projective coordinates

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11: 12.17 Physical Applications

§12.17 Physical Applications

… ►in Cartesian coordinates $x,y,z$ of three-dimensional space (§1.5(ii)). By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta$, defined by … ►In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. … …

12: Ingram Olkin

… ►Olkin was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. During this time he served as Associate Editor with responsibilities for all aspects of the project. The original committee was reconstituted in 2015; for details see About the Project.

13: Nico M. Temme

… ►He is also one of the Associate Editors for Special Functions in the DLMF project and a Senior Associate Editor of the DLMF. … ►Temme was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. During this time he served as Associate Editor with responsibilities for all aspects of the project. The original committee was reconstituted in 2015; for details see About the Project. …

14: Gloria Wiersma

… ►She was one of the LaTeX editors for the DLMF project.

15: Funding

… ►The NIST DLMF project has been funded, in part, by the Knowledge & Distributed Intelligence Program of the National Science Foundation. …

16: 14.31 Other Applications

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§14.31(i) Toroidal Functions

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§14.31(ii) Conical Functions

►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). … ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

17: 34.10 Zeros

… ►In a $\mathit{3}j$ symbol, if the three angular momenta $j_1,j_2,j_3$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3}j$ symbol is zero. …However, the $\mathit{3}j$ and $\mathit{6}j$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …

18: Preface

… ► [This preface was written for the original release of the DLMF in 2010. For an up-to-date account of the current status of the project, see About the Project.] … ►A summary of the responsibilities of these groups may help in understanding the structure and results of this project. … ►The editors acknowledge the many other individuals who contributed to the project in a variety of ways. …Undoubtedly, the editors have overlooked some individuals who contributed, as is inevitable in a large long-lasting project. … ►The project was funded in part by NSF Award 9980036, administered by the NSF’s Knowledge and Distributed Intelligence Program. …

19: 28.27 Addition Theorems

… ►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …

20: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(i) Oblate Spheroidal Coordinates

►Oblate spheroidal coordinates $\xi ,\eta ,\varphi$ are related to Cartesian coordinates $x,y,z$ by … ►

§30.14(ii) Metric Coefficients

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§30.14(iii) Laplacian

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