Differential Equations, 3rd Edition

6Power Series Solutions and Special Functions

DOI: 10.1201/9781003214526-6

• Power series basics

• Convergence of power series

• Series solutions of first-order equations

• Series solutions of second-order equations

• Ordinary points

• Regular singular points

• Frobenius method

• The hypergeometric equation

6.1 Introduction and Review of Power Series

It is useful to classify the functions that we know, or will soon know, in an informal way. The polynomials are functions of the form

$$\begin{array}{l}{a}_0+a_{1}x+a_2{x}^2+\cdots +a_{k-1}{x}^{k-1}+a_k{x}^k,\hfill \end{array}$$

where $a_0,a_1,\dots ,a_k$ are constants. This is a polynomial of degree k. A rational function is a quotient of polynomials. For example,

$$\begin{array}{l}r(x)=\frac{3x^3-x+4}{x^2+5x+1}\hfill \end{array}$$

is a rational function.

A transcendental function is one that is not a polynomial or a rational function. ...