8 Series Solutions and Special Functions

Linear equations with variable coefficients rarely have solutions in elementary functions. The equations that arise from separating variables in the partial differential equations of physics, such as Laplace’s equation, the heat equation and the wave equation in polar, cylindrical and spherical coordinates, are of this kind, and their solutions, the Bessel functions and Legendre polynomials among them, are the special functions of mathematical physics. This section constructs them as power series, first near ordinary points, where the coefficients are analytic, then near regular singular points, where Frobenius’s method allows a controlled singularity.

We consider \(y'' + p(t)y' + q(t)y = 0\). A point \(t_0\) is ordinary if \(p\) and \(q\) are analytic there (given by convergent power series in \(t - t_0\)), and singular otherwise.

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