Download Mathematical methods for physicists : a comprehensive guide by George B Arfken; Hans-Jurgen Weber; Frank E Harris PDF

By George B Arfken; Hans-Jurgen Weber; Frank E Harris

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Show that ξ2 agrees with ξ1 through (A−1 )2 . Find the difference in the coefficients of the (A−1 )3 term. 18 1 1 dt arctan t , t (a) (b) ln x − 0 dx . 1 + x2 0 Note. 12. 4 MATHEMATICAL INDUCTION We are occasionally faced with the need to establish a relation which is valid for a set of integer values, in situations where it may not initially be obvious how to proceed. However, it may be possible to show that if the relation is valid for an arbitrary value of some index n, then it is also valid if n is replaced by n + 1.

2 (a) 5 ζ (3) = − 4 ∞ n=2 1 . n 3 (n 2 − 1) SERIES OF FUNCTIONS We extend our concept of infinite series to include the possibility that each term u n may be a function of some variable, u n = u n (x). 32) as does the series sum, defined as the limit of the partial sums: ∞ u n (x) = S(x) = lim sn (x). 33) So far we have concerned ourselves with the behavior of the partial sums as a function of n. Now we consider how the foregoing quantities depend on x. The key concept here is that of uniform convergence.

1)!! = 1. 77) 36 Chapter 1 Mathematical Preliminaries where c is the velocity of light. Using Eq. 69) with m = −1/2 and x = −v 2 /c2 , and evaluating the binomial coefficients using Eq. 74), we have E = mc v2 1 − 2 1− 2 c 2 3 v2 + − 2 8 c 1 3 v2 = mc2 + mv 2 + mv 2 2 2 8 c + 2 3 5 v2 − − 2 16 c 5 v2 mv 2 − 2 16 c + ··· 2 + ··· . 78) The first term, mc2 , is identified as the rest-mass energy. Then 1 v2 3 v2 5 E kinetic = mv 2 1 + − + 2 4 c2 8 c2 2 + ··· . 79) For particle velocity v c, the expression in the brackets reduces to unity and we see that the kinetic portion of the total relativistic energy agrees with the classical result.

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