About the Project

q-cosine function

AdvancedHelp

(0.002 seconds)

10 matching pages

1: 17.3 q -Elementary and q -Special Functions
β–Ί
q -Cosine Functions
β–Ί
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 ,
β–Ί
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
β–Ί
28.2.1 w ′′ + ( a 2 ⁒ q ⁒ cos ⁑ ( 2 ⁒ z ) ) ⁒ w = 0 .
3: 28.20 Definitions and Basic Properties
β–Ί
28.20.1 w ′′ ( a 2 ⁒ q ⁒ cosh ⁑ ( 2 ⁒ z ) ) ⁒ w = 0 ,
4: 28.14 Fourier Series
β–Ί
28.14.2 ce ν ⁑ ( z , q ) = m = c 2 ⁒ m ν ⁑ ( q ) ⁒ cos ⁑ ( ν + 2 ⁒ m ) ⁒ z ,
5: 28.4 Fourier Series
β–Ί
28.4.1 ce 2 ⁒ n ⁑ ( z , q ) = m = 0 A 2 ⁒ m 2 ⁒ n ⁑ ( q ) ⁒ cos ⁑ 2 ⁒ m ⁒ z ,
β–Ί
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
β–Ί
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‹― ,
β–Ί
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
β–Ί
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 ) ,
β–Ί
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
β–Ί
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–Ίβ–ΊFor 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–ΊThis dilatation transformation, which does require analyticity of q ⁒ ( x ) in (1.18.28), or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of ⟨ ( z T ) 1 ⁒ f , f ⟩ . … β–ΊSurprisingly, 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. …