harmonic functions

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1—10 of 33 matching pages

1: Tom H. Koornwinder

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …

2: 1.9 Calculus of a Complex Variable

… ►

Harmonic Functions

►If

f(z)=u(x,y)+\mathrm{i}v(x,y)

is analytic in an open domain

D

, then

u

and

v

are harmonic in

D

, that is, … ►

Mean Value Property

►For

u(z)

harmonic, … ►

Poisson Integral

…

3: 12.17 Physical Applications

… ►Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. … ►For this topic and other boundary-value problems see Boyd (1973), Hillion (1997), Magnus (1941), Morse and Feshbach (1953a, b), Müller (1988), Ott (1985), Rice (1954), and Shanmugam (1978). ►Lastly, parabolic cylinder functions arise in the description of ultra cold atoms in harmonic trapping potentials; see Busch et al. (1998) and Edwards et al. (1999).

4: 14.30 Spherical and Spheroidal Harmonics

§14.30 Spherical and Spheroidal Harmonics

… ►Most mathematical properties of

Y_{{l},{m}}\left(\theta,\phi\right)

can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. … ►

14.30.3

Y_{{l},{m}}\left(\theta,\phi\right)=\frac{(-1)^{l+m}}{2^{l}l!}\left(\frac{(l-m% )!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\left(\sin\theta\right)^{m}\*\left% (\frac{\mathrm{d}}{\mathrm{d}(\cos\theta)}\right)^{l+m}\left(\sin\theta\right)% ^{2l}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : 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

… ►

14.30.8

\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\,\mathrm{d}% \theta\,\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E8
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

… ►

5: 15.17 Mathematical Applications

… ►

§15.17(iii) Group Representations

►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …

6: 1.10 Functions of a Complex Variable

… ►

Harmonic Functions

…

7: 17.17 Physical Applications

… ►See Kassel (1995). …

8: 34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►

§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

►For the polynomials

P_{l}

see §18.3, and for the function

Y_{{l},{m}}

see §14.30. … ►

34.3.22

\int_{0}^{2\pi}\!\int_{0}^{\pi}Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{% l_{2}},{m_{2}}}\left(\theta,\phi\right)Y_{{l_{3}},{m_{3}}}\left(\theta,\phi% \right)\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi=\left(\frac{(2l_{1}+1)(2l_% {2}+1)(2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §14.30(ii), §34.3(vii)
Permalink:: http://dlmf.nist.gov/34.3.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

…

9: Donald St. P. Richards

… ►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. …

10: 25.16 Mathematical Applications

… ►

25.16.13

\sum_{n=1}^{\infty}\left(\frac{H_{n}}{n}\right)^{2}=\frac{17}{4}\zeta\left(4% \right),

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $H_{\NVar{n}}$ : harmonic number, $n$ : nonnegative integer and $h(n)$ : harmonic number
Keywords:: harmonic number, infinite series, special value
Source:: Flajolet and Salvy (1998, (4-1), p. 24)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.16.E13
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

…

harmonic functions

1—10 of 33 matching pages

1: Tom H. Koornwinder

2: 1.9 Calculus of a Complex Variable

Harmonic Functions

Mean Value Property

Poisson Integral

3: 12.17 Physical Applications

4: 14.30 Spherical and Spheroidal Harmonics

§14.30 Spherical and Spheroidal Harmonics

5: 15.17 Mathematical Applications

§15.17(iii) Group Representations

6: 1.10 Functions of a Complex Variable

Harmonic Functions

7: 17.17 Physical Applications

8: 34.3 Basic Properties: 3 ⁢ j Symbol

§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

9: Donald St. P. Richards

10: 25.16 Mathematical Applications

8: 34.3 Basic Properties: $\mathit{3j}$ Symbol