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1: 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 . …

2: 30.6 Functions of Complex Argument

… ►The solutions ►

${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),$

… ►

${\mathrm{𝑃𝑠}}_n^m(z,0)=P_n^m\left(z\right),$

… ►

30.6.3 $$𝒲\left\{{\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),{\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)\right\}=\frac{{(-1)}^m(n+m)!}{(1-z^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2),$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/30.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

… ►

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

ⓘ

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

…

3: 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 ►

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

…

4: 30.7 Graphics

… ►

►►

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

►

►►

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

►

►►

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

►

►►

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

►

►

►

Figure 30.7.9: ${\mathrm{𝖯𝗌}}_2^0(x,{\gamma }^2)$, $-1\le x\le 1$, $-50\le {\gamma }^2\le 50$. Magnify 3D Help

…

5: 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). …

6: 30.5 Functions of the Second Kind

… ►

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

ⓘ

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

…

7: 30.16 Methods of Computation

… ►If $|{\gamma }^2|$ is large, then we can use the asymptotic expansions referred to in §30.9 to approximate ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$. ►If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then we can compute ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime }(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime }(0)=1$ if $n-m$ is odd. ►If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ can be found by summing (30.8.1). … ►

30.16.9 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)=\underset{d\to \mathrm{\infty }}{lim}\sum _{j=1}^d{(-1)}^{j-p}{e}_{j,d}{𝖯}_{n+2(j-p)}^m\left(x\right).$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/30.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(ii), §30.16 and Ch.30

…

8: 30.18 Software

… ►

SWF2: ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$.

…

9: 30.11 Radial Spheroidal Wave Functions

… ►

§30.11(v) Connection with the $\mathrm{𝑃𝑠}$ and $\mathrm{𝑄𝑠}$ Functions

►

30.11.8 $$S_n^{m(1)}(z,\gamma )=K_n^m(\gamma ){\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),$$

ⓘ

Symbols:

$S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $K_n^m(\gamma )$: connection coefficient and ${\gamma }^2$: real parameter

A&S Ref:

21.10.1 (in different form)

Permalink:

http://dlmf.nist.gov/30.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(v), §30.11 and Ch.30

… ►

30.11.10 $$K_n^m(\gamma )=\frac{\sqrt{\pi }}{2}{\left(\frac{\gamma }{2}\right)}^m\frac{{(-1)}^m{a}_{n,\frac{1}{2}(m-n)}^{-m}({\gamma }^2)}{\mathrm{\Gamma }\left(\frac{3}{2}+m\right)A_n^{-m}({\gamma }^2){\mathrm{𝖯𝗌}}_n^m(0,{\gamma }^2)},$$ $n-m$ even,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ , $K_n^m(\gamma )$: connection coefficient, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Referenced by:

§30.16(iii)

Permalink:

http://dlmf.nist.gov/30.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(v), §30.11 and Ch.30

… ►

30.11.11 $$K_n^m(\gamma )=\frac{\sqrt{\pi }}{2}{\left(\frac{\gamma }{2}\right)}^{m+1}\frac{{(-1)}^m{a}_{n,\frac{1}{2}(m-n+1)}^{-m}({\gamma }^2)}{\mathrm{\Gamma }\left(\frac{5}{2}+m\right)A_n^{-m}({\gamma }^2)({d{\mathrm{𝖯𝗌}}_n^m(z,{\gamma }^2)/dz|}_{z=0})},$$ $n-m$ odd.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ , $K_n^m(\gamma )$: connection coefficient, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Referenced by:

§30.16(iii)

Permalink:

http://dlmf.nist.gov/30.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(v), §30.11 and Ch.30

… ►

30.11.12 $$A_n^{-m}({\gamma }^2)S_n^{m(1)}(z,\gamma )=\frac{1}{2}{\mathrm{i}}^{m+n}{\gamma }^m\frac{(n-m)!}{(n+m)!}{z}^m{(1-z^{-2})}^{\frac{1}{2}m}{\int }_{-1}^1{\mathrm{e}}^{-\mathrm{i}\gamma zt}{(1-t^2)}^{\frac{1}{2}m}{\mathrm{𝖯𝗌}}_n^m(t,{\gamma }^2)dt.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(vi), §30.11 and Ch.30

…

10: 30.8 Expansions in Series of Ferrers Functions

… ►

30.8.1 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)=\sum _{k=-R}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){𝖯}_{n+2k}^m\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $R=\lfloor (n-m)/2\rfloor$: limit and $a_{n,k}^m({\gamma }^2)$: coefficients

A&S Ref:

21.7.1 (in different form)

Referenced by:

§30.16(ii), §30.16(ii)

Permalink:

http://dlmf.nist.gov/30.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

… ►

30.8.2 $$a_{n,k}^m({\gamma }^2)={(-1)}^k\left(n+2k+\frac{1}{2}\right)\frac{(n-m+2k)!}{(n+m+2k)!}{\int }_{-1}^1{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2){𝖯}_{n+2k}^m\left(x\right)dx.$$

ⓘ

Defines:

$a_{n,k}^m({\gamma }^2)$: coefficients (locally)

Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Referenced by:

§30.11(i)

Permalink:

http://dlmf.nist.gov/30.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(i), §30.8 and Ch.30

…