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11: 14.30 Spherical and Spheroidal Harmonics

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14.30.1 $$Y_{l,m}(\theta ,\varphi )={\left(\frac{(l-m)!(2l+1)}{4\pi (l+m)!}\right)}^{1/2}{\mathrm{e}}^{\mathrm{i}m\varphi }{𝖯}_l^m\left(\cos \theta \right),$$

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Defines:

$Y_{l,m}(\theta ,\varphi )$: spherical harmonic

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(i), §14.30(ii), §14.30(ii), §14.30(iv)

Permalink:

http://dlmf.nist.gov/14.30.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(i), §14.30 and Ch.14

… ►Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics $R_n^m(x)={\mathrm{e}}^{-\mathrm{i}\pi n/2}{P}_n^m\left(\mathrm{i}x\right)$ and $T_n^m(x)=\mathrm{i}{\mathrm{e}}^{\mathrm{i}\pi n/2}{Q}_n^m\left(\mathrm{i}x\right)$ which are real when $x>0$ and $n=0,1,2,\mathrm{\dots }$. … ►

14.30.3 $$Y_{l,m}(\theta ,\varphi )=\frac{{(-1)}^{l+m}}{2^{l}l!}{\left(\frac{(l-m)!(2l+1)}{4\pi (l+m)!}\right)}^{1/2}{\mathrm{e}}^{\mathrm{i}m\varphi }{\left(\sin \theta \right)}^m{\left(\frac{d}{d(\cos \theta )}\right)}^{l+m}{\left(\sin \theta \right)}^{2l}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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14.30.4

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii), Erratum (V1.0.25) for Section 14.30

Permalink:

http://dlmf.nist.gov/14.30.E4

Encodings:

pMML, png, TeX

Correction (effective with 1.0.25):

Previously this equation was only given for $m\ge 0$. In this update we have extended the definition of spherical harmonics to include negative integer values such that $|m|\le l$. Hence in the second part of this equation, we have replaced $m$ with $|m|$.

Suggested 2019-10-15 by Ching-Li Chai

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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14.30.5

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\notin$: not an element of, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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12: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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A vortex filament moving without change of form. J. Fluid Mech. 112, pp. 397–409.

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External Links:

ISSN 0022-1120, Document, MathReview, ZentralBlatt

Referenced by:

§19.35(ii).

Encodings:

BibTeX

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

BibTeX

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13: 1.11 Zeros of Polynomials

… ►The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. A similar relation holds for the changes in sign of the coefficients of $f(-z)$, and hence for the number of negative zeros of $f(z)$. … ►Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros. … ►The sum and product of the roots are respectively $-b/a$ and $c/a$. … ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. …

14: 20.7 Identities

… ►See Lawden (1989, pp. 19–20). … ►In the following equations ${\tau }^{\prime }=-1/\tau$, and all square roots assume their principal values. ►

20.7.30 $${(-\mathrm{i}\tau )}^{1/2}{\theta }_1\left(z\right|\tau )=-\mathrm{i}\exp \left(\mathrm{i}{\tau }^{\prime }{z}^2/\pi \right){\theta }_1\left(z{\tau }^{\prime }\right|{\tau }^{\prime }),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $z$: complex and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(viii), §20.7 and Ch.20

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20.7.31 $${(-\mathrm{i}\tau )}^{1/2}{\theta }_2\left(z\right|\tau )=\exp \left(\mathrm{i}{\tau }^{\prime }{z}^2/\pi \right){\theta }_4\left(z{\tau }^{\prime }\right|{\tau }^{\prime }),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $z$: complex and $\tau$: lattice parameter

Referenced by:

(20.10.1)

Permalink:

http://dlmf.nist.gov/20.7.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(viii), §20.7 and Ch.20

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20.7.34 $$\frac{{\theta }_1(z,q^2){\theta }_3(z,q^2)}{{\theta }_1(z,\mathrm{i}q)}=\frac{{\theta }_2(z,q^2){\theta }_4(z,q^2)}{{\theta }_2(z,\mathrm{i}q)}={\mathrm{i}}^{-1/4}\sqrt{\frac{{\theta }_2(0,q^2){\theta }_4(0,q^2)}{2}}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{i}$: imaginary unit, $z$: complex and $q$: nome

Referenced by:

§20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/20.7.E34

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. See Walker (2012).

See also:

Annotations for §20.7(ix), §20.7 and Ch.20

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