About the Project

for%20ratios

AdvancedHelp

(0.002 seconds)

1—10 of 33 matching pages

1: 9.9 Zeros
9.9.6 a k = T ( 3 8 π ( 4 k 1 ) ) ,
9.9.7 Ai ( a k ) = ( 1 ) k 1 V ( 3 8 π ( 4 k 1 ) ) ,
9.9.8 a k = U ( 3 8 π ( 4 k 3 ) ) ,
9.9.9 Ai ( a k ) = ( 1 ) k 1 W ( 3 8 π ( 4 k 3 ) ) .
2: 7.8 Inequalities
7.8.5 x 2 2 x 2 + 1 x 2 ( 2 x 2 + 5 ) 4 x 4 + 12 x 2 + 3 x 𝖬 ( x ) < 2 x 4 + 9 x 2 + 4 4 x 4 + 20 x 2 + 15 < x 2 + 1 2 x 2 + 3 , x 0 .
3: 20.7 Identities
20.7.17 θ 2 ( 2 z | 2 τ ) = A θ 1 ( 1 4 π z | τ ) θ 1 ( 1 4 π + z | τ ) ,
20.7.18 θ 3 ( 2 z | 2 τ ) = A θ 3 ( 1 4 π z | τ ) θ 3 ( 1 4 π + z | τ ) ,
§20.7(vii) Derivatives of Ratios of Theta Functions
See Lawden (1989, pp. 19–20). …
20.7.34 θ 1 ( z , q 2 ) θ 3 ( z , q 2 ) θ 1 ( z , i q ) = θ 2 ( z , q 2 ) θ 4 ( z , q 2 ) θ 2 ( z , i q ) = i 1 / 4 θ 2 ( 0 , q 2 ) θ 4 ( 0 , q 2 ) 2 .
4: 22.3 Graphics
See accompanying text
Figure 22.3.13: sn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
See accompanying text
Figure 22.3.14: cn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
See accompanying text
Figure 22.3.15: dn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
See accompanying text
Figure 22.3.28: Density plot of | sn ( 20 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . Grayscale, running from 0 (black) to 10 (white), with | sn ( 20 , k ) | > 10 truncated to 10. … Magnify
5: 36.2 Catastrophes and Canonical Integrals
36.2.28 Ψ ( E ) ( 0 , 0 , z ) = Ψ ( E ) ( 0 , 0 , z ) ¯ = 2 π π z 27 exp ( 2 27 i z 3 ) ( J 1 / 6 ( 2 27 z 3 ) + i J 1 / 6 ( 2 27 z 3 ) ) , z 0 ,
6: 36.4 Bifurcation Sets
x = 9 20 z 2 .
x = 3 20 z 2 ,
36.4.11 x + i y = z 2 exp ( 2 3 i π m ) , m = 0 , 1 , 2 .
7: Bibliography G
  • W. Gautschi (1964a) Algorithm 222: Incomplete beta function ratios. Comm. ACM 7 (3), pp. 143–144.
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • B. N. Gupta (1970) On Mill’s ratio. Proc. Cambridge Philos. Soc. 67, pp. 363–364.
  • 8: 25.6 Integer Arguments
    25.6.2 ζ ( 2 n ) = ( 2 π ) 2 n 2 ( 2 n ) ! | B 2 n | , n = 1 , 2 , 3 , .
    25.6.3 ζ ( n ) = B n + 1 n + 1 , n = 1 , 2 , 3 , .
    25.6.6 ζ ( 2 k + 1 ) = ( 1 ) k + 1 ( 2 π ) 2 k + 1 2 ( 2 k + 1 ) ! 0 1 B 2 k + 1 ( t ) cot ( π t ) d t , k = 1 , 2 , 3 , .
    25.6.11 ζ ( 0 ) = 1 2 ln ( 2 π ) .
    25.6.12 ζ ′′ ( 0 ) = 1 2 ( ln ( 2 π ) ) 2 + 1 2 γ 2 1 24 π 2 + γ 1 ,
    9: 20.10 Integrals
    20.10.1 0 x s 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 2 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
    20.10.2 0 x s 1 ( θ 3 ( 0 | i x 2 ) 1 ) d x = π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
    20.10.3 0 x s 1 ( 1 θ 4 ( 0 | i x 2 ) ) d x = ( 1 2 1 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 0 .
    20.10.4 0 e s t θ 1 ( β π 2 | i π t 2 ) d t = 0 e s t θ 2 ( ( 1 + β ) π 2 | i π t 2 ) d t = s sinh ( β s ) sech ( s ) ,
    20.10.5 0 e s t θ 3 ( ( 1 + β ) π 2 | i π t 2 ) d t = 0 e s t θ 4 ( β π 2 | i π t 2 ) d t = s cosh ( β s ) csch ( s ) .
    10: 9.7 Asymptotic Expansions
    Lastly, for x > 0 we define
    9.7.3 χ ( x ) π 1 / 2 Γ ( 1 2 x + 1 ) / Γ ( 1 2 x + 1 2 ) .
    9.7.4 χ ( x ) ( 1 2 π x ) 1 / 2 .
    Numerical values of χ ( n ) are given in Table 9.7.1 for n = 1 ( 1 ) 20 to 2D. …