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1: 4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
2: 14.13 Trigonometric Expansions
§14.13 Trigonometric Expansions
For other trigonometric expansions see Erdélyi et al. (1953a, pp. 146–147).
3: 28.26 Asymptotic Approximations for Large q
28.26.4 Fc m ( z , h ) 1 + s 8 h cosh 2 z + 1 2 11 h 2 ( s 4 + 86 s 2 + 105 cosh 4 z - s 4 + 22 s 2 + 57 cosh 2 z ) + 1 2 14 h 3 ( - s 5 + 14 s 3 + 33 s cosh 2 z - 2 s 5 + 124 s 3 + 1122 s cosh 4 z + 3 s 5 + 290 s 3 + 1627 s cosh 6 z ) + ,
28.26.5 Gc m ( z , h ) sinh z cosh 2 z ( s 2 + 3 2 5 h + 1 2 9 h 2 ( s 3 + 3 s + 4 s 3 + 44 s cosh 2 z ) + 1 2 14 h 3 ( 5 s 4 + 34 s 2 + 9 - s 6 - 47 s 4 + 667 s 2 + 2835 12 cosh 2 z + s 6 + 505 s 4 + 12139 s 2 + 10395 12 cosh 4 z ) ) + .
4: 10.19 Asymptotic Expansions for Large Order
J ν ( ν sech α ) e ν ( tanh α - α ) ( 2 π ν tanh α ) 1 2 k = 0 U k ( coth α ) ν k ,
Y ν ( ν sech α ) - e ν ( α - tanh α ) ( 1 2 π ν tanh α ) 1 2 k = 0 ( - 1 ) k U k ( coth α ) ν k ,
J ν ( ν sech α ) ( sinh ( 2 α ) 4 π ν ) 1 2 e ν ( tanh α - α ) k = 0 V k ( coth α ) ν k ,
Y ν ( ν sech α ) ( sinh ( 2 α ) π ν ) 1 2 e ν ( α - tanh α ) k = 0 ( - 1 ) k V k ( coth α ) ν k .
J ν ( ν sec β ) ( sin ( 2 β ) π ν ) 1 2 ( - sin ξ k = 0 V 2 k ( i cot β ) ν 2 k - i cos ξ k = 0 V 2 k + 1 ( i cot β ) ν 2 k + 1 ) ,
5: 28.8 Asymptotic Expansions for Large q
28.8.11 P m ( x ) 1 + s 2 3 h cos 2 x + 1 h 2 ( s 4 + 86 s 2 + 105 2 11 cos 4 x - s 4 + 22 s 2 + 57 2 11 cos 2 x ) + ,
28.8.12 Q m ( x ) sin x cos 2 x ( 1 2 5 h ( s 2 + 3 ) + 1 2 9 h 2 ( s 3 + 3 s + 4 s 3 + 44 s cos 2 x ) ) + .
6: 6.12 Asymptotic Expansions
The asymptotic expansions of Si ( z ) and Ci ( z ) are given by (6.2.19), (6.2.20), together with
6.12.3 f ( z ) 1 z ( 1 - 2 ! z 2 + 4 ! z 4 - 6 ! z 6 + ) ,
6.12.4 g ( z ) 1 z 2 ( 1 - 3 ! z 2 + 5 ! z 4 - 7 ! z 6 + ) ,
When 1 4 π | ph z | < 1 2 π the remainders are bounded in magnitude by csc ( 2 | ph z | ) times the first neglected terms. …
7: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.13 2 K cs ( 2 K t , k ) = lim N n = - N N ( - 1 ) n π tan ( π ( t - n τ ) ) = lim N n = - N N ( - 1 ) n ( lim M m = - M M 1 t - m - n τ ) .
8: 14.20 Conical (or Mehler) Functions
§14.20(v) Trigonometric Expansion
9: 12.10 Uniform Asymptotic Expansions for Large Parameter
12.10.18 U ( - 1 2 μ 2 , μ t 2 ) 2 g ( μ ) ( 1 - t 2 ) 1 4 ( cos κ s = 0 ( - 1 ) s 𝒜 ~ 2 s ( t ) μ 4 s - sin κ s = 0 ( - 1 ) s 𝒜 ~ 2 s + 1 ( t ) μ 4 s + 2 ) ,
12.10.19 U ( - 1 2 μ 2 , μ t 2 ) μ 2 g ( μ ) ( 1 - t 2 ) 1 4 ( sin κ s = 0 ( - 1 ) s ~ 2 s ( t ) μ 4 s + cos κ s = 0 ( - 1 ) s ~ 2 s + 1 ( t ) μ 4 s + 2 ) ,
12.10.20 V ( - 1 2 μ 2 , μ t 2 ) 2 g ( μ ) Γ ( 1 2 + 1 2 μ 2 ) ( 1 - t 2 ) 1 4 ( cos χ s = 0 ( - 1 ) s 𝒜 ~ 2 s ( t ) μ 4 s - sin χ s = 0 ( - 1 ) s 𝒜 ~ 2 s + 1 ( t ) μ 4 s + 2 ) ,
12.10.21 V ( - 1 2 μ 2 , μ t 2 ) μ 2 g ( μ ) ( 1 - t 2 ) 1 4 Γ ( 1 2 + 1 2 μ 2 ) ( sin χ s = 0 ( - 1 ) s ~ 2 s ( t ) μ 4 s + cos χ s = 0 ( - 1 ) s ~ 2 s + 1 ( t ) μ 4 s + 2 ) ,
10: 12.14 The Function W ( a , x )
Other expansions, involving cos ( 1 4 x 2 ) and sin ( 1 4 x 2 ) , can be obtained from (12.4.3) to (12.4.6) by replacing a by - i a and z by x e π i / 4 ; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). …
12.14.25 W ( 1 2 μ 2 , μ t 2 ) 2 - 1 2 e - 1 4 π μ 2 l ( μ ) ( t 2 - 1 ) 1 4 ( cos σ s = 0 ( - 1 ) s 𝒜 2 s ( t ) μ 4 s - sin σ s = 0 ( - 1 ) s 𝒜 2 s + 1 ( t ) μ 4 s + 2 ) ,
12.14.26 W ( 1 2 μ 2 , - μ t 2 ) 2 1 2 e 1 4 π μ 2 l ( μ ) ( t 2 - 1 ) 1 4 ( sin σ s = 0 ( - 1 ) s 𝒜 2 s ( t ) μ 4 s + cos σ s = 0 ( - 1 ) s 𝒜 2 s + 1 ( t ) μ 4 s + 2 ) ,
12.14.34 W ( - 1 2 μ 2 , μ t 2 ) l ( μ ) ( t 2 + 1 ) 1 4 ( cos σ ¯ s = 0 ( - 1 ) s u ¯ 2 s ( t ) ( t 2 + 1 ) 3 s μ 4 s - sin σ ¯ s = 0 ( - 1 ) s u ¯ 2 s + 1 ( t ) ( t 2 + 1 ) 3 s + 3 2 μ 4 s + 2 ) ,
12.14.35 W ( - 1 2 μ 2 , μ t 2 ) - μ 2 l ( μ ) ( t 2 + 1 ) 1 4 ( sin σ ¯ s = 0 ( - 1 ) s v ¯ 2 s ( t ) ( t 2 + 1 ) 3 s μ 4 s + cos σ ¯ s = 0 ( - 1 ) s v ¯ 2 s + 1 ( t ) ( t 2 + 1 ) 3 s + 3 2 μ 4 s + 2 ) ,