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Unformatted text preview: 32 Power Series Solutions II: Generalizations and Theory A major goal in this chapter is to confirm the claims made in theorems 31.1 and 31.3 regarding the validity of the algebraic method. Along the way, we will also expand both the set of differential equations for which this method can be considered and our definitions of “regular” and “singular” points. As a bonus, we’ll also obtain formulas that, at least in some cases, can simplify the computation of the terms of the power series solutions. 32.1 Equations with Analytic Coefficients In the previous chapter, we discussed an algebraic method for finding a general power series solution about a point x to any differential equation of the form A ( x ) y ′ + B ( x ) y = or A ( x ) y ′′ + B ( x ) y ′ + C ( x ) y = where A ( x ) , B ( x ) and C ( x ) are polynomials with A ( x ) negationslash= 0 . Note that these polynomials can be written as A ( x ) = N summationdisplay k = a k ( x − x ) k with a negationslash= , B ( x ) = N summationdisplay k = b k ( x − x ) k and C ( x ) = N summationdisplay k = c k ( x − x ) k where N is the highest power appearing in these polynomials. Now, I know just what you are wondering: Must N be finite? Or will our algebraic method still work if N = ∞ ? That is, can we use our algebraic method to find power series solutions about x to A ( x ) y ′ + B ( x ) y = and A ( x ) y ′′ + B ( x ) y ′ + C ( x ) y = when A ( x ) , B ( x ) and C ( x ) are functions expressible as power series about x (i.e., when A , B and C are functions analytic at x ), with A ( x ) negationslash= 0 . And the answer to this question is yes , at least in theory. Simply replace the coefficients in the differential equations with their power series about x , and follow the steps already outlined in 8/14/2011 665 666 Power Series Solutions II: Generalizations and Theory sections 31.2 and 31.3 (possibly using the formula from theorem 30.7 on page 612 for multiplying infinite series). There are, of course, some further questions you are bound to be asking regarding these power series solutions and the finding of them. In particular: 1. What will be the radii of convergence for the resulting power series solutions? and 2. Are there any shortcuts to what could clearly be a rather lengthy and tedious set of calculations. For the answers, read on. 32.2 Ordinary and Singular Points, the Radius of Analyticity, and the Reduced Form Introducing Complex Variables To properly address at least one of our questions, and to simplify the statements of our theorems, it will help to start viewing the coefficients of our differential equations as functions of a complex variable z . We actually did this in the last chapter when we referred to a point z s in the complex plane for which A ( z s ) = 0 . But A was a polynomial then, and viewing polynomials as functions of a complex variable is so easy that we hardly noted doing so. Viewing other functions (such as exponentials, logarithms and trigonometric functions) as functions of a complex variable may...
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 Summer '09
 Differential Equations, Algebra, Equations, Derivative, Power Series, Taylor Series, Analytic function

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