About the Project

change of variables

AdvancedHelp

(0.003 seconds)

11—20 of 93 matching pages

11: 9.6 Relations to Other Functions
9.6.1 ζ = 2 3 z 3 / 2 ,
9.6.10 z = ( 3 2 ζ ) 2 / 3 ,
9.6.11 J ± 1 / 3 ( ζ ) = 1 2 3 / z ( 3 Ai ( z ) Bi ( z ) ) ,
9.6.12 J ± 2 / 3 ( ζ ) = 1 2 ( 3 / z ) ( ± 3 Ai ( z ) + Bi ( z ) ) ,
9.6.13 I ± 1 / 3 ( ζ ) = 1 2 3 / z ( 3 Ai ( z ) + Bi ( z ) ) ,
12: 22.9 Cyclic Identities
22.9.7 κ = dn ( 2 K ( k ) / 3 , k ) ,
22.9.8 s 1 , 3 ( 4 ) s 2 , 3 ( 4 ) + s 2 , 3 ( 4 ) s 3 , 3 ( 4 ) + s 3 , 3 ( 4 ) s 1 , 3 ( 4 ) = κ 2 1 k 2 ,
22.9.13 s 1 , 3 ( 4 ) s 2 , 3 ( 4 ) s 3 , 3 ( 4 ) = 1 1 κ 2 ( s 1 , 3 ( 4 ) + s 2 , 3 ( 4 ) + s 3 , 3 ( 4 ) ) ,
22.9.14 c 1 , 3 ( 4 ) c 2 , 3 ( 4 ) c 3 , 3 ( 4 ) = κ 2 1 κ 2 ( c 1 , 3 ( 4 ) + c 2 , 3 ( 4 ) + c 3 , 3 ( 4 ) ) ,
22.9.15 d 1 , 3 ( 2 ) d 2 , 3 ( 2 ) d 3 , 3 ( 2 ) = κ 2 + k 2 1 1 κ 2 ( d 1 , 3 ( 2 ) + d 2 , 3 ( 2 ) + d 3 , 3 ( 2 ) ) ,
13: 22.2 Definitions
With
22.2.4 sn ( z , k ) = θ 3 ( 0 , q ) θ 2 ( 0 , q ) θ 1 ( ζ , q ) θ 4 ( ζ , q ) = 1 ns ( z , k ) ,
22.2.5 cn ( z , k ) = θ 4 ( 0 , q ) θ 2 ( 0 , q ) θ 2 ( ζ , q ) θ 4 ( ζ , q ) = 1 nc ( z , k ) ,
22.2.6 dn ( z , k ) = θ 4 ( 0 , q ) θ 3 ( 0 , q ) θ 3 ( ζ , q ) θ 4 ( ζ , q ) = 1 nd ( z , k ) ,
14: 9.5 Integral Representations
9.5.6 Ai ( z ) = 3 2 π 0 exp ( t 3 3 z 3 3 t 3 ) d t , | ph z | < 1 6 π .
9.5.7 Ai ( z ) = e ζ π 0 exp ( z 1 / 2 t 2 ) cos ( 1 3 t 3 ) d t , | ph z | < π .
9.5.8 Ai ( z ) = e ζ ζ 1 / 6 π ( 48 ) 1 / 6 Γ ( 5 6 ) 0 e t t 1 / 6 ( 2 + t ζ ) 1 / 6 d t , | ph z | < 2 3 π .
15: 15.12 Asymptotic Approximations
where
15.12.6 ζ = arccosh z .
15.12.10 ζ = arccosh ( 1 4 z 1 ) ,
15.12.11 β = ( 3 2 ζ + 9 4 ln ( 2 + e ζ 2 + e ζ ) ) 1 / 3 ,
15.12.13 G 0 ( ± β ) = ( 2 + e ± ζ ) c b ( 1 / 2 ) ( 1 + e ± ζ ) a c + ( 1 / 2 ) ( z 1 e ± ζ ) a + ( 1 / 2 ) β e ζ e ζ .
16: 8.18 Asymptotic Expansions of I x ( a , b )
Let
8.18.2 ξ = ln x .
8.18.4 a F k + 1 = ( k + b a ξ ) F k + k ξ F k 1 ,
8.18.6 ( 1 e t t ) b 1 = k = 0 d k ( t ξ ) k .
17: 9.8 Modulus and Phase
In terms of Bessel functions, and with ξ = 2 3 | x | 3 / 2 ,
9.8.9 | x | 1 / 2 M 2 ( x ) = 1 2 ξ ( J 1 / 3 2 ( ξ ) + Y 1 / 3 2 ( ξ ) ) ,
9.8.10 | x | 1 / 2 N 2 ( x ) = 1 2 ξ ( J 2 / 3 2 ( ξ ) + Y 2 / 3 2 ( ξ ) ) ,
9.8.11 θ ( x ) = 2 3 π + arctan ( Y 1 / 3 ( ξ ) / J 1 / 3 ( ξ ) ) ,
9.8.12 ϕ ( x ) = 1 3 π + arctan ( Y 2 / 3 ( ξ ) / J 2 / 3 ( ξ ) ) .
18: 7.7 Integral Representations
7.7.13 f ( z ) = ( 2 π ) 3 / 2 2 π i c i c + i ζ s Γ ( s ) Γ ( s + 1 2 ) Γ ( s + 3 4 ) Γ ( 1 4 s ) d s ,
7.7.14 g ( z ) = ( 2 π ) 3 / 2 2 π i c i c + i ζ s Γ ( s ) Γ ( s + 1 2 ) Γ ( s + 1 4 ) Γ ( 3 4 s ) d s .
In (7.7.13) and (7.7.14) the integration paths are straight lines, ζ = 1 16 π 2 z 4 , and c is a constant such that 0 < c < 1 4 in (7.7.13), and 0 < c < 3 4 in (7.7.14). …
19: 29.15 Fourier Series and Chebyshev Series
29.15.8 𝑠𝐸 2 n + 1 m ( z , k 2 ) = p = 0 n A 2 p + 1 cos ( ( 2 p + 1 ) ϕ ) .
29.15.13 𝑐𝐸 2 n + 1 m ( z , k 2 ) = p = 0 n B 2 p + 1 sin ( ( 2 p + 1 ) ϕ ) .
29.15.23 𝑠𝑐𝐸 2 n + 2 m ( z , k 2 ) = p = 0 n B 2 p + 2 sin ( ( 2 p + 2 ) ϕ ) .
20: 12.11 Zeros
12.11.5 p 0 ( ζ ) = t ( ζ ) ,
12.11.6 p 1 ( ζ ) = t 3 6 t 24 ( t 2 1 ) 2 + 5 48 ( ( t 2 1 ) ζ 3 ) 1 2 .
12.11.8 q 0 ( ζ ) = t ( ζ ) .