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Unformatted text preview: 30 Series Solutions: Preliminaries (A Brief Review of Infinite Series, Power Series and a Little Complex Variables) At this point, you should have no problem in solving any differential equation of the form a d 2 y dx 2 + b dy dx + cy = or ax 2 d 2 y dx 2 + bx dy dx + cy = when a , b and c are all constants. You’ve even solved a few (in chapters 11 and 13) in which a , b and/or c where not constants. Unfortunately, the methods used in those chapters are somewhat limited. More general methods do exist, and, in the next few chapters, we will discuss some of the more important of these in which solutions are described in terms of “power series” and “modified power series”. Ideally, you are already wellenough acquainted with infinite series and power series to jump right into the discussion of the next chapter. As a precaution, though, you may want to skim through this chapter. It is a brief review of infinite series with an emphasis on power series, along with a brief discussion of using complex variables in these series. As much as possible, we’ll limit our discussion to topics that will be needed in the next few chapters, including a few that probably were not emphasized during your first exposure to power series. 30.1 Infinite Series Basic Basics Recall that, in the language of mathematics, an infinite series is a summation with infinitely many terms. For example, ∞ summationdisplay k = 1 1 k = 1 + 1 2 + 1 3 + 1 4 + 1 5 + ··· is the infamous harmonic series . More generally, an infinite series is anything of the form ∞ summationdisplay k = γ α k = α γ + α γ + 1 + α γ + 2 + α γ + 3 + α γ + 4 + ··· 8/14/2011 601 602 Brief Review of Infinite Series and Power Series where γ , the starting index, is some integer (often, it’s 0 or 1 ), and the α k ’s are things that can be added together. These α k ’s may be numbers, functions or even matrices. For the moment, we will assume them to be numbers (as in the harmonic series, above). Given an arbitrary infinite series ∞ summationdisplay k = γ α k = α γ + α γ + 1 + α γ + 2 + α γ + 3 + α γ + 4 + ··· and any integer N ≥ γ , we define the corresponding N th partial sum S N by 1 S N = sum of all terms from a γ to a N = N summationdisplay k = γ α k = α γ + α γ + 1 + α γ + 2 + ··· + α N . Observe that lim N →∞ S N = lim N →∞ N summationdisplay k = γ α k = ∞ summationdisplay k = γ α k = α γ + α γ + 1 + α γ + 2 + α γ + 3 + α γ + 4 + ··· . Naturally, the usefulness of an infinite series usually depends on whether it actually adds up to some finite value; that is, whether lim N →∞ N summationdisplay k = γ α k is some finite value. If the above limit does exist as a finite value, then we say our series converges , and call that value the sum of that series (freely using ∑ ∞ k = γ α k to denote both the series and its sum). On the other hand, if this limit does not exist as a finite value, then we say the seriesits sum)....
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 Summer '09
 Differential Equations, Equations, Power Series, analytic functions

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