About the Project

Lauricella%0Afunction

AdvancedHelp

The term"afunct" was not found.Possible alternative term: "funct".

(0.004 seconds)

1—10 of 699 matching pages

1: 19.15 Advantages of Symmetry
Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s F D (Carlson (1961b)). The function R a ( b 1 , b 2 , , b n ; z 1 , z 2 , , z n ) (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in F D , and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. …
2: 19.25 Relations to Other Functions
Assume 0 x y z , x < z , ( x , y ) ( 0 , 0 ) and p > 0 . …with α 0 . …
§19.25(vii) Hypergeometric Function
For these results and extensions to the Appell function F 1 16.13) and Lauricella’s function F D see Carlson (1963). ( F 1 and F D are equivalent to the R -function of 3 and n variables, respectively, but lack full symmetry.) …
3: Bille C. Carlson
In his paper Lauricella’s hypergeometric function F D (1963), he defined the R -function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …
4: Bibliography C
  • B. C. Carlson (1963) Lauricella’s hypergeometric function F D . J. Math. Anal. Appl. 7 (3), pp. 452–470.
  • J. P. Coleman (1980) A Fortran subroutine for the Bessel function J n ( x ) of order 0 to 10 . Comput. Phys. Comm. 21 (1), pp. 109–118.
  • 5: 33.24 Tables
  • Abramowitz and Stegun (1964, Chapter 14) tabulates F 0 ( η , ρ ) , G 0 ( η , ρ ) , F 0 ( η , ρ ) , and G 0 ( η , ρ ) for η = 0.5 ( .5 ) 20 and ρ = 1 ( 1 ) 20 , 5S; C 0 ( η ) for η = 0 ( .05 ) 3 , 6S.

  • Curtis (1964a) tabulates P ( ϵ , r ) , Q ( ϵ , r ) 33.1), and related functions for = 0 , 1 , 2 and ϵ = 2 ( .2 ) 2 , with x = 0 ( .1 ) 4 for ϵ < 0 and x = 0 ( .1 ) 10 for ϵ 0 ; 6D.

  • 6: 26.15 Permutations: Matrix Notation
    The set 𝔖 n 26.13) can be identified with the set of n × n matrices of 0’s and 1’s with exactly one 1 in each row and column. The permutation σ corresponds to the matrix in which there is a 1 at the intersection of row j with column σ ( j ) , and 0’s in all other positions. … Define r 0 ( B ) = 1 . … The Ferrers board of shape ( b 1 , b 2 , , b n ) , 0 b 1 b 2 b n , is the set B = { ( j , k ) |  1 j n , 1 k b j } . …If B is the Ferrers board of shape ( 0 , 1 , 2 , , n 1 ) , then …
    7: 24.2 Definitions and Generating Functions
    B 2 n + 1 = 0 ,
    ( 1 ) n + 1 B 2 n > 0 , n = 1 , 2 , .
    E 2 n + 1 = 0 ,
    ( 1 ) n E 2 n > 0 .
    E ~ n ( x ) = E n ( x ) , 0 x < 1 ,
    8: 11.14 Tables
  • Abramowitz and Stegun (1964, Chapter 12) tabulates 𝐇 n ( x ) , 𝐇 n ( x ) Y n ( x ) , and I n ( x ) 𝐋 n ( x ) for n = 0 , 1 and x = 0 ( .1 ) 5 , x 1 = 0 ( .01 ) 0.2 to 6D or 7D.

  • Agrest et al. (1982) tabulates 𝐇 n ( x ) and e x 𝐋 n ( x ) for n = 0 , 1 and x = 0 ( .001 ) 5 ( .005 ) 15 ( .01 ) 100 to 11D.

  • Abramowitz and Stegun (1964, Chapter 12) tabulates 0 x ( I 0 ( t ) 𝐋 0 ( t ) ) d t and ( 2 / π ) x t 1 𝐇 0 ( t ) d t for x = 0 ( .1 ) 5 to 5D or 7D; 0 x ( 𝐇 0 ( t ) Y 0 ( t ) ) d t ( 2 / π ) ln x , 0 x ( I 0 ( t ) 𝐋 0 ( t ) ) d t ( 2 / π ) ln x , and x t 1 ( 𝐇 0 ( t ) Y 0 ( t ) ) d t for x 1 = 0 ( .01 ) 0.2 to 6D.

  • Agrest et al. (1982) tabulates 0 x 𝐇 0 ( t ) d t and e x 0 x 𝐋 0 ( t ) d t for x = 0 ( .001 ) 5 ( .005 ) 15 ( .01 ) 100 to 11D.

  • Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function 𝐇 n ( x , α ) for n = 0 , 1 , x = 0 ( .2 ) 10 , and α = 0 ( .2 ) 1.4 , 1 2 π , together with surface plots.

  • 9: 20.4 Values at z = 0
    §20.4 Values at z = 0
    20.4.1 θ 1 ( 0 , q ) = θ 2 ( 0 , q ) = θ 3 ( 0 , q ) = θ 4 ( 0 , q ) = 0 ,
    20.4.6 θ 1 ( 0 , q ) = θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( 0 , q ) .
    20.4.7 θ 1 ′′ ( 0 , q ) = θ 2 ′′′ ( 0 , q ) = θ 3 ′′′ ( 0 , q ) = θ 4 ′′′ ( 0 , q ) = 0 .
    20.4.12 θ 1 ′′′ ( 0 , q ) θ 1 ( 0 , q ) = θ 2 ′′ ( 0 , q ) θ 2 ( 0 , q ) + θ 3 ′′ ( 0 , q ) θ 3 ( 0 , q ) + θ 4 ′′ ( 0 , q ) θ 4 ( 0 , q ) .
    10: 4.31 Special Values and Limits
    Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 π i .
    z 0 1 2 π i π i 3 2 π i
    cosh z 1 0 1 0
    coth z 0 0 1
    4.31.1 lim z 0 sinh z z = 1 ,
    4.31.2 lim z 0 tanh z z = 1 ,
    4.31.3 lim z 0 cosh z 1 z 2 = 1 2 .