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1: 30.16 Methods of Computation

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30.16.2 $${\alpha }_{j,d+1}\le {\alpha }_{j,d},$$

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Symbols:

$d$: dimension and ${\alpha }_{j,k}$: real eigenvalues

Permalink:

http://dlmf.nist.gov/30.16.E2

Encodings:

pMML, png, TeX

See also:

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

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30.16.3 $${\lambda }_n^m\left({\gamma }^2\right)=\underset{d\to \mathrm{\infty }}{lim}{\alpha }_{p,d},$$ $p=\lfloor \frac{1}{2}(n-m)\rfloor +1$.

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Symbols:

$\lfloor x\rfloor$: floor of $x$ , ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\alpha }_{j,k}$: real eigenvalues and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.16.E3

Encodings:

pMML, png, TeX

See also:

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

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30.16.4 $${\alpha }_{p,d}-{\lambda }_n^m\left({\gamma }^2\right)=O\left(\frac{{\gamma }^{4d}}{4^{2d+1}{((m+2d-1)!(m+2d+1)!)}^2}\right),$$ $d\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\alpha }_{j,k}$: real eigenvalues and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.16.E4

Encodings:

pMML, png, TeX

See also:

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

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30.16.7 $$\sum _{j=1}^d{e}_{j,d}^2\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1}=\frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}.$$

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Symbols:

$!$: factorial (as in $n!$), $m$: nonnegative integer, $n\ge m$: integer degree and $d$: dimension

Permalink:

http://dlmf.nist.gov/30.16.E7

Encodings:

pMML, png, TeX

See also:

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

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30.16.8 $$a_{n,k}^m({\gamma }^2)=\underset{d\to \mathrm{\infty }}{lim}{e}_{k+p,d},$$

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Symbols:

$m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Permalink:

http://dlmf.nist.gov/30.16.E8

Encodings:

pMML, png, TeX

See also:

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

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2: 37.11 Spherical Harmonics

… ►In the case of dimension $d=3$ see §14.30 for spherical harmonics and (1.5.17) for the Laplacian. … ►

37.11.3 $${𝕊}^{d-1}=\{𝐱\in {ℝ}^d\mid x_1^2+\mathrm{\cdots }+x_d^2=1\},$$ $d\ge 2$.

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Defines:

${𝕊}^{d-1}$: unit sphere of dimension $d-1$ (locally)

Symbols:

$\in$: element of, $ℝ$: real line, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11, Part 37.III and Ch.37

… ►Its dimension is ►

37.11.5 $$N_n^d=dim{\mathscr{H}}_n^{0,d}=dim{\mathscr{H}}_n^d=\frac{2n+d-2}{n+d-2}\left(\genfrac{}{}{0.0pt}{}{n+d-2}{n}\right),$$ $n\in {\mathbb{N}}_0$, $d\ge 2$.

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\in$: element of, $\mathbb{N}$: set of all positive integers, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${\mathbb{N}}_0$: set of all nonnegative integers, ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$ and ${\mathscr{H}}_n^d$: complex linear space

Proved:

Dunkl and Xu (2014, Theorem 4.1.2)(proved)

Referenced by:

§37.12(i)

Permalink:

http://dlmf.nist.gov/37.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11, Part 37.III and Ch.37

… ►

Reduction to Lower Dimension

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3: About Color Map

… ►We use color to augment these vizualizations, either to reinforce the recognition of the height, or to convey an extra dimension to represent the phase of complex valued functions. …

4: 37.13 General Orthogonal Polynomials of $d$ Variables

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37.13.2 $$dim{𝒱}_n^d=\left(\genfrac{}{}{0.0pt}{}{n+d-1}{d}\right).$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ and ${𝒱}_n^d$: vector space of $d$-variable polynomials of degree $n$

Permalink:

http://dlmf.nist.gov/37.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13, Part 37.III and Ch.37

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37.13.8 $$R_{Y,k,n}(r\xi )=p_k^{(n-2k+\frac{1}{2}d-1)}(r^2)r^{n-2k}Y(\xi ),$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y\in {\mathscr{H}}_{n-2k}^{0,d}$, $r\ge 0$, $\xi \in {𝕊}^{d-1}$.

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Symbols:

$\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$

Referenced by:

§37.13(i), §37.15(iii)

Permalink:

http://dlmf.nist.gov/37.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13(i), §37.13, Part 37.III and Ch.37

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37.13.9 $${\int }_{{ℝ}^d}{R}_{Y_1,k,n}(𝐱)R_{Y_2,k,n}(𝐱)W(𝐱)d𝐱=\frac{1}{2}{\omega }_d{\int }_0^{\mathrm{\infty }}{\left(p_k^{(n-2k+\frac{1}{2}d-1)}(x)\right)}^{2}w(x)x^{n-2k+\frac{1}{2}d-1}dx{⟨Y_1,Y_2⟩}_{{𝕊}^{d-1}},$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y_1,Y_2\in {\mathscr{H}}_{n-2k}^{0,d}$.

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $ℝ$: real line, $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ , ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$ , ${\omega }_d$: surface area of unit sphere ${𝕊}^{d-1}$ , ${⟨f,g⟩}_{{𝕊}^{d-1}}$: inner product on $(d-1)$-dimensional sphere, $W$: weight function in $d$-variables and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.13(i), §37.13, Part 37.III and Ch.37

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5: 1.6 Vectors and Vector-Valued Functions

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Green’s Theorem

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Green’s Theorem (for Volume)

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6: 37.19 Other Orthogonal Polynomials of $d$ Variables

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37.19.3 $${⟨f,g⟩}_{w_{\kappa }}={\int }_{{𝕊}^{d-1}}f(\xi )g(\xi )w_{\kappa }(\xi )d\sigma ,$$

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Defines:

${⟨f,g⟩}_{w_{\kappa }}$: inner product for Dunkl harmonics (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $d$: positive integer, usually $\ge 2$ and ${𝕊}^{d-1}$: unit sphere of dimension $d-1$

Sources:

Dunkl (1989, Theorem 1.2); Dunkl and Xu (2014, Lemma 7.1.4)

Permalink:

http://dlmf.nist.gov/37.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(i), §37.19, Part 37.III and Ch.37

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37.19.7 $$⟨f,g⟩={\int }_{{𝔹}^d}{\nabla }^{s}f(𝐱){\nabla }^{s}g(𝐱)d𝐱+\sum _{k=0}^{\lfloor \frac{s}{2}\rfloor -1}{\int }_{{𝕊}^{d-1}}{\mathrm{\Delta }}^{k}f(\xi ){\mathrm{\Delta }}^{k}g(\xi )d\sigma (\xi ),$$

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Defines:

$⟨f,g⟩$: Sobolev inner product on the ball (locally)

Symbols:

$dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $k$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${𝔹}^d$: unit ball

Permalink:

http://dlmf.nist.gov/37.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.19(iv), §37.19, Part 37.III and Ch.37

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7: 21.2 Definitions

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21.2.1 $$\theta \left(𝐳\right|𝛀)=\sum _{𝐧\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝐧\cdot 𝛀\cdot 𝐧+𝐧\cdot 𝐳\right)}.$$

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Defines:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer and $𝛀$: a Riemann matrix

Referenced by:

§21.2(ii)

Permalink:

http://dlmf.nist.gov/21.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(i), §21.2 and Ch.21

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8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Example 1: In one and two dimensions any $q(x)$ with a ‘Dip, or Well’ has a partly discrete spectrum

►Suppose that $X$ is the whole real line in one dimension, and that $q(x)$, in (1.18.28) has (non-oscillatory) limits of $0$ at both $\pm \mathrm{\infty }$, and thus a continuous spectrum on $𝝈\ge 0$. … … ►Put $n_{+}=dim{N}_z$ ($\mathrm{\Im }z>0$) and $n_{-}=dim{N}_z$ (), the deficiency indices for $T$. … ►Thus $N_z$ has dimension 0, 1 or 2. …

9: 37.15 Orthogonal Polynomials on the Ball

… ►

37.15.7 $$R_{Y,k,n}^{\alpha }(r\xi )=R_k^{(\alpha ,n-2k+\frac{1}{2}d-1)}\left(2r^2-1\right)r^{n-2k}Y(\xi ),$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y\in {\mathscr{H}}_{n-2k}^{0,d}$, $r\ge 0$, $\xi \in {𝕊}^{d-1}$.

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Symbols:

$R_n^{(\alpha ,\beta )}\left(x\right)$: Standardized Jacobi polynomial, $\in$: element of, $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$

Referenced by:

(37.17.13), §37.17(v)

Permalink:

http://dlmf.nist.gov/37.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iii), §37.15, Part 37.III and Ch.37

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37.15.8 $${⟨R_{Y_1,k,n}^{\alpha },R_{Y_2,k,n}^{\alpha }⟩}_{\alpha }=\frac{\left(\alpha +n-k+\frac{1}{2}d\right)k!{\left(\frac{1}{2}d\right)}_{n-k}}{\left(\alpha +n+\frac{1}{2}d\right){\left(\alpha +1\right)}_k{\left(\alpha +\frac{1}{2}d+1\right)}_{n-k}}{⟨Y_1,Y_2⟩}_{{𝕊}^{d-1}},$$ $0\le k\le \lfloor \frac{1}{2}n\rfloor$, $Y_1,Y_2\in {\mathscr{H}}_{n-2k}^{0,d}$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ , ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$ , ${⟨f,g⟩}_{{𝕊}^{d-1}}$: inner product on $(d-1)$-dimensional sphere and ${⟨f,g⟩}_{\alpha }$: inner product on $d$-dimensional ball

Proved:

Dunkl and Xu (2014, Proposition 5.2.1); There replace $\mu =\alpha +\frac{1}{2}$(proved)

Permalink:

http://dlmf.nist.gov/37.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iii), §37.15, Part 37.III and Ch.37

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37.15.14 $$C_n^{(\alpha +\frac{1}{2}d)}(⟨𝐱,𝐲⟩)=\sum _{|𝝂|=n}\frac{2^n{\left(\alpha +\frac{1}{2}d\right)}_n}{𝝂!}{𝐲}^{𝝂}{V}_{𝝂}^{(\alpha +\frac{1}{2})}(𝐱),$$ $𝐲\in {𝕊}^{d-1}$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${𝕊}^{d-1}$: unit sphere of dimension $d-1$

Proof sketch:

Use (18.12.4) and (37.15.13).

Permalink:

http://dlmf.nist.gov/37.15.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iv), §37.15(iv), §37.15, Part 37.III and Ch.37

… ►

37.15.19 $${\mathbb{P}}_n^0(𝐱,𝐲)=\frac{2n+d}{d}{\int }_{{𝕊}^{d-1}}{C}_n^{(\frac{1}{2}d)}\left(⟨𝐱,\xi ⟩\right)C_n^{(\frac{1}{2}d)}\left(⟨𝐲,\xi ⟩\right)d\sigma (\xi ),$$ $𝐱,𝐲\in {𝔹}^d$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\in$: element of, $⟨𝐮,𝐯⟩$: inner product over vectors, $\int$: integral, $(a,b)$: open interval, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and ${𝔹}^d$: unit ball

Proved:

Dunkl and Xu (2014, Theorem 5.2.9); There replace $\mu =\alpha +\frac{1}{2}$.(proved)

Permalink:

http://dlmf.nist.gov/37.15.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(vi), §37.15, Part 37.III and Ch.37

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10: 10.73 Physical Applications

… ►Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …