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11: 4.15 Graphics
§4.15(i) Real Arguments
Figure 4.15.7 illustrates the conformal mapping of the strip 1 2 π < z < 1 2 π onto the whole w -plane cut along the real axis from to 1 and 1 to , where w = sin z and z = arcsin w (principal value). …
See accompanying text
Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify
§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).
12: 19.10 Relations to Other Functions
arctan ( x / y ) = x R C ( y 2 , y 2 + x 2 ) ,
arctanh ( x / y ) = x R C ( y 2 , y 2 x 2 ) ,
arcsin ( x / y ) = x R C ( y 2 x 2 , y 2 ) ,
arcsinh ( x / y ) = x R C ( y 2 + x 2 , y 2 ) ,
For relations to the Gudermannian function gd ( x ) and its inverse gd 1 ( x ) 4.23(viii)), see (19.6.8) and …
13: 4.46 Tables
§4.46 Tables
14: 4.45 Methods of Computation
Inverse Trigonometric Functions
Hyperbolic and Inverse Hyperbolic Functions
The inverses arcsinh , arccosh , and arctanh can be computed from the logarithmic forms given in §4.37(iv), with real arguments. For arccsch , arcsech , and arccoth we have (4.37.7)–(4.37.9). … Similarly for the hyperbolic and inverse hyperbolic functions; compare (4.28.1)–(4.28.7), §4.37(iv), and (4.37.7)–(4.37.9). …
15: 4.39 Continued Fractions
§4.39 Continued Fractions
4.39.2 arcsinh z 1 + z 2 = z 1 + 1 2 z 2 3 + 1 2 z 2 5 + 3 4 z 2 7 + 3 4 z 2 9 + ,
4.39.3 arctanh z = z 1 z 2 3 4 z 2 5 9 z 2 7 ,
For these and other continued fractions involving inverse hyperbolic functions see Lorentzen and Waadeland (1992, pp. 569–571). …
16: 27.5 Inversion Formulas
§27.5 Inversion Formulas
which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … Special cases of Möbius inversion pairs are: … Other types of Möbius inversion formulas include: …
17: 4.40 Integrals
§4.40 Integrals
§4.40(iv) Inverse Hyperbolic Functions
4.40.11 arcsinh x d x = x arcsinh x ( 1 + x 2 ) 1 / 2 .
4.40.14 arccsch x d x = x arccsch x + arcsinh x , 0 < x < ,
4.40.15 arcsech x d x = x arcsech x + arcsin x , 0 < x < 1 ,
18: 27.17 Other Applications
§27.17 Other Applications
Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …
19: Mark J. Ablowitz
for appropriate data they can be linearized by the Inverse Scattering Transform (IST) and they possess solitons as special solutions. …Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering. …
20: 4.25 Continued Fractions
§4.25 Continued Fractions
4.25.3 arcsin z 1 z 2 = z 1 1 2 z 2 3 1 2 z 2 5 3 4 z 2 7 3 4 z 2 9 ,
4.25.4 arctan z = z 1 + z 2 3 + 4 z 2 5 + 9 z 2 7 + 16 z 2 9 + ,
4.25.5 e 2 a arctan ( 1 / z ) = 1 + 2 a z a + a 2 + 1 3 z + a 2 + 4 5 z + a 2 + 9 7 z + ,
See Lorentzen and Waadeland (1992, pp. 560–571) for other continued fractions involving inverse trigonometric functions. …