# harmonic functions

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## 1—10 of 32 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
###### HarmonicFunctions
If $f(z)=u(x,y)+iv(x,y)$ is analytic in an open domain $D$, then $u$ and $v$ are harmonic in $D$, that is, …
###### Mean Value Property
For $u(z)$ harmonic, …
##### 3: 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 $Y_{{l},{m}}\left(\theta,\phi\right)=\frac{(-1)^{l+m}}{2^{l}l!}\left(\frac{(l-m% )!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\left(\sin\theta\right)^{m}\*\left% (\frac{\mathrm{d}}{\mathrm{d}(\cos\theta)}\right)^{l+m}\left(\sin\theta\right)% ^{2l}.$
14.30.8 $\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\mathrm{d}% \theta\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.$
##### 4: 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).
##### 5: 15.17 Mathematical Applications
###### §15.17(iii) Group Representations
For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function15.9(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. …
##### 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 $\int_{0}^{2\pi}\!\int_{0}^{\pi}Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{% l_{2}},{m_{2}}}\left(\theta,\phi\right)Y_{{l_{3}},{m_{3}}}\left(\theta,\phi% \right)\sin\theta\mathrm{d}\theta\mathrm{d}\phi=\left(\frac{(2l_{1}+1)(2l_{2}+% 1)(2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.$
##### 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: 1.17 Integral and Series Representations of the Dirac Delta
1.17.25 $\delta\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta\left(\phi_{1}-\phi_{2}% \right)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}Y_{{\ell},{m}}\left(\theta_% {1},\phi_{1}\right)\overline{Y_{{\ell},{m}}\left(\theta_{2},\phi_{2}\right)}.$