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As shown by Morse and Feshbach (1953) and Arfken (1970), the Helmholtz differential equation is separable in oblate spheroidal coordinates.

The scale factors are h_u=h_v=sqrt(u^2+v^2), h_theta=uv and the separation functions are f_1(u)=u, f_2(v)=v, f_3(theta)=1, given a Stäckel determinant of S=u^2+v^2. The ...

As shown by Morse and Feshbach (1953) and Arfken (1970), the Helmholtz differential equation is separable in prolate spheroidal coordinates.

A differential ideal is an ideal I in the ring of smooth forms on a manifold M. That is, it is closed under addition, scalar multiplication, and wedge product with an ...

(1) or (2) The solutions are Jacobi polynomials P_n^((alpha,beta))(x) or, in terms of hypergeometric functions, as y(x)=C_1_2F_1(-n,n+1+alpha+beta,1+alpha,1/2(x-1)) ...

In bispherical coordinates, Laplace's equation becomes (1) Attempt separation of variables by plugging in the trial solution f(u,v,phi)=sqrt(coshv-cosu)U(u)V(v)Psi(psi), (2) ...

In toroidal coordinates, Laplace's equation becomes (1) Attempt separation of variables by plugging in the trial solution f(u,v,phi)=sqrt(coshu-cosv)U(u)V(v)Psi(psi), (2) ...

A determinant used to determine in which coordinate systems the Helmholtz differential equation is separable (Morse and Feshbach 1953). A determinant S=|Phi_(mn)|=|Phi_(11) ...

The twistor equation states that del _(A^')^((A)phi^(B...E))=0, where the parentheses denote symmetrization, in a Lorentz transformation, primed spinors transform under the ...

Spinor fields describing particles of zero rest mass satisfy the so-called zero rest mass equations. Examples of zero rest mass particles include the neutrino (a fermion) and ...