x\le 8$
and
$-4\le \nu \le 4$
.
Symbols:
${\mathbf{M}}_{\nu}\left(z\right)$
: modified Struve function
,
$x$
: real variable
and
$\nu $
: real or complex order
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F18
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.19:
$|{\mathbf{M}}_{-\frac{1}{2}}\left(x+\mathrm{i}y\right)|$
(principal value) for
$-3\le x\le 3$
and
$-3\le y\le 3$
. There is a cut along the negative real axis.
Symbols:
${\mathbf{M}}_{\nu}\left(z\right)$
: modified Struve function
and
$x$
: real variable
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F19
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 11.3.20:
$|{\mathbf{M}}_{\frac{1}{2}}\left(x+\mathrm{i}y\right)|$
(principal value) for
$-3\le x\le 3$
and
$-3\le y\le 3$
. There is a cut along the negative real axis.
Symbols:
${\mathbf{M}}_{\nu}\left(z\right)$
: modified Struve function
and
$x$
: real variable
Keywords:
Struve functions and modified Struve functions
Permalink:
http://dlmf.nist.gov/11.3.F20
Encodings:
VRML
,
X3D
,
pdf
,
png