Bases of Special Functions and Their Domains of ConvergenceA systematic approach to expansions of analytic functions in series of special functions is presented. Many expansions of this kind are identified with eigenfunction expansions for differential operators in the complex domain. Central ponits of our theory are the construction of biorthogonal canonical systems of eigen - and associated functions and the determination of the domains of convergence of the corresponding eigenfunction expansions. |
Contents
Introduction | 9 |
Foundations of the theory | 15 |
First order differential systems with a regular singular point | 43 |
Copyright | |
7 other sections not shown
Common terms and phrases
a₁ A₁(z ao(z associated functions associated vectors assume b₁ Bessel bilinear form biorthogonal canonical systems circle compact subsets completes the proof converges absolutely uniformly corresponding CSRF defined according differential equation differential operator differential system Dy(z domain Ŝ domains of convergence easily verifies eigenfunctions and associated eigenvectors and associated expansions in series Floquet eigenvalue problem form a fundamental formal adjoint Fréchet spaces functions of mathematical fundamental system Furthermore generalised Fourier coefficients H²(Slog holomorphic function holomorphic with respect immediate consequence isolated point Lemma m-fold products mapped biholomorphically mathematical physics matrix matrix-valued function obtain open subset order differential equation radius regular singular point Remark ring-shaped region root function second order differential series converges absolutely special functions terms of eigenfunctions Ulog uniformly on compact Wa(z Whittaker functions Wronskian determinant y₁ αι απ