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30
Spheroidal Wave Functions
Properties
30.6
Functions of Complex Argument
30.8
Expansions in Series of Ferrers Functions
§30.7
Graphics
Permalink:
http://dlmf.nist.gov/30.7
Contents
§30.7(i)
Eigenvalues
§30.7(ii)
Functions of the First Kind
§30.7(iii)
Functions of the Second Kind
§30.7(iv)
Functions of Complex Argument
§30.7(i)
Eigenvalues
Notes:
Figures
30.7.1
–
30.7.4
were produced at NIST with the aid of Maple procedures provided by the author.
Keywords:
spheroidal differential equation
Permalink:
http://dlmf.nist.gov/30.7.i
Figure 30.7.1:
Eigenvalues
${\lambda}_{n}^{0}\left({\gamma}^{2}\right)$
,
$n=0,1,2,3$
,
$-10\le {\gamma}^{2}\le 10$
.
Symbols:
${\lambda}_{n}^{m}\left({\gamma}^{2}\right)$
: eigenvalues of the spheroidal differential equation
,
$n\ge m$
: integer degree
and
${\gamma}^{2}$
: real parameter
Referenced by:
§30.7(i)
Permalink:
http://dlmf.nist.gov/30.7.F1
Encodings:
pdf
,
png
Figure 30.7.2:
Eigenvalues
${\lambda}_{n}^{1}\left({\gamma}^{2}\right)$
$n=1,2,3,4$
,
$-10\le {\gamma}^{2}\le 10$
.
Symbols:
${\lambda}_{n}^{m}\left({\gamma}^{2}\right)$
: eigenvalues of the spheroidal differential equation
,
$n\ge m$
: integer degree
and
${\gamma}^{2}$
: real parameter
Permalink:
http://dlmf.nist.gov/30.7.F2
Encodings:
pdf
,
png
Figure 30.7.3:
Eigenvalues
${\lambda}_{n}^{5}\left({\gamma}^{2}\right)$
,
$n=5,6,7,8$
,
$-40\le {\gamma}^{2}\le 40$
.
Symbols:
${\lambda}_{n}^{m}\left({\gamma}^{2}\right)$
: eigenvalues of the spheroidal differential equation
,
$n\ge m$
: integer degree
and
${\gamma}^{2}$
: real parameter
Permalink:
http://dlmf.nist.gov/30.7.F3
Encodings:
pdf
,
png
Figure 30.7.4:
Eigenvalues
${\lambda}_{n}^{10}\left({\gamma}^{2}\right)$
,
$n=10,11,12,13$
,
$-50\le {\gamma}^{2}\le 150$
.
Symbols:
${\lambda}_{n}^{m}\left({\gamma}^{2}\right)$
: eigenvalues of the spheroidal differential equation
,
$n\ge m$
: integer degree
and
${\gamma}^{2}$
: real parameter
Referenced by:
§30.7(i)
Permalink:
http://dlmf.nist.gov/30.7.F4
Encodings:
pdf
,
png
§30.7(ii)
Functions of the First Kind
Notes:
Figures
30.7.5
–
30.7.10
were produced at NIST with the aid of Maple procedures provided by the author.
Keywords:
spheroidal wave functions
Permalink:
http://dlmf.nist.gov/30.7.ii
Figure 30.7.5:
${\mathsf{Ps}}_{n}^{0}(x,4)$
,
$n=0,1,2,3$
,
$-1\le x\le 1$
.
Symbols:
${\mathsf{Ps}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the first kind
,
$x$
: real variable
and
$n\ge m$
: integer degree
Referenced by:
§30.7(ii)
Permalink:
http://dlmf.nist.gov/30.7.F5
Encodings:
pdf
,
png
Figure 30.7.6:
${\mathsf{Ps}}_{n}^{0}(x,-4)$
,
$n=0,1,2,3$
,
$-1\le x\le 1$
.
Symbols:
${\mathsf{Ps}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the first kind
,
$x$
: real variable
and
$n\ge m$
: integer degree
Permalink:
http://dlmf.nist.gov/30.7.F6
Encodings:
pdf
,
png
Figure 30.7.7:
${\mathsf{Ps}}_{n}^{1}(x,30)$
,
$n=1,2,3,4$
,
$-1\le x\le 1$
.
Symbols:
${\mathsf{Ps}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the first kind
,
$x$
: real variable
and
$n\ge m$
: integer degree
Permalink:
http://dlmf.nist.gov/30.7.F7
Encodings:
pdf
,
png
Figure 30.7.8:
${\mathsf{Ps}}_{n}^{1}(x,-30)$
,
$n=1,2,3,4$
,
$-1\le x\le 1$
.
Symbols:
${\mathsf{Ps}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the first kind
,
$x$
: real variable
and
$n\ge m$
: integer degree
Permalink:
http://dlmf.nist.gov/30.7.F8
Encodings:
pdf
,
png
Visualization Help
Figure 30.7.9:
${\mathsf{Ps}}_{2}^{0}(x,{\gamma}^{2})$
,
$-1\le x\le 1$
,
$-50\le {\gamma}^{2}\le 50$
.
Symbols:
${\mathsf{Ps}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the first kind
,
$x$
: real variable
and
${\gamma}^{2}$
: real parameter
Permalink:
http://dlmf.nist.gov/30.7.F9
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 30.7.10:
${\mathsf{Ps}}_{3}^{1}(x,{\gamma}^{2})$
,
$-1\le x\le 1$
,
$-50\le {\gamma}^{2}\le 50$
.
Symbols:
${\mathsf{Ps}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the first kind
,
$x$
: real variable
and
${\gamma}^{2}$
: real parameter
Referenced by:
§30.7(ii)
Permalink:
http://dlmf.nist.gov/30.7.F10
Encodings:
VRML
,
X3D
,
pdf
,
png
§30.7(iii)
Functions of the Second Kind
Notes:
Figures
30.7.11
–
30.7.15
were produced at NIST with the aid of Maple procedures provided by the author.
Permalink:
http://dlmf.nist.gov/30.7.iii
Figure 30.7.11:
${\mathsf{Qs}}_{n}^{0}(x,4)$
,
$n=0,1,2,3$
,
$$
.
Symbols:
${\mathsf{Qs}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the second kind
,
$x$
: real variable
and
$n\ge m$
: integer degree
Referenced by:
§30.7(iii)
Permalink:
http://dlmf.nist.gov/30.7.F11
Encodings:
pdf
,
png
Figure 30.7.12:
${\mathsf{Qs}}_{n}^{0}(x,-4)$
,
$n=0,1,2,3$
,
$$
.
Symbols:
${\mathsf{Qs}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the second kind
,
$x$
: real variable
and
$n\ge m$
: integer degree
Permalink:
http://dlmf.nist.gov/30.7.F12
Encodings:
pdf
,
png
Figure 30.7.13:
${\mathsf{Qs}}_{n}^{1}(x,4)$
, for
$n=1,2,3,4$
,
$$
.
Symbols:
${\mathsf{Qs}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the second kind
,
$x$
: real variable
and
$n\ge m$
: integer degree
Permalink:
http://dlmf.nist.gov/30.7.F13
Encodings:
pdf
,
png
Figure 30.7.14:
${\mathsf{Qs}}_{n}^{1}(x,-4)$
,
$n=1,2,3,4$
,
$$
.
Symbols:
${\mathsf{Qs}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the second kind
,
$x$
: real variable
and
$n\ge m$
: integer degree
Permalink:
http://dlmf.nist.gov/30.7.F14
Encodings:
pdf
,
png
Visualization Help
Figure 30.7.15:
$$
.
Symbols:
${\mathsf{Qs}}_{n}^{m}(x,{\gamma}^{2})$
: spheroidal wave function of the second kind
,
$x$
: real variable
and
${\gamma}^{2}$
: real parameter
Referenced by:
§30.7(iii)
Permalink:
http://dlmf.nist.gov/30.7.F15
Encodings:
VRML
,
X3D
,
pdf
,
png
§30.7(iv)
Functions of Complex Argument
Notes:
Figures
30.7.16
–
30.7.21
were produced at NIST with the aid of Maple procedures provided by the author.
Keywords:
spheroidal wave functions
Referenced by:
§30.1
Permalink:
http://dlmf.nist.gov/30.7.iv
Visualization Help
Figure 30.7.16:
$|{\mathit{Ps}}_{0}^{0}(x+\mathrm{i}y,4)|$
,
$-2\le x\le 2$
,
$-2\le y\le 2$
.
Symbols:
${\mathit{Ps}}_{n}^{m}(z,{\gamma}^{2})$
: spheroidal wave function of complex argument
,
$x$
: real variable
and
$y$
: real variable
Referenced by:
§30.7(iv)
Permalink:
http://dlmf.nist.gov/30.7.F16
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 30.7.17:
$|{\mathit{Ps}}_{0}^{0}(x+\mathrm{i}y,-4)|$
,
$-2\le x\le 2$
,
$-2\le y\le 2$
.
Symbols:
${\mathit{Ps}}_{n}^{m}(z,{\gamma}^{2})$
: spheroidal wave function of complex argument
,
$x$
: real variable
and
$y$
: real variable
Permalink:
http://dlmf.nist.gov/30.7.F17
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 30.7.18:
$|{\mathit{Ps}}_{1}^{1}(x+\mathrm{i}y,4)|$
,
$-2\le x\le 2$
,
$-2\le y\le 2$
.
Symbols:
${\mathit{Ps}}_{n}^{m}(z,{\gamma}^{2})$
: spheroidal wave function of complex argument
,
$x$
: real variable
and
$y$
: real variable
Permalink:
http://dlmf.nist.gov/30.7.F18
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 30.7.19:
$|{\mathit{Ps}}_{1}^{1}(x+\mathrm{i}y,-4)|$
,
$-2\le x\le 2$
,
$-2\le y\le 2$
.
Symbols:
${\mathit{Ps}}_{n}^{m}(z,{\gamma}^{2})$
: spheroidal wave function of complex argument
,
$x$
: real variable
and
$y$
: real variable
Permalink:
http://dlmf.nist.gov/30.7.F19
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 30.7.20:
$|{\mathit{Qs}}_{0}^{0}(x+\mathrm{i}y,4)|$
,
$-2\le x\le 2$
,
$-2\le y\le 2$
.
Symbols:
${\mathit{Qs}}_{n}^{m}(z,{\gamma}^{2})$
: spheroidal wave function of complex argument
,
$x$
: real variable
and
$y$
: real variable
Permalink:
http://dlmf.nist.gov/30.7.F20
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 30.7.21:
$|{\mathit{Qs}}_{0}^{0}(x+\mathrm{i}y,-4)|$
,
$-1.8\le x\le 1.8$
,
$-2\le y\le 2$
.
Symbols:
${\mathit{Qs}}_{n}^{m}(z,{\gamma}^{2})$
: spheroidal wave function of complex argument
,
$x$
: real variable
and
$y$
: real variable
Referenced by:
§30.7(iv)
Permalink:
http://dlmf.nist.gov/30.7.F21
Encodings:
VRML
,
X3D
,
pdf
,
png