3 - Series Solutions: Frobenius Theory

Summary

3.1 Introduction

In Chapter 1 on first- and second-order equations, we have seen that the solutions of linear first-order equations can be obtained in explicit form by converting the problem essentially to an integral calculus problem. We have also seen that there is no procedure in general to obtain the solutions of linear second-order equations with variable coefficients in explicit form. Nevertheless, we could obtain valuable information about the solutions by exploiting the linearity, superposition principle and so on. In this chapter, we consider a class of linear second-order equations whose solutions may be written down in explicit form. Since these solutions will be in the form of an infinite (power) series, eliciting the qualitative behaviour of solutions near some specified point or at infinity will be an important aspect. The results of this chapter are collectively called Frobenius theory. Some important equations, such as Bessel's equation, Hermite equation, Chebyshev equation, Laguerre equation, Airy equation and so on, are included in the class of equations considered here. Owing to the importance of these equations, which appear in applications frequently, the major properties of their solutions have been tabulated in mathematical handbooks. The interested reader may refer to Ref. [1]. We restrict the discussion to the real domain, though it is more advantageous to work in a complex domain. For example, the function (1 + x²)⁻¹ as a function of the real variable x is smooth and has no singularity. However, when x is considered in the complex domain, this function has singularities at the points x = ±i. The discussion in the complex domain also requires tools from complex analysis. For these interesting and important results for the equations in the complex domain, the interested reader is referred to Refs. [7], [15], [29], [54], among others.

3.2 Real Analytic Functions

The class of equations we consider will have analytic coefficients. Roughly speaking, analyticity means convergent power series. We are familiar with power series in the context of Taylor's series and Maclaurin's series in calculus.