Someone argues as follows about Bessel's equation of order , : "The indicial equation is
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Special functions and orthogonal polynomials
Conditions of Validity, Ch Series solutions, singular points and where the special functions come from
Problem
Someone argues as follows about Bessel's equation of order 3, x2y′′+xy′+(x2−9)y=0: "The indicial equation is s2−9=0, with roots s=3 and s=−3. These differ by the integer 6, but the general theory of the Frobenius method only rules out a plain series for the smaller root when the coefficients actually blow up in the recursion; nothing says they must. So I build the series for s=−3 term by term exactly as for s=3, call the result J−3(x), and since it started from a different exponent it must be linearly independent of J3(x). Together J3 and J−3 form a complete pair of solutions." Find the error in this argument.