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1: 30.5 Functions of the Second Kind

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30.5.1 $${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$$ $n=m,m+1,m+2,\mathrm{\dots }$.

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Symbols:

${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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30.5.2 $${\mathrm{𝖰𝗌}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m+1}{\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$$

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Symbols:

${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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30.5.3 $${\mathrm{𝖰𝗌}}_n^m(x,0)={𝖰}_n^m\left(x\right);$$

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Symbols:

: Ferrers function of the second kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer and $n\ge m$: integer degree

Permalink:

http://dlmf.nist.gov/30.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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30.5.4 $$𝒲\left\{{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),{\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\right\}=\frac{(n+m)!}{(1-x^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2)(\ne 0),$$

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Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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2: 30.4 Functions of the First Kind

… ►The eigenfunctions of (30.2.1) that correspond to the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are denoted by ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, $n=m,m+1,m+2,\mathrm{\dots }$. …the sign of ${\mathrm{𝖯𝗌}}_n^m(0,{\gamma }^2)$ being ${(-1)}^{(n+m)/2}$ when $n-m$ is even, and the sign of ${d{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)/dx|}_{x=0}$ being ${(-1)}^{(n+m-1)/2}$ when $n-m$ is odd. ►When ${\gamma }^2>0$ ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the prolate angular spheroidal wave function, and when ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the oblate angular spheroidal wave function. If $\gamma =0$, ${\mathrm{𝖯𝗌}}_n^m(x,0)$ reduces to the Ferrers function ${𝖯}_n^m\left(x\right)$: … ► ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ has exactly $n-m$ zeros in the interval . …

3: Software Index

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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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4: 30.1 Special Notation

… ►The main functions treated in this chapter are the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ and the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$, and $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. … ►Flammer (1957) and Abramowitz and Stegun (1964) use ${\lambda }_{mn}(\gamma )$ for ${\lambda }_n^m\left({\gamma }^2\right)+{\gamma }^2$, $R_{mn}^{(j)}(\gamma ,z)$ for $S_n^{m(j)}(z,\gamma )$, and ►

$S_{mn}^{(1)}(\gamma ,x)=d_{mn}(\gamma ){\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$

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$S_{mn}^{(2)}(\gamma ,x)=d_{mn}(\gamma ){\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$

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5: 30.7 Graphics

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Figure 30.7.5: ${\mathrm{𝖯𝗌}}_n^0(x,4)$, $n=0,1,2,3$, $-1\le x\le 1$. Magnify

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Figure 30.7.6: ${\mathrm{𝖯𝗌}}_n^0(x,-4)$, $n=0,1,2,3$, $-1\le x\le 1$. Magnify

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Figure 30.7.7: ${\mathrm{𝖯𝗌}}_n^1(x,30)$, $n=1,2,3,4$, $-1\le x\le 1$. Magnify

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Figure 30.7.8: ${\mathrm{𝖯𝗌}}_n^1(x,-30)$, $n=1,2,3,4$, $-1\le x\le 1$. Magnify

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Figure 30.7.11: ${\mathrm{𝖰𝗌}}_n^0(x,4)$, $n=0,1,2,3$, . Magnify

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6: 14.22 Graphics

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Figure 14.22.1: $P_{1/2}^0\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. There is a cut along the real axis from $-\mathrm{\infty }$ to $-1$. Magnify 3D Help

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Figure 14.22.2: $P_{1/2}^{-1/2}\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. There is a cut along the real axis from $-\mathrm{\infty }$ to $1$. Magnify 3D Help

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Figure 14.22.3: $P_{1/2}^{-1}\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. There is a cut along the real axis from $-\mathrm{\infty }$ to $1$. Magnify 3D Help

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Figure 14.22.4: ${𝑸}_0^0\left(x+\mathrm{i}y\right)$, $-5\le x\le 5$, $-5\le y\le 5$. There is a cut along the real axis from $-1$ to $1$. Magnify 3D Help

7: 30.10 Series and Integrals

… ►Integrals and integral equations for ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …

8: 4.37 Inverse Hyperbolic Functions

… ►These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively. … ►

§4.37(iv) Logarithmic Forms

… ►On the cuts … ►On the part of the cuts from $-1$ to $1$ … ►On the cuts …

9: 30.6 Functions of Complex Argument

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30.6.4 $${\mathrm{𝑃𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

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Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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30.6.5 $${\mathrm{𝑄𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m\left({\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\mp \frac{1}{2}\mathrm{i}\pi {\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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10: 4.2 Definitions

… ►Most texts extend the definition of the principal value to include the branch cut …where $k$ is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense. ►In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by $z=x+\mathrm{i}0$, the other set corresponding to the “lower side” and denoted by $z=x-\mathrm{i}0$. …Consequently $\ln z$ is two-valued on the cut, and discontinuous across the cut. … ►This is an analytic function of $z$ on $ℂ\setminus (-\mathrm{\infty },0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in \mathbb{Z}$. …