Legendre polynomials equation, spherical coordinates

For k 0. Eq. (5) reduces to the Laplace equation, which is used to calculate the potential inside the particle that is modeled as dielectric sphere with central multipoles. For this case the solution in spherical coordinates is given by a series of Legendre polynomials Pi (x) and related powers of r,... 

Axisymmetric irrotational (i.e., potential) flow of an incompressible ideal fluid past a stationary gas bubble exhibits no vorticity. Hence, V x v = 0. This problem can be solved using the stream fnnction approach rather than the scalar velocity potential method. Develop the appropriate equation that governs the solution to the stream function f for two-dimensional axisymmetric potential flow in spherical coordinates. Which Legendre polynomial describes the angular dependence of the stream function ... 

The first of these equations defines the spherical Bessel functions of radial coordinate r, the second leads to the Legendre polynomials for scattering angle Q, while the last defines sinus functions of the azimuthal angle Note that for rotational symmetry = 1) the constant m vanishes (m = 0). 

Spherical harmonics Y 9,(l>) are the angular contribution to the solution of Laplace s equation (or Helmholtz differential equation) in spherical coordinates (i.e. Eqs. (C.9) and (C.IO)). They are hence the product of the associated Legendre polynomial of cos0 and the general sine of the azimuth (/> ... 

Specifically these are the associated Legendre polynomials of the first kind and are usually written as fimctions of cosO rather than 6. They are named after Adrien-Marie Legendre (1752-1833), who discovered them as a general family of solutions to differential equations in spherical coordinates while he was working on a mathematical description of the motions of stars. His colleague, Simon-Pierre Laplace (1749-1827), then drew on the Legendre polynomials to formulate the three-dimensional spherical harmonics. 

A wavefunction Yi m for a specific state of orbital angular momentum, i.e., orthogonal functions of the angular coordinates which satisfy the differential equation = —1(1 + 1)K, where is the Legendre operator. The functions are polynomials in sin 6 and cos. Spherical harmonics are the angular factors in centrosymmetric atomic orbitals. 

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