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axisymmetric Bessel function boundary value problems coefficients components consider constants continuously differentiable cosx cylinder defined by equation denotes derived determination discussed dual integral equations dual series equations dual series relations easily shown electrified disk equa equations 00 equivalent Erdelyi-Kober operators expression into equation follows from equation formula fractional integral Fredholm equation Fredholm integral equation free term function A(u given by equation half-space Hankel transform harmonic function Hence interchange the order inversion theorem Jacobi polynomial kernel Laplace's equation means of equation method mixed boundary conditions mixed boundary value obtain the relation operator pair of dual pair of equations plane potential function potential theory radius reduces result right hand side satisfies the mixed satisfy the dual second kind side of equation Similarly Sneddon solve spherical cap Srivastav substitute from equation Substituting this expression tend to zero tion Tranter triple integral equations unknown function written xel2 xudu