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1: 19.2 Definitions
§19.2(i) General Elliptic Integrals
2: 19.35 Other Applications
§19.35(i) Mathematical
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute π to high precision (Borwein and Borwein (1987, p. 26)). …
3: 19.14 Reduction of General Elliptic Integrals
§19.14 Reduction of General Elliptic Integrals
4: 19.29 Reduction of General Elliptic Integrals
§19.29 Reduction of General Elliptic Integrals
§19.29(i) Reduction Theorems
§19.29(ii) Reduction to Basic Integrals
(19.2.3) can be written …
19.29.33 ( x - y ) 2 U = ( x 4 + a 4 + y 4 + a 4 ) 2 - ( x 2 - y 2 ) 2 .
5: 19.1 Special Notation
l , m , n

nonnegative integers.

6: 19.20 Special Cases
The general lemniscatic case is … The general lemniscatic case is …
7: Bibliography
  • G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen (2000) Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.
  • 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.
  • 9: Bibliography B
  • W. Bartky (1938) Numerical calculation of a generalized complete elliptic integral. Rev. Mod. Phys. 10, pp. 264–269.
  • R. Bulirsch (1969b) Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.
  • P. J. Bushell (1987) On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.
  • 10: 23.6 Relations to Other Functions
    23.6.20 e 3 = - K 2 3 ω 1 2 ( 1 + k 2 ) .
    Also, 𝕃 1 , 𝕃 2 , 𝕃 3 are the lattices with generators ( 4 K , 2 i K ) , ( 2 K - 2 i K , 2 K + 2 i K ) , ( 2 K , 4 i K ) , respectively. …