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1: 1.5 Calculus of Two or More Variables
§1.5(ii) Coordinate Systems
Polar Coordinates
Cylindrical Coordinates
Spherical Coordinates
For applications and other coordinate systems see §§12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14. …
2: 10.73 Physical Applications
Bessel functions of the first kind, J n ( x ) , arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation 2 V = 0 , or by the Helmholtz equation ( 2 + k 2 ) ψ = 0 . … In cylindrical coordinates r , ϕ , z , (§1.5(ii) we have … Bessel functions enter in the study of the scattering of light and other electromagnetic radiation, not only from cylindrical surfaces but also in the statistical analysis involved in scattering from rough surfaces. … On separation of variables into cylindrical coordinates, the Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) and K n ( x ) , all appear. … The functions j n ( x ) , y n ( x ) , h n ( 1 ) ( x ) , and h n ( 2 ) ( x ) arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates ρ , θ , ϕ 1.5(ii)): …
3: Bibliography H
  • P. I. Hadži (1972) Certain sums that contain cylindrical functions. Bul. Akad. Štiince RSS Moldoven. 1972 (3), pp. 75–77, 94 (Russian).
  • P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
  • M. H. Halley, D. Delande, and K. T. Taylor (1993) The combination of R -matrix and complex coordinate methods: Application to the diamagnetic Rydberg spectra of Ba and Sr. J. Phys. B 26 (12), pp. 1775–1790.
  • 4: Bibliography S
  • L. Shen (1981) The elliptical microstrip antenna with circular polarization. IEEE Trans. Antennas and Propagation 29 (1), pp. 90–94.
  • K. M. Siegel and F. B. Sleator (1954) Inequalities involving cylindrical functions of nearly equal argument and order. Proc. Amer. Math. Soc. 5 (3), pp. 337–344.
  • R. Spigler (1980) Some results on the zeros of cylindrical functions and of their derivatives. Rend. Sem. Mat. Univ. Politec. Torino 38 (1), pp. 67–85 (Italian. English summary).
  • 5: 36.13 Kelvin’s Ship-Wave Pattern
    In a reference frame where the ship is at rest we use polar coordinates r and ϕ with ϕ = 0 in the direction of the velocity of the water relative to the ship. …
    6: 29.18 Mathematical Applications
    §29.18(i) Sphero-Conal Coordinates
    when transformed to sphero-conal coordinates r , β , γ : …
    29.18.4 u ( r , β , γ ) = u 1 ( r ) u 2 ( β ) u 3 ( γ ) ,
    §29.18(ii) Ellipsoidal Coordinates
    The wave equation (29.18.1), when transformed to ellipsoidal coordinates α , β , γ : …
    7: 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 ξ , η , ζ , 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. … …
    8: 22.18 Mathematical Applications
    In polar coordinates, x = r cos ϕ , y = r sin ϕ , the lemniscate is given by r 2 = cos ( 2 ϕ ) , 0 ϕ 2 π . …
    9: 14.31 Other Applications
    §14.31(i) Toroidal Functions
    §14.31(ii) Conical Functions
    The conical functions P - 1 2 + i τ m ( x ) appear in boundary-value problems for the Laplace equation in toroidal coordinates14.19(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)). …
    10: 28.27 Addition Theorems
    Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …