The Discriminant of Hills Equation
The Discriminant of Hills Equation
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We wish to estimate the absolute value of ^ n=l \^-n — X(X-n2) ^ for X ^^ oo . In the case of a finite sum, ve would have |d(X)| = ^(X )- Now all we can do is to prove (5. 21) |d(X)| = (^{\-^^^) ( X^ oo).
2 To do this, let X = cd and observe that / 2 i I e n g g sin nx dx = - 2(-l; — and therefore 2 2 '^n'='-n (x> -n 17 - XD(X) = - i I I ^ (-l)"^ n gj^S_^ sin nxl e '^^ dx = - I I k r 10)" y (51 n g^g_^ Sin nx I dx ~n~-n n=l Integrating by parts, we find ^^^^^ ^ ~ ^ I ^^^'"^ " ^■^ I ^ "^ ^n^-n ^...°^ "^ ' ^ Because of (5. 20), the integrand is a continuous function of x which is boiinded independently of od. This proves (5-21).
Similar arguments can be applied in order to extend the estimates (5. L) and (5-2) to the case where infinitely many of the g are different from zero. However, we shall not go into the details, which are rather tedious.
18 6. Some relations between the characteristic values . We shall prove the following results: Theorem 3 . Let the roots of A(X) + 2 and of A(X) - 2 be denoted as In Section 2 .
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To quickly assess the difficulty of the text, read a short excerpt:
We wish to estimate the absolute value of ^ n=l \^-n — X(X-n2) ^ for X ^^ oo . In the case of a finite sum, ve would have |d(X)| = ^(X )- Now all we can do is to prove (5. 21) |d(X)| = (^{\-^^^) ( X^ oo).
2 To do this, let X = cd and observe that / 2 i I e n g g sin nx dx = - 2(-l; — and therefore 2 2 '^n'='-n (x> -n 17 - XD(X) = - i I I ^ (-l)"^ n gj^S_^ sin nxl e '^^ dx = - I I k r 10)" y (51 n g^g_^ Sin nx I dx ~n~-n n=l Integrating by parts, we find ^^^^^ ^ ~ ^ I ^^^'"^ " ^■^ I ^ "^ ^n^-n ^...°^ "^ ' ^ Because of (5. 20), the integrand is a continuous function of x which is boiinded independently of od. This proves (5-21).
Similar arguments can be applied in order to extend the estimates (5. L) and (5-2) to the case where infinitely many of the g are different from zero. However, we shall not go into the details, which are rather tedious.
18 6. Some relations between the characteristic values . We shall prove the following results: Theorem 3 . Let the roots of A(X) + 2 and of A(X) - 2 be denoted as In Section 2 .
What to read after The Discriminant of Hills Equation?
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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