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1: 17.3 q -Elementary and q -Special Functions
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q -Cosine Functions
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17.3.5 cos q ⁑ ( x ) = 1 2 ⁒ ( e q ⁑ ( i ⁒ x ) + e q ⁑ ( i ⁒ x ) ) = n = 0 ( 1 q ) 2 ⁒ n ⁒ ( 1 ) n ⁒ x 2 ⁒ n ( q ; q ) 2 ⁒ n ,
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17.3.6 Cos q ⁑ ( x ) = 1 2 ⁒ ( E q ⁑ ( i ⁒ x ) + E q ⁑ ( i ⁒ x ) ) = n = 0 ( 1 q ) 2 ⁒ n ⁒ q n ⁒ ( 2 ⁒ n 1 ) ⁒ ( 1 ) n ⁒ x 2 ⁒ n ( q ; q ) 2 ⁒ n .
2: 28.2 Definitions and Basic Properties
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28.2.1 w ′′ + ( a 2 ⁒ q ⁒ cos ⁑ ( 2 ⁒ z ) ) ⁒ w = 0 .
3: 28.20 Definitions and Basic Properties
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28.20.1 w ′′ ( a 2 ⁒ q ⁒ cosh ⁑ ( 2 ⁒ z ) ) ⁒ w = 0 ,
4: 28.14 Fourier Series
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28.14.2 ce ν ⁑ ( z , q ) = m = c 2 ⁒ m ν ⁑ ( q ) ⁒ cos ⁑ ( ν + 2 ⁒ m ) ⁒ z ,
5: 28.4 Fourier Series
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28.4.1 ce 2 ⁒ n ⁑ ( z , q ) = m = 0 A 2 ⁒ m 2 ⁒ n ⁑ ( q ) ⁒ cos ⁑ 2 ⁒ m ⁒ z ,
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28.4.2 ce 2 ⁒ n + 1 ⁑ ( z , q ) = m = 0 A 2 ⁒ m + 1 2 ⁒ n + 1 ⁑ ( q ) ⁒ cos ⁑ ( 2 ⁒ m + 1 ) ⁒ z ,
6: 28.6 Expansions for Small q
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28.6.21 2 1 / 2 ⁒ ce 0 ⁑ ( z , q ) = 1 1 2 ⁒ q ⁒ cos ⁑ 2 ⁒ z + 1 32 ⁒ q 2 ⁒ ( cos ⁑ 4 ⁒ z 2 ) 1 128 ⁒ q 3 ⁒ ( 1 9 ⁒ cos ⁑ 6 ⁒ z 11 ⁒ cos ⁑ 2 ⁒ z ) + β‹― ,
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28.6.22 ce 1 ⁑ ( z , q ) = cos ⁑ z 1 8 ⁒ q ⁒ cos ⁑ 3 ⁒ z + 1 128 ⁒ q 2 ⁒ ( 2 3 ⁒ cos ⁑ 5 ⁒ z 2 ⁒ cos ⁑ 3 ⁒ z cos ⁑ z ) 1 1024 ⁒ q 3 ⁒ ( 1 9 ⁒ cos ⁑ 7 ⁒ z 8 9 ⁒ cos ⁑ 5 ⁒ z 1 3 ⁒ cos ⁑ 3 ⁒ z + 2 ⁒ cos ⁑ z ) + β‹― ,
7: 28.10 Integral Equations
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28.10.9 0 Ο€ / 2 J 0 ⁑ ( 2 ⁒ q ⁒ ( cos 2 ⁑ Ο„ sin 2 ⁑ ΞΆ ) ) ⁒ ce 2 ⁒ n ⁑ ( Ο„ , q ) ⁒ d Ο„ = w II ⁑ ( 1 2 ⁒ Ο€ ; a 2 ⁒ n ⁑ ( q ) , q ) ⁒ ce 2 ⁒ n ⁑ ( ΞΆ , q ) ,
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28.10.10 0 Ο€ J 0 ⁑ ( 2 ⁒ q ⁒ ( cos ⁑ Ο„ + cos ⁑ ΞΆ ) ) ⁒ ce n ⁑ ( Ο„ , q ) ⁒ d Ο„ = w II ⁑ ( Ο€ ; a n ⁑ ( q ) , q ) ⁒ ce n ⁑ ( ΞΆ , q ) .
8: 28.32 Mathematical Applications
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28.32.4 2 K z 2 2 K ΢ 2 = 2 ⁒ q ⁒ ( cos ⁑ ( 2 ⁒ z ) cos ⁑ ( 2 ⁒ ΢ ) ) ⁒ K .
9: 28.9 Zeros
§28.9 Zeros
β–ΊFor real q each of the functions ce 2 ⁒ n ⁑ ( z , q ) , se 2 ⁒ n + 1 ⁑ ( z , q ) , ce 2 ⁒ n + 1 ⁑ ( z , q ) , and se 2 ⁒ n + 2 ⁑ ( z , q ) has exactly n zeros in 0 < z < 1 2 ⁒ Ο€ . They are continuous in q . For q the zeros of ce 2 ⁒ n ⁑ ( z , q ) and se 2 ⁒ n + 1 ⁑ ( z , q ) approach asymptotically the zeros of 𝐻𝑒 2 ⁒ n ⁑ ( q 1 / 4 ⁒ ( Ο€ 2 ⁒ z ) ) , and the zeros of ce 2 ⁒ n + 1 ⁑ ( z , q ) and se 2 ⁒ n + 2 ⁑ ( z , q ) approach asymptotically the zeros of 𝐻𝑒 2 ⁒ n + 1 ⁑ ( q 1 / 4 ⁒ ( Ο€ 2 ⁒ z ) ) . …Furthermore, for q > 0 ce m ⁑ ( z , q ) and se m ⁑ ( z , q ) also have purely imaginary zeros that correspond uniquely to the purely imaginary z -zeros of J m ⁑ ( 2 ⁒ q ⁒ cos ⁑ z ) 10.21(i)), and they are asymptotically equal as q 0 and | ⁑ z | . …
10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
β–ΊThe Fourier cosine and sine transform pairs (1.14.9) & (1.14.11) and (1.14.10) & (1.14.12) can be easily obtained from (1.18.57) as for Ξ½ = ± 1 2 the Bessel functions reduce to the trigonometric functions, see (10.16.1).β–ΊFor f ⁑ ( x ) even in x this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for f ⁑ ( x ) odd the Fourier sine transform pair (1.14.10) & (1.14.12). These latter results also correspond to use of the Ξ΄ ⁑ ( x y ) as defined in (1.17.12_1) and (1.17.12_2). β–ΊMore generally, continuous spectra may occur in sets of disjoint finite intervals [ Ξ» a , Ξ» b ] ( 0 , ) , often called bands, when q ⁒ ( x ) is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). Should q ⁒ ( x ) be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10). For example, replacing 2 ⁒ q ⁒ cos ⁑ ( 2 ⁒ z ) of (28.2.1) by Ξ» ⁒ cos ⁑ ( 2 ⁒ Ο€ ⁒ Ξ± ⁒ n + ΞΈ ) , n β„€ gives an almost Mathieu equation which for appropriate Ξ± has such properties. β–Ί
Example 1: In one and two dimensions any q ⁒ ( x ) with a ‘Dip, or Well’ has a partly discrete spectrum
β–ΊSuppose that X is the whole real line in one dimension, and that q ⁒ ( x ) , in (1.18.28) has (non-oscillatory) limits of 0 at both ± , and thus a continuous spectrum on 𝝈 0 . What then is the condition on q ⁒ ( x ) to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if q ⁒ ( x ) 0 then there is only a continuous spectrum. Surprisingly, if q ⁒ ( x ) < 0 on any interval on the real line, even if positive elsewhere, as long as X q ⁒ ( x ) ⁒ d x 0 , see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding L 2 ⁑ ( X ) eigenfunction. Thus, and this is a case where q ⁒ ( x ) is not continuous, if q ⁒ ( x ) = Ξ± ⁒ Ξ΄ ⁑ ( x a ) , Ξ± > 0 , there will be an L 2 eigenfunction localized in the vicinity of x = a , with a negative eigenvalue, thus disjoint from the continuous spectrum on [ 0 , ) . Similar results hold for two, but not higher, dimensional quantum systems. See Brownstein (2000) and Yang and de Llano (1989) for numerical examples, based on variational calculations, and Simon (1976) and Chadan et al. (2003) for rigorous mathematical discussion.