Infinite Determinants Associated With Hills Equation
Infinite Determinants Associated With Hills Equation
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11) is an analytic function of (0, it suffices to show that it vanishes for all real values of m. We shall prove this by expressing the left-hand side of (3. 11) in terms of the u (x, co) which satisfy the recurrence relations 2 3'u_ (3. 12) ax , 5^ + U CO u^ + U q(x)u^, - c n n-x (];3. 12) can be derived easily from (2. 9) and (2. 10)]. It follows from (3. 5) and (3. 7) that (3. 13) u^(x, (o) -/ g^(x, 0)e^^'*de.
Therefore we have for n ^ 2: (3. 1U) a'u.
dx (/-x ax 2icd& de since any terra deri...ved by differentiating the integral in (3. 13) with respect -15- to its limits vanishes for n > 2. For the same reason we find from an in- tegration by parts that (3. 15) - / ~i^ exp(2icoe) d© - U oi^u (x, ©).
Equations (3. L5), (3. 13), (3. 12) show that (3. 11) and (3. 12) are equivalent. Since (3. 12) is true, the proof of Theorem II has been completedo Keferences [i] Kevanlirma, R. , Eindeutige analytische Funktionen; Berlin, 1936.
[fj Paley, R. E. A. C. , and Wiener, N. , Fourier transforms in the complex domain; American Mathematical Society Publications, Volume 19, 193^+.
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To quickly assess the difficulty of the text, read a short excerpt:
11) is an analytic function of (0, it suffices to show that it vanishes for all real values of m. We shall prove this by expressing the left-hand side of (3. 11) in terms of the u (x, co) which satisfy the recurrence relations 2 3'u_ (3. 12) ax , 5^ + U CO u^ + U q(x)u^, - c n n-x (];3. 12) can be derived easily from (2. 9) and (2. 10)]. It follows from (3. 5) and (3. 7) that (3. 13) u^(x, (o) -/ g^(x, 0)e^^'*de.
Therefore we have for n ^ 2: (3. 1U) a'u.
dx (/-x ax 2icd& de since any terra deri...ved by differentiating the integral in (3. 13) with respect -15- to its limits vanishes for n > 2. For the same reason we find from an in- tegration by parts that (3. 15) - / ~i^ exp(2icoe) d© - U oi^u (x, ©).
Equations (3. L5), (3. 13), (3. 12) show that (3. 11) and (3. 12) are equivalent. Since (3. 12) is true, the proof of Theorem II has been completedo Keferences [i] Kevanlirma, R. , Eindeutige analytische Funktionen; Berlin, 1936.
[fj Paley, R. E. A. C. , and Wiener, N. , Fourier transforms in the complex domain; American Mathematical Society Publications, Volume 19, 193^+.
What to read after Infinite Determinants Associated With 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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