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analytic arbitrary axis basis Bessel functions Bessel's equation boundary conditions boundary value problem commute complex number consider constant continuous function contour converge corresponding critical point curves cylinder defined definition denoted dimensional space eigen functions eigen values element Equating the coefficient Example finite Fourier transform given Green's function Helmholtz equation Hermitian operator infinite series interval inverse irreducible representations Jv(x known Laplace's equation Legendre equation Legendre polynomials linear operator linear vector space linearly independent linearly independent solutions obtained orthonormal physical Pi(x potential properties prove quantum mechanics radius real numbers recurrence relations regular singular point right hand side rotation satisfies the boundary scalar product Similarly sin2 spherical polar coordinates Sturm-Liouville subgroup Substituting surface tensor of rank theorem unitary matrix variable verified written yi(x z-plane zero