Part5_TOC - Contents V Power Series and Modied Power Series...

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Contents V Power Series and Modified Power Series Solutions 599 30 Series Solutions: Preliminaries 601 30.1 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 Basic Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 The Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Absolute Convergence and Convergence Tests . . . . . . . . . . . . . . . . 605 30.2 Power Series and Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . 607 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 Convergence and the Radius of Convergence . . . . . . . . . . . . . . . . . 608 Algebra with Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 Calculus with Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . 613 Partial Sum Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 617 30.3 Elementary Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 619 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 Power Series and Analytic Functions . . . . . . . . . . . . . . . . . . . . . 620 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 31 Power Series Solutions I: Basic Computational Methods 625 31.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 General Power Series Solutions . . . . . . . . . . . . . . . . . . . . . . . . 625 The Two Methods, Briefly . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 31.2 The Algebraic Method with First-Order Equations . . . . . . . . . . . . . . . . 627 Details of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 Practical Advice on Using the Method . . . . . . . . . . . . . . . . . . . . . 632 Initial-Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 On the Validity of the Power Series Solution . . . . . . . . . . . . . . . . . 637 31.3 The Algebraic Method with Second-Order Equations . . . . . . . . . . . . . . 640 Details of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 Practical Advice on Using the Method . . . . . . . . . . . . . . . . . . . . . 645 Initial-Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 On the Validity of the Power Series Solutions . . . . . . . . . . . . . . . . . 646 31.4 The Taylor Series Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 The Basic Idea (Expanded) . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 The Steps in the Taylor Series Method . . . . . . . . . . . . . . . . . . . . . 649 8/14/2011 iii
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iv Validity of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 31.5 Appendix: Using Induction .
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Part5_TOC - Contents V Power Series and Modied Power Series...

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