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an+j analytic function anisotropic ap(L Bn+j boundary Cauchy domain Cauchy's principal value characteristic Eq characteristic equation closed form commutative operators complex plane concludes the proof continued fraction Bl contour integral convergence denominator difference equation differential operator eigenfunction eigenvalue problem elements of S0 Equivalence Transformation exactly two solutions exists expansion theorem explicitly f(jU,jU finite number Fourier coefficient func function space functional analysis given Hence identity Integral Equation Inversion Theorem isotropic scattering Lemma limit linear operator mathematical neutron obtain Op(L operator L0 p(Le pair of elements plane geometry point spectrum pointwise in x,ju polynomial positive previous section projection operator Proof of Statement pure imaginary real axis real or pure resolution right-hand side satisfy scattering function second term sense of Cauchy's sequence spectral theory STATIONARY TRANSPORT OPERATOR Substituting Theorem 3.1 tion transport Eq transport equation vanishes variables verified x.jU yields z-plane zero