The Frobenius method for complex roots of the indicial equation

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In the real domain with a(x)and b(x)analytic at x= 0 and the roots of the indicial equation complex conjugate, it is demonstrated that the real and imaginary parts of any complex solution of the differential equation x2y” + xa(x)y’ + b(x)y =0obtained, for example, by the Frobenius method using any one root, constitute a real basis for the solution space. With the above result in mind, and using the Frobenius method, explicit series representations for the two real linearly independent solutions are obtained for the particular class of equations x2y+x(a0+a1xn)y'+(b0 + b1xn)y=0, where nis a non‐negative integer and a0, a1, b0 and b1 are real constants, with the additional proviso that a0 and b0 are chosen to guarantee complex roots. The structure of the solutions and the question of convergence are examined, and the general results are applied to two concrete examples.