Dr. Katz DEq Homework Solutions 53

Dr. Katz DEq Homework Solutions 53 - 1 ` •...

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Algebraic Solutions of Differential Equations 53 that, unless otherwise specified, all indexing variables run over the set {1, . ..,hi. We must verify the congruence (3.5.0.3) in the case (3.5.4.0) ~ =-c (1)/x ~ (2) A. .. /x ~ (b) = I] ~ (v). v Using the substitutions (3.5.1) and the identity (3.5.2.3)just as in the case b = 2, we see that the congruence in question is equivalent to the con- gruence Lie (P)e- 1(2 (- 1)'+~f(i) I-I tr(v)) i v*i (3.5.4.1) ---~ (- 1) '+l PP-l(f(i)) H t7(v) modulo dF(V, ~?~/s(log O~)). i v*i Expanding the left member of (3.5.4.1) be, Leibniz's rule, and re- membering (3.5.3.7), (3.5.4.1) becomes p--1
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Unformatted text preview: (- 1) `+' • (- 1)kPP-~-k(f(i)) (3.5.4.2) ' k=l Lie (p)k( [I a (v)) ~ dF(V, f~/s(log Dr) ) . v*i Our next task is to expand each of the terms (3.5.4.3) Lie (p)k ( 1-I a (v)) v~:i using Leibniz's rule. To facilitate this, we introduce the notations (3.5.4.4) t~ = (t'l . .... (b), a b-tuple of non-negative integers (3.5.4.5) Ill =~ (, (3.5.4.6) ([~1) = (IfD' (~)' i (3.5.4.7) S(f) = {ilf~.O} (3.5.4.8) for any nonempty subset A c{1 . ... ,b}-{i} we denote by sgn (A, i) the sign of the permutation A, {1 . .... b}- {i}-A~ {l, . .., b}- {i} f (We make the convention that subsets of Z are to be enumerated in increasing order.)...
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