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Bulirsch elliptic integrals

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1: 19.2 Definitions
§19.2(iii) Bulirsch’s Integrals
K ( k ) = cel ( k c , 1 , 1 , 1 ) ,
E ( k ) = cel ( k c , 1 , 1 , k c 2 ) ,
D ( k ) = cel ( k c , 1 , 0 , 1 ) ,
F ( ϕ , k ) = el1 ( x , k c ) = el2 ( x , k c , 1 , 1 ) ,
2: 19.1 Special Notation
l , m , n nonnegative integers.
R - a ( b 1 , b 2 , , b n ; z 1 , z 2 , , z n ) is a multivariate hypergeometric function that includes all the functions in (19.1.3). …
3: 19.25 Relations to Other Functions
§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals
19.25.19 cel ( k c , p , a , b ) = a R F ( 0 , k c 2 , 1 ) + 1 3 ( b - p a ) R J ( 0 , k c 2 , 1 , p ) ,
4: 19.39 Software
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 ) ). …
5: 19.36 Methods of Computation
Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). …
6: Bibliography B
  • R. Bulirsch (1969a) An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind. Numer. Math. 13 (3), pp. 266–284.
  • R. Bulirsch (1969b) Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.
  • R. Bulirsch (1965a) Numerical calculation of elliptic integrals and elliptic functions. II. Numer. Math. 7 (4), pp. 353–354.
  • R. Bulirsch (1965b) Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.
  • 7: Bibliography R
  • K. Reinsch and W. Raab (2000) Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.
  • 8: Bibliography M
  • P. Midy (1975) An improved calculation of the general elliptic integral of the second kind in the neighbourhood of x = 0 . Numer. Math. 25 (1), pp. 99–101.