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1: 4.15 Graphics
§4.15(i) Real Arguments
§4.15(iii) Complex Arguments: Surfaces
The corresponding surfaces for arccos ( x + i y ) , arccot ( x + i y ) , arcsec ( x + i y ) can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).
2: 10.22 Integrals
Trigonometric Arguments
3: 4.45 Methods of Computation
The inverses arcsinh , arccosh , and arctanh can be computed from the logarithmic forms given in §4.37(iv), with real arguments. …
4: 23.12 Asymptotic Approximations
23.12.2 ζ ( z ) = π 2 4 ω 1 2 ( z 3 + 2 ω 1 π cot ( π z 2 ω 1 ) - 8 ( z - ω 1 π sin ( π z ω 1 ) ) q 2 + O ( q 4 ) ) ,
5: 4.21 Identities
§4.21(iii) Multiples of the Argument
6: 4.29 Graphics
§4.29(i) Real Arguments
See accompanying text
Figure 4.29.6: Principal values of arccsch x and arcsech x . … Magnify
§4.29(ii) Complex Arguments
The conformal mapping w = sinh z is obtainable from Figure 4.15.7 by rotating both the w -plane and the z -plane through an angle 1 2 π , compare (4.28.8). The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …
7: 19.7 Connection Formulas
sin θ = 1 + k 2 sin ϕ 1 + k 2 sin 2 ϕ ,
Imaginary-Argument Transformation
With sinh ϕ = tan ψ , … The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of Π ( ϕ , α 2 , k ) when α 2 > csc 2 ϕ (see (19.6.5) for the complete case). Let c = csc 2 ϕ α 2 . …
8: 19.11 Addition Theorems
19.11.2 E ( θ , k ) + E ( ϕ , k ) = E ( ψ , k ) + k 2 sin θ sin ϕ sin ψ .
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 ) .
19.11.9 tan θ = 1 / ( k tan ϕ ) .
9: 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 ) .
10: 4.35 Identities
4.35.14 2 sinh u sinh v = cosh ( u + v ) - cosh ( u - v ) ,
4.35.16 2 sinh u cosh v = sinh ( u + v ) + sinh ( u - v ) .
§4.35(iii) Multiples of the Argument
4.35.34 sinh z = sinh x cos y + i cosh x sin y ,
4.35.35 cosh z = cosh x cos y + i sinh x sin y ,