By Pope C.
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4. If p(x) or q(x) themselves have worse singularities than poles, the solutions will be even more pathological. 98) where ν is a constant, given, parameter in the equation. As we have already discussed, this equation has a regular singular point at x = 0, and an irregular singular point at x = ∞. We shall perform a series expansion around x = 0. Since this is a regular singular point, we therefore seek solutions of the form an xn+σ . 98), we obtain [(n + σ)2 − ν 2 ] an xn+σ + n≥0 an xn+σ+2 = 0 .
106) where σ = ±ν. In either case, at large n we see that the absolute value of the ratio tends to x2 /n2 , and thus the ratio becomes zero for any fixed x, no matter how large. Thus the radius of convergence is infinite. Notice that this accords perfectly with the fact that the next nearest singular point of the Bessel equation is at x = ∞. Thus we should expect the series to converge for any finite x. We can easily see that with ν taking a generic value (neither integer nor half-integer), our two solutions corresponding to the two roots of the indicial equation σ 2 −ν 2 = 0, namely σ = +ν and σ = −ν, give linearly-independent solutions.
Continuing to higher orders, we thus obtain all the an for n ≥ 2 in terms of a0 and a1 . Since the two initial coefficients a0 and a1 are arbitrary, these parameterise the twodimensional space of solutions of the second-order ODE. 50) where y1 and y2 are the two independent solutions determined by our series expansions. ) Solving for the various higher coefficients an as described above, one finds that the two solutions are given by y1 = 1 − 12 q(a) (x − a)2 + 16 [(q(a)p(a) − q ′ (a)] (x − a)3 + · · · , y2 = (x − a) − 12 p(a) (x − a)2 + 16 [p2 (a) − p′ (a) − q(a)] (x − a)3 + · · · .