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In parabolic cylindrical coordinates, the scale factors are h_u=h_v=sqrt(u^2+v^2), h_z=1 and the separation functions are f_1(u)=f_2(v)=f_3(z)=1, giving Stäckel determinant ...

In two-dimensional polar coordinates, the Helmholtz differential equation is 1/rpartial/(partialr)(r(partialF)/(partialr))+1/(r^2)(partial^2F)/(partialtheta^2)+k^2F=0. (1) ...

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

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 ...

There are two important theorems known as Herbrand's theorem. The first arises in ring theory. Let an ideal class be in A if it contains an ideal whose lth power is ...

The second-order ordinary differential equation (d^2y)/(dx^2)-2x(dy)/(dx)+lambday=0. (1) This differential equation has an irregular singularity at infty. It can be solved ...

A Hermitian metric on a complex vector bundle assigns a Hermitian inner product to every fiber bundle. The basic example is the trivial bundle pi:U×C^k->U, where U is an open ...

Lambda_0(phi|m)=(F(phi|1-m))/(K(1-m))+2/piK(m)Z(phi|1-m), where phi is the Jacobi amplitude, m is the parameter, Z is the Jacobi zeta function, and F(phi|m^') and K(m) are ...

The Hh-function is a function closely related to the normal distribution function. It can be defined using the auxilary functions Z(x) = 1/(sqrt(2pi))e^(-x^2/2) (1) Q(x) = ...

Let H be a Hilbert space and (e_i)_(i in I) is an orthonormal basis for H. The set S(H) of all operators T for which sum_(i in I)||Te_i||^2<infty is a self-adjoint ideal of ...