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11: 19.14 Reduction of General Elliptic Integrals
19.14.5 sin 2 ϕ = γ α U 2 + γ ,
19.14.7 sin 2 ϕ = ( γ α ) x 2 a 1 a 2 + γ x 2 .
19.14.8 sin 2 ϕ = γ α b 1 b 2 y 2 + γ .
19.14.9 sin 2 ϕ = ( γ α ) ( x 2 y 2 ) γ ( x 2 y 2 ) a 1 ( a 2 + b 2 x 2 ) .
19.14.10 sin 2 ϕ = ( γ α ) ( y 2 x 2 ) γ ( y 2 x 2 ) a 1 ( a 2 + b 2 y 2 ) .
12: 19.11 Addition Theorems
19.11.1 F ( θ , k ) + F ( ϕ , k ) = F ( ψ , k ) ,
19.11.2 E ( θ , k ) + E ( ϕ , k ) = E ( ψ , k ) + k 2 sin θ sin ϕ sin ψ .
19.11.9 tan θ = 1 / ( k tan ϕ ) .
13: 33.26 Software
§33.26(iii) Complex Variable and/or Parameters
14: 8.28 Software
§8.28(iii) Incomplete Gamma Functions for Complex Argument and Parameter
§8.28(v) Incomplete Beta Functions for Complex Argument and Parameters
§8.28(vii) Generalized Exponential Integral for Complex Argument and/or Parameter
15: 12.21 Software
§12.21(iii) Complex Arguments and Parameters
16: 13.32 Software
§13.32(iii) Complex Argument and/or Parameters
17: 7.22 Methods of Computation
The computation of these functions can be based on algorithms for the complementary error function with complex argument; compare (7.19.3). …
18: 5.3 Graphics
§5.3(ii) Complex Argument
19: 19.4 Derivatives and Differential Equations
19.4.8 ( k k 2 D k 2 + ( 1 3 k 2 ) D k k ) F ( ϕ , k ) = k sin ϕ cos ϕ ( 1 k 2 sin 2 ϕ ) 3 / 2 ,
19.4.9 ( k k 2 D k 2 + k 2 D k + k ) E ( ϕ , k ) = k sin ϕ cos ϕ 1 k 2 sin 2 ϕ .
20: Bibliography T
  • I. J. Thompson and A. R. Barnett (1985) COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.
  • I. J. Thompson and A. R. Barnett (1986) Coulomb and Bessel functions of complex arguments and order. J. Comput. Phys. 64 (2), pp. 490–509.
  • I. J. Thompson and A. R. Barnett (1987) Modified Bessel functions I ν ( z ) and K ν ( z ) of real order and complex argument, to selected accuracy. Comput. Phys. Comm. 47 (2-3), pp. 245–257.
  • I. J. Thompson (2004) Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 159 (3), pp. 241–242.
  • J. Todd (1954) Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.