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1: 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 …
2: 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
3: 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.
  • 4: 28.32 Mathematical Applications
    §28.32(ii) Paraboloidal Coordinates
    The general paraboloidal coordinate system is linked with Cartesian coordinates via …
    5: 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. … …
    6: Bibliography
  • F. M. Arscott (1967) The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.