Hills Equation Ii Transformations Approximation Examples
Hills Equation Ii Transformations Approximation Examples
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-'/") Qf(\i~''' ) -terms may be differentiated with respect to |i, 5/2 where the having a derivative of the order of (i Again ^ the left-hand sides in (6. 2k) to (6. 31) are functions of |i of order of growth —, and will admit product representations of the type (6. 21), e. G. (for X^ 4 0^^ 4 0^ -^n ^ °^ (6. 35) y'(i:/2, -n) = c(l+nA)fr (l+^^^„) ^ n=l (6. 3i^) y^U/2, -^) = c IT (1+fi/c^) n=l where c, c are constants. Now we can calculate an asymptotic expansion for y^(rt/2, -n) A 1°S in two dif...ferent ways, using (6. 31) and (6. 32) or using (6. 33) and (6. 5^) By equating these two expansions, we find 00 00 n=l I n + l^n 0-7 n n {[1 + 7 ) (m- + cr ) o ' — r" 'n^ n n=l h9 or 03 Q n=l n=l 03 (7 -a J n=i -P . ^(. -5/2) which proves (6. I7). Equation (6. 16) can be proved in the same manner.
7. Coexistence 7. 1. Introduction .
According to Floquet's Theorem (Section 1. 2); Hill's equation will; in general, have only one periodic solution (and its constant multiples) of period n or 2n.
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To quickly assess the difficulty of the text, read a short excerpt:
-'/") Qf(\i~''' ) -terms may be differentiated with respect to |i, 5/2 where the having a derivative of the order of (i Again ^ the left-hand sides in (6. 2k) to (6. 31) are functions of |i of order of growth —, and will admit product representations of the type (6. 21), e. G. (for X^ 4 0^^ 4 0^ -^n ^ °^ (6. 35) y'(i:/2, -n) = c(l+nA)fr (l+^^^„) ^ n=l (6. 3i^) y^U/2, -^) = c IT (1+fi/c^) n=l where c, c are constants. Now we can calculate an asymptotic expansion for y^(rt/2, -n) A 1°S in two dif...ferent ways, using (6. 31) and (6. 32) or using (6. 33) and (6. 5^) By equating these two expansions, we find 00 00 n=l I n + l^n 0-7 n n {[1 + 7 ) (m- + cr ) o ' — r" 'n^ n n=l h9 or 03 Q n=l n=l 03 (7 -a J n=i -P . ^(. -5/2) which proves (6. I7). Equation (6. 16) can be proved in the same manner.
7. Coexistence 7. 1. Introduction .
According to Floquet's Theorem (Section 1. 2); Hill's equation will; in general, have only one periodic solution (and its constant multiples) of period n or 2n.
What to read after Hills Equation Ii Transformations Approximation Examples?
You can find similar books in the "Read Also" column, or choose other free books by W Magnus to read onlineMoreLess
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