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1: 19.2 Definitions
Assume 1 sin 2 ϕ ( , 0 ] and 1 k 2 sin 2 ϕ ( , 0 ] , except that one of them may be 0, and 1 α 2 sin 2 ϕ { 0 } . …The integral for E ( ϕ , k ) is well defined if k 2 = sin 2 ϕ = 1 , and the Cauchy principal value (§1.4(v)) of Π ( ϕ , α 2 , k ) is taken if 1 α 2 sin 2 ϕ vanishes at an interior point of the integration path. … If 1 < k 1 / sin ϕ , then k c is pure imaginary. …
§19.2(iv) A Related Function: R C ( x , y )
For the special cases of R C ( x , x ) and R C ( 0 , y ) see (19.6.15). …
2: Bibliography S
  • I. J. Schoenberg (1971) Norm inequalities for a certain class of C  functions. Israel J. Math. 10, pp. 364–372.
  • R. S. Scorer (1950) Numerical evaluation of integrals of the form I = x 1 x 2 f ( x ) e i ϕ ( x ) 𝑑 x and the tabulation of the function Gi ( z ) = ( 1 / π ) 0 sin ( u z + 1 3 u 3 ) 𝑑 u . Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.
  • M. J. Seaton (2002b) FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.
  • L. Shen (1998) On an identity of Ramanujan based on the hypergeometric series F 1 2 ( 1 3 , 2 3 ; 1 2 ; x ) . J. Number Theory 69 (2), pp. 125–134.
  • M. M. Shepherd and J. G. Laframboise (1981) Chebyshev approximation of ( 1 + 2 x ) exp ( x 2 ) erfc x in 0 x < . Math. Comp. 36 (153), pp. 249–253.
  • 3: 10.34 Analytic Continuation
    10.34.2 K ν ( z e m π i ) = e m ν π i K ν ( z ) π i sin ( m ν π ) csc ( ν π ) I ν ( z ) .
    10.34.4 K ν ( z e m π i ) = csc ( ν π ) ( ± sin ( m ν π ) K ν ( z e ± π i ) sin ( ( m 1 ) ν π ) K ν ( z ) ) .
    4: 4.42 Solution of Triangles
    4.42.4 a sin A = b sin B = c sin C ,
    4.42.8 cos a = cos b cos c + sin b sin c cos A ,
    4.42.9 sin A sin a = sin B sin b = sin C sin c ,
    4.42.11 cos a cos C = sin a cot b sin C cot B ,
    4.42.12 cos A = cos B cos C + sin B sin C cos a .
    5: 19.11 Addition Theorems
    sin ψ = ( sin θ cos ϕ ) Δ ( ϕ ) + ( sin ϕ cos θ ) Δ ( θ ) 1 k 2 sin 2 θ sin 2 ϕ ,
    cos ψ = cos θ cos ϕ ( sin θ sin ϕ ) Δ ( θ ) Δ ( ϕ ) 1 k 2 sin 2 θ sin 2 ϕ ,
    19.11.6_5 R C ( γ δ , γ ) = 1 δ arctan ( δ sin θ sin ϕ sin ψ α 2 1 α 2 cos θ cos ϕ cos ψ ) .
    19.11.14 sin ψ = ( sin 2 θ ) Δ ( θ ) / ( 1 k 2 sin 4 θ ) ,
    cos ψ = ( cos ( 2 θ ) + k 2 sin 4 θ ) / ( 1 k 2 sin 4 θ ) ,
    6: 19.23 Integral Representations
    19.23.1 R F ( 0 , y , z ) = 0 π / 2 ( y cos 2 θ + z sin 2 θ ) 1 / 2 d θ ,
    19.23.3 R D ( 0 , y , z ) = 3 0 π / 2 ( y cos 2 θ + z sin 2 θ ) 3 / 2 sin 2 θ d θ .
    19.23.6 4 π R F ( x , y , z ) = 0 2 π 0 π sin θ d θ d ϕ ( x sin 2 θ cos 2 ϕ + y sin 2 θ sin 2 ϕ + z cos 2 θ ) 1 / 2 ,
    With l 1 , l 2 , l 3 denoting any permutation of sin θ cos ϕ , sin θ sin ϕ , cos θ , …
    7: 25.5 Integral Representations
    25.5.11 ζ ( s ) = 1 2 + 1 s 1 + 2 0 sin ( s arctan x ) ( 1 + x 2 ) s / 2 ( e 2 π x 1 ) d x .
    25.5.12 ζ ( s ) = 2 s 1 s 1 2 s 0 sin ( s arctan x ) ( 1 + x 2 ) s / 2 ( e π x + 1 ) d x .
    25.5.15 ζ ( s ) = 1 s 1 + sin ( π s ) π 0 ( ln ( 1 + x ) ψ ( 1 + x ) ) x s d x ,
    25.5.16 ζ ( s ) = 1 s 1 + sin ( π s ) π ( s 1 ) 0 ( 1 1 + x ψ ( 1 + x ) ) x 1 s d x ,
    25.5.20 ζ ( s ) = Γ ( 1 s ) 2 π i ( 0 + ) z s 1 e z 1 d z , s 1 , 2 , ,
    8: 19.5 Maclaurin and Related Expansions
    where F 1 2 is the Gauss hypergeometric function (§§15.1 and 15.2(i)). …
    19.5.4_1 F ( ϕ , k ) = m = 0 ( 1 2 ) m sin 2 m + 1 ϕ ( 2 m + 1 ) m ! F 1 2 ( m + 1 2 , 1 2 m + 3 2 ; sin 2 ϕ ) k 2 m = sin ϕ F 1 ( 1 2 ; 1 2 , 1 2 ; 3 2 ; sin 2 ϕ , k 2 sin 2 ϕ ) ,
    19.5.4_2 E ( ϕ , k ) = m = 0 ( 1 2 ) m sin 2 m + 1 ϕ ( 2 m + 1 ) m ! F 1 2 ( m + 1 2 , 1 2 m + 3 2 ; sin 2 ϕ ) k 2 m = sin ϕ F 1 ( 1 2 ; 1 2 , 1 2 ; 3 2 ; sin 2 ϕ , k 2 sin 2 ϕ ) ,
    where F 1 ( α ; β , β ; γ ; x , y ) is an Appell function (§16.13). … 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). …
    9: 1.6 Vectors and Vector-Valued Functions
    The divergence of a differentiable vector-valued function 𝐅 = F 1 𝐢 + F 2 𝐣 + F 3 𝐤 is … The curl of 𝐅 is … Sufficient conditions for this result to hold are that F 1 ( x , y ) and F 2 ( x , y ) are continuously differentiable on S , and C is piecewise differentiable. … For a sphere x = ρ sin θ cos ϕ , y = ρ sin θ sin ϕ , z = ρ cos θ , … For a vector-valued function 𝐅 , …
    10: 7.4 Symmetry
    7.4.4 F ( z ) = F ( z ) .
    C ( z ) = C ( z ) ,
    C ( i z ) = i C ( z ) ,
    g ( z ) = 2 sin ( 1 4 π + 1 2 π z 2 ) g ( z ) .