About the Project
NIST

integer parameters

AdvancedHelp

(0.006 seconds)

1—10 of 270 matching pages

1: 14.34 Software
  • Adams and Swarztrauber (1997). Integer parameters. Fortran.

  • Braithwaite (1973). Integer parameters. Fortran.

  • Delic (1979a). Integer parameters. Fortran.

  • Gil and Segura (1998). Integer parameters. Fortran.

  • Lozier and Smith (1981). Integer parameters. Fortran.

  • 2: 28.11 Expansions in Series of Mathieu Functions
    28.11.3 1 = 2 n = 0 A 0 2 n ( q ) ce 2 n ( z , q ) ,
    28.11.4 cos 2 m z = n = 0 A 2 m 2 n ( q ) ce 2 n ( z , q ) , m 0 ,
    28.11.5 cos ( 2 m + 1 ) z = n = 0 A 2 m + 1 2 n + 1 ( q ) ce 2 n + 1 ( z , q ) ,
    28.11.6 sin ( 2 m + 1 ) z = n = 0 B 2 m + 1 2 n + 1 ( q ) se 2 n + 1 ( z , q ) ,
    28.11.7 sin ( 2 m + 2 ) z = n = 0 B 2 m + 2 2 n + 2 ( q ) se 2 n + 2 ( z , q ) .
    3: 28.19 Expansions in Series of me ν + 2 n Functions
    28.19.2 f ( z ) = n = - f n me ν + 2 n ( z , q ) ,
    28.19.3 f n = 1 π 0 π f ( z ) me ν + 2 n ( - z , q ) d z .
    28.19.4 e i ν z = n = - c - 2 n ν + 2 n ( q ) me ν + 2 n ( z , q ) ,
    4: 28.4 Fourier Series
    28.4.9 2 ( A 0 2 n ( q ) ) 2 + m = 1 ( A 2 m 2 n ( q ) ) 2 = 1 ,
    28.4.10 m = 0 ( A 2 m + 1 2 n + 1 ( q ) ) 2 = 1 ,
    28.4.11 m = 0 ( B 2 m + 1 2 n + 1 ( q ) ) 2 = 1 ,
    28.4.12 m = 0 ( B 2 m + 2 2 n + 2 ( q ) ) 2 = 1 .
    28.4.17 A 2 m 2 n ( - q ) = ( - 1 ) n - m A 2 m 2 n ( q ) ,
    5: 28.23 Expansions in Series of Bessel Functions
    28.23.6 Mc 2 m ( j ) ( z , h ) = ( - 1 ) m ( ce 2 m ( 0 , h 2 ) ) - 1 = 0 ( - 1 ) A 2 2 m ( h 2 ) 𝒞 2 ( j ) ( 2 h cosh z ) ,
    28.23.7 Mc 2 m ( j ) ( z , h ) = ( - 1 ) m ( ce 2 m ( 1 2 π , h 2 ) ) - 1 = 0 A 2 2 m ( h 2 ) 𝒞 2 ( j ) ( 2 h sinh z ) ,
    28.23.8 Mc 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m ( ce 2 m + 1 ( 0 , h 2 ) ) - 1 = 0 ( - 1 ) A 2 + 1 2 m + 1 ( h 2 ) 𝒞 2 + 1 ( j ) ( 2 h cosh z ) ,
    28.23.9 Mc 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m + 1 ( ce 2 m + 1 ( 1 2 π , h 2 ) ) - 1 coth z = 0 ( 2 + 1 ) A 2 + 1 2 m + 1 ( h 2 ) 𝒞 2 + 1 ( j ) ( 2 h sinh z ) ,
    28.23.10 Ms 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m ( se 2 m + 1 ( 0 , h 2 ) ) - 1 tanh z = 0 ( - 1 ) ( 2 + 1 ) B 2 + 1 2 m + 1 ( h 2 ) 𝒞 2 + 1 ( j ) ( 2 h cosh z ) ,
    6: 28.28 Integrals, Integral Representations, and Integral Equations
    28.28.20 2 π 0 π 𝒞 2 ( j ) ( 2 h R ) cos ( 2 ϕ ) ce 2 m ( t , h 2 ) d t = ε ( - 1 ) + m A 2 2 m ( h 2 ) Mc 2 m ( j ) ( z , h ) ,
    28.28.38 α ^ n , m ( s ) = 1 2 π 0 2 π cos t se n ( t , h 2 ) se m ( t , h 2 ) d t = ( - 1 ) p 2 i π se n ( 0 , h 2 ) se m ( 0 , h 2 ) h Ds 2 ( n , m , 0 ) .
    7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
    28.24.2 ε s Mc 2 m ( j ) ( z , h ) = ( - 1 ) m = 0 ( - 1 ) A 2 2 m ( h 2 ) A 2 s 2 m ( h 2 ) ( J - s ( h e - z ) 𝒞 + s ( j ) ( h e z ) + J + s ( h e - z ) 𝒞 - s ( j ) ( h e z ) ) ,
    28.24.3 Mc 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m = 0 ( - 1 ) A 2 + 1 2 m + 1 ( h 2 ) A 2 s + 1 2 m + 1 ( h 2 ) ( J - s ( h e - z ) 𝒞 + s + 1 ( j ) ( h e z ) + J + s + 1 ( h e - z ) 𝒞 - s ( j ) ( h e z ) ) ,
    28.24.4 Ms 2 m + 1 ( j ) ( z , h ) = ( - 1 ) m = 0 ( - 1 ) B 2 + 1 2 m + 1 ( h 2 ) B 2 s + 1 2 m + 1 ( h 2 ) ( J - s ( h e - z ) 𝒞 + s + 1 ( j ) ( h e z ) - J + s + 1 ( h e - z ) 𝒞 - s ( j ) ( h e z ) ) ,
    28.24.5 Ms 2 m + 2 ( j ) ( z , h ) = ( - 1 ) m = 0 ( - 1 ) B 2 + 2 2 m + 2 ( h 2 ) B 2 s + 2 2 m + 2 ( h 2 ) ( J - s ( h e - z ) 𝒞 + s + 2 ( j ) ( h e z ) - J + s + 2 ( h e - z ) 𝒞 - s ( j ) ( h e z ) ) ,
    For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).
    8: 8.28 Software
    §8.28(vi) Generalized Exponential Integral for Real Argument and Integer Parameter
    9: 31.14 General Fuchsian Equation
    31.14.1 d 2 w d z 2 + ( j = 1 N γ j z - a j ) d w d z + ( j = 1 N q j z - a j ) w = 0 , j = 1 N q j = 0 .
    31.14.3 w ( z ) = ( j = 1 N ( z - a j ) - γ j / 2 ) W ( z ) ,
    31.14.4 d 2 W d z 2 = j = 1 N ( γ ~ j ( z - a j ) 2 + q ~ j z - a j ) W , j = 1 N q ~ j = 0 ,
    10: 28.22 Connection Formulas
    28.22.5 g e , 2 m ( h ) = ( - 1 ) m 2 π ce 2 m ( 1 2 π , h 2 ) A 0 2 m ( h 2 ) ,
    28.22.6 g e , 2 m + 1 ( h ) = ( - 1 ) m + 1 2 π ce 2 m + 1 ( 1 2 π , h 2 ) h A 1 2 m + 1 ( h 2 ) ,
    28.22.7 g o , 2 m + 1 ( h ) = ( - 1 ) m 2 π se 2 m + 1 ( 1 2 π , h 2 ) h B 1 2 m + 1 ( h 2 ) ,