About the Project

elliptic integrals

AdvancedHelp

(0.013 seconds)

21—30 of 132 matching pages

21: 29.4 Graphics
See accompanying text
Figure 29.4.13: 𝐸𝑐 1.5 m ( x , 0.5 ) for 2 K x 2 K , m = 0 , 1 , 2 . K = 1.85407 . Magnify
See accompanying text
Figure 29.4.14: 𝐸𝑠 1.5 m ( x , 0.5 ) for 2 K x 2 K , m = 1 , 2 , 3 . K = 1.85407 . Magnify
See accompanying text
Figure 29.4.15: 𝐸𝑐 1.5 m ( x , 0.1 ) for 2 K x 2 K , m = 0 , 1 , 2 . K = 1.61244 . Magnify
See accompanying text
Figure 29.4.16: 𝐸𝑠 1.5 m ( x , 0.1 ) for 2 K x 2 K , m = 1 , 2 , 3 . K = 1.61244 . Magnify
See accompanying text
Figure 29.4.17: 𝐸𝑐 1.5 m ( x , 0.9 ) for 2 K x 2 K , m = 0 , 1 , 2 . K = 2.57809 . Magnify
22: 29.16 Asymptotic Expansions
The approximations for Lamé polynomials hold uniformly on the rectangle 0 z K , 0 z K , when n k and n k assume large real values. …
23: 19.1 Special Notation
l , m , n nonnegative integers.
K ( k ) ,
E ( k ) ,
F ( ϕ , k ) ,
E ( ϕ , k ) ,
24: 19.39 Software
Unless otherwise stated, the functions are K ( k ) and E ( k ) , with 0 k 2 ( = m ) 1 . … For other software, sometimes with Π ( α 2 , k ) and complex variables, see the Software Index. … Unless otherwise stated, the variables are real, and the functions are F ( ϕ , k ) and E ( ϕ , k ) . For research software see Bulirsch (1965b, function el2 ), Bulirsch (1969b, function el3 ), Jefferson (1961), and Neuman (1969a, functions E ( ϕ , k ) and Π ( ϕ , k 2 , k ) ). For other software, sometimes with Π ( ϕ , α 2 , k ) and complex variables, see the Software Index. …
25: 19.9 Inequalities
Further inequalities for K ( k ) and E ( k ) can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996). … Sharper inequalities for F ( ϕ , k ) are: … Inequalities for both F ( ϕ , k ) and E ( ϕ , k ) involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). …
26: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.2 2 K k sn ( 2 K t , k ) = n = π sin ( π ( t ( n + 1 2 ) τ ) ) = n = ( m = ( 1 ) m t m ( n + 1 2 ) τ ) ,
22.12.8 2 K dc ( 2 K t , k ) = n = π sin ( π ( t + 1 2 n τ ) ) = n = ( m = ( 1 ) m t + 1 2 m n τ ) ,
22.12.11 2 K ns ( 2 K t , k ) = n = π sin ( π ( t n τ ) ) = n = ( m = ( 1 ) m t m n τ ) ,
22.12.12 2 K ds ( 2 K t , k ) = n = ( 1 ) n π sin ( π ( t n τ ) ) = n = ( m = ( 1 ) m + n t m n τ ) ,
27: 22.5 Special Values
Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its z -derivative (or at a pole, the residue), for values of z that are integer multiples of K , i K . …
Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.
z
0 K K + i K i K 2 K 2 K + 2 i K 2 i K
Table 22.5.2: Other special values of Jacobian elliptic functions.
z
1 2 K 1 2 ( K + i K ) 1 2 i K
3 2 K 3 2 ( K + i K ) 3 2 i K
Expansions for K , K as k 0 or 1 are given in §§19.5, 19.12. …
28: 19.37 Tables
Functions K ( k ) and E ( k )
Functions K ( k ) , K ( k ) , and i K ( k ) / K ( k )
Function exp ( π K ( k ) / K ( k ) ) ( = q ( k ) )
Functions F ( ϕ , k ) and E ( ϕ , k )
29: 22.1 Special Notation
x , y real variables.
K , K K ( k ) , K ( k ) = K ( k ) (complete elliptic integrals of the first kind (§19.2(ii))).
τ i K / K .
30: 19.5 Maclaurin and Related Expansions
19.5.5 q = exp ( π K ( k ) / K ( k ) ) = r + 8 r 2 + 84 r 3 + 992 r 4 + , r = 1 16 k 2 , 0 k 1 .
Coefficients of terms up to λ 49 are given in Lee (1990), along with tables of fractional errors in K ( k ) and E ( k ) , 0.1 k 2 0.9999 , obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … An infinite series for ln K ( k ) is equivalent to the infinite product … Series expansions of F ( ϕ , k ) and E ( ϕ , k ) are surveyed and improved in Van de Vel (1969), and the case of F ( ϕ , k ) is summarized in Gautschi (1975, §1.3.2). …