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On the surface of a sphere, attempt separation of variables in spherical coordinates by writing F(theta,phi)=Theta(theta)Phi(phi), (1) then the Helmholtz differential ...

In three dimensions, the spherical harmonic differential equation is given by ...

Using the notation of Byerly (1959, pp. 252-253), Laplace's equation can be reduced to (1) where alpha = cint_c^lambda(dlambda)/(sqrt((lambda^2-b^2)(lambda^2-c^2))) (2) = ...

The ordinary differential equation (x^py^')^'+/-x^sigmay^n=0.

In conical coordinates, Laplace's equation can be written ...

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 ...

In conical coordinates, Laplace's equation can be written ...

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 ...

Poisson's equation is del ^2phi=4pirho, (1) where phi is often called a potential function and rho a density function, so the differential operator in this case is L^~=del ...

Apéry's numbers are defined by A_n = sum_(k=0)^(n)(n; k)^2(n+k; k)^2 (1) = sum_(k=0)^(n)([(n+k)!]^2)/((k!)^4[(n-k)!]^2) (2) = _4F_3(-n,-n,n+1,n+1;1,1,1;1), (3) where (n; k) ...