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Unformatted text preview: 1 Mathematical Physics 371 NAME: Homework 11: Weber & Arfken Chapter 11: Legendre and Spherical Harmonics Due as announced on the course web page in the tentative schedule. Homework solutions will be posted sometime after the due date in the tentative schedule. The solutions are intended to be (but not necessarily are) super-perfect and often go beyond a fully correct answer. 005 qfull 02571 2 3 0 easy math: intro infinite series Extra keywords: WA-267-5.2.3 and other material 1. Infinite series are summations of infinitely many real numbers. Of course, what we really mean by an infinite series is that it is the limit of a sequence of partial sums. Say we have sum S N = N summationdisplay n =1 a n . The infinite series is S = lim n →∞ S n which we conventionally write S = ∞ summationdisplay n =1 a n . Note that the summation can also start from a zeroth term and does so in many cases. If a sequence of sums approaches a finite limit (i.e., a finite single value including zero), the series is convergent. The classic convergent series is the geometric series for | x | < 1: i.e., S = ∞ summationdisplay n =0 x n (WA-258). The geometric series is a special case of power series which are defined by S = ∞ summationdisplay n =0 a n x n (WA-291). If a sequence of sums approaches an infinite limit, the series is divergent. The geometric series with | x | > 1 and x = 1 is divergent (WA-258). The classic divergent series is the harmonic series: S = ∞ summationdisplay n =1 1 n = ∞ (WA-259). However, it diverges very slowly: S N =1 , 000 , 000 = 1 , 000 , 000 summationdisplay n =1 1 n = 14 . 392726 . . . (WA-266). If a divergent series turns up in a physical analysis for a quantity that is actually finite (which is always/almost always the case), then the divergent series is not the correct result. There also can be oscillatory series that neither converge or diverge: e.g., the geometric series for x = − 1: S = 1 − 1 + 1 − 1 + 1 − . . . (WA-261). Oscillatory series have some mathematical interest, but have never had much application in the empirical sciences (WA-261): if they turn up, one probably has the wrong result. There are what also what are called asymptotic or semi-convergent series which are very useful (WA-314). 2 If the absolute values of the terms of series converge, then the series is said to be absolutely convergent (WA-271). The terms of an absolutely convergent series can be summed in any order with the same result. A conditionally convergent series is one that is not absolutely convergent, but converges because there is a cancelation between positive and negative terms (WA-271). Unlike absolutely convergent series, conditionally convergent do not converge to a unique value independent of the order of summation. It can be shown that a conditionally convergent series will converge to any value desired or even diverge depending on the order of summation (WA-272). In this problem (which we are slowly converging to), we will not consider conditional covergence problems.we are slowly converging to), we will not consider conditional covergence problems....
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.
- Spring '10