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paraboloid of revolution

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1: 12.17 Physical Applications
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. … …
2: 13.28 Physical Applications
§13.28(i) Exact Solutions of the Wave Equation
The reduced wave equation 2 w = k 2 w in paraboloidal coordinates, x = 2 ξ η cos ϕ , y = 2 ξ η sin ϕ , z = ξ η , can be solved via separation of variables w = f 1 ( ξ ) f 2 ( η ) e i p ϕ , where …
3: Sidebar 5.SB1: Gamma & Digamma Phase Plots
In the upper half of the image, the poles of Γ ( z ) are clearly visible at negative integer values of z : the phase changes by 2 π around each pole, showing a full revolution of the color wheel. …
4: 28.31 Equations of Whittaker–Hill and Ince
§28.31(iii) Paraboloidal Wave Functions
With (28.31.10) and (28.31.11), …are called paraboloidal wave functions. … More important are the double orthogonality relations for p 1 p 2 or m 1 m 2 or both, given by …
Asymptotic Behavior
5: Bibliography U
  • K. M. Urwin (1964) Integral equations for paraboloidal wave functions. I. Quart. J. Math. Oxford Ser. (2) 15, pp. 309–315.
  • K. M. Urwin (1965) Integral equations for the paraboloidal wave functions. II. Quart. J. Math. Oxford Ser. (2) 16, pp. 257–262.
  • 6: 28.32 Mathematical Applications
    §28.32(ii) Paraboloidal Coordinates
    The general paraboloidal coordinate system is linked with Cartesian coordinates via …
    7: 14.31 Other Applications
    The conical functions 𝖯 1 2 + i τ m ( x ) 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)). …
    8: 30.14 Wave Equation in Oblate Spheroidal Coordinates
    The coordinate surfaces ξ = const . are oblate ellipsoids of revolution with focal circle z = 0 , x 2 + y 2 = c 2 . The coordinate surfaces η = const . are halves of one-sheeted hyperboloids of revolution with the same focal circle. …
    9: 30.13 Wave Equation in Prolate Spheroidal Coordinates
    The coordinate surfaces ξ = const . are prolate ellipsoids of revolution with foci at x = y = 0 , z = ± c . The coordinate surfaces η = const . are sheets of two-sheeted hyperboloids of revolution with the same foci. …
    10: 1.6 Vectors and Vector-Valued Functions
    For a surface of revolution, y = f ( x ) , x [ a , b ] , about the x -axis, …