# sectorial harmonics

(0.001 seconds)

## 11—20 of 35 matching pages

##### 11: Errata
• Section 14.30

In regard to the definition of the spherical harmonics $Y_{{l},{m}}$, the domain of the integer $m$ originally written as $0\leq m\leq l$ has been replaced with the more general $|m|\leq l$. Because of this change, in the sentence just below (14.30.2), “tesseral for $m and sectorial for $m=l$” has been replaced with “tesseral for $|m| and sectorial for $|m|=l$”. Furthermore, in (14.30.4), $m$ has been replaced with $|m|$.

Reported by Ching-Li Chai on 2019-10-05

• ##### 12: 1.7 Inequalities
###### §1.7(iii) Means
1.7.7 $H\leq G\leq A,$
##### 13: 1.17 Integral and Series Representations of the Dirac Delta
###### Spherical Harmonics (§14.30)
1.17.25 $\delta\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta\left(\phi_{1}-\phi_{2}% \right)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}Y_{{\ell},{m}}\left(\theta_% {1},\phi_{1}\right)\overline{Y_{{\ell},{m}}\left(\theta_{2},\phi_{2}\right)}.$
##### 14: 18.39 Physical Applications
For a harmonic oscillator, the potential energy is given by …
##### 15: 1.9 Calculus of a Complex Variable
###### Mean Value Property
For $u(z)$ harmonic, …
###### Poisson Integral
is harmonic in $|z|. …
##### 16: Bibliography T
• A. Terras (1988) Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.
• O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.
• T. Ton-That, K. I. Gross, D. St. P. Richards, and P. J. Sally (Eds.) (1995) Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.
• ##### 17: 1.10 Functions of a Complex Variable
###### Harmonic Functions
If $u(z)$ is harmonic in $D$, $z_{0}\in D$, and $u(z)\leq u(z_{0})$ for all $z\in D$, then $u(z)$ is constant in $D$. Moreover, if $D$ is bounded and $u(z)$ is continuous on $\overline{D}$ and harmonic in $D$, then $u(z)$ is maximum at some point on $\partial D$. …
##### 18: Bibliography G
• W. Gautschi (1974) A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.
• A. Gil and J. Segura (1998) A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.
• A. Gil and J. Segura (2000) Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.
• A. Gil and J. Segura (2001) DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Comm. 139 (2), pp. 186–191.
• ##### 19: Bibliography H
• E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.
• L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.
• ##### 20: Bibliography W
• G. Weiss (1965) Harmonic Analysis. In Studies in Real and Complex Analysis, I. I. Hirschman (Ed.), Studies in Mathematics, Vol. 3, pp. 124–178.
• E. T. Whittaker (1902) On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.