harmonic functions

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1—10 of 42 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: 37.11 Spherical Harmonics

… ►

Zonal Functions

… ►A spherical harmonic

Y\in\mathcal{H}_{n}^{0,d}

is zonal iff … ► … ►See §37.12. … ►See Braaksma and Meulenbeld (1968). …

9: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►

37.19.4

w_{\kappa}(\mathbf{x})=\prod_{\mathbf{v}\in R_{+}}{\left|\left\langle\mathbf{x% },\mathbf{v}\right\rangle\right|}^{2\kappa_{\mathbf{v}}}.

ⓘ

Symbols:: $\in$ : element of, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $\mathbf{x}$ : vector in $d$ -dimensional Euclidean space
Sources:: Dunkl (1989, p. 168); Dunkl and Xu (2014, Definition 7.1.1)
Referenced by:: §37.19(ii)
Permalink:: http://dlmf.nist.gov/37.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §37.19(i), §37.19, Part 37.III and Ch.37

… ►In the simplest case when the group is

\mathbb{Z}_{2}^{d}

, the weight function is given by …

10: 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

…

harmonic functions

1—10 of 42 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: 37.11 Spherical Harmonics

Zonal Functions

9: 37.19 Other Orthogonal Polynomials of d Variables

10: 34.3 Basic Properties: 3 ⁢ j Symbol

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

9: 37.19 Other Orthogonal Polynomials of $d$ Variables

10: 34.3 Basic Properties: $\mathit{3j}$ Symbol