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11: 4.15 Graphics
§4.15(i) Real Arguments
See accompanying text
Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify
§4.15(iii) Complex Arguments: Surfaces
4.15.1 cos ( x + i y ) = sin ( x + 1 2 π + i y ) ,
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: 22.10 Maclaurin Series
22.10.4 sn ( z , k ) = sin z - k 2 4 ( z - sin z cos z ) cos z + O ( k 4 ) ,
22.10.5 cn ( z , k ) = cos z + k 2 4 ( z - sin z cos z ) sin z + O ( k 4 ) ,
22.10.7 sn ( z , k ) = tanh z - k 2 4 ( z - sinh z cosh z ) sech 2 z + O ( k 4 ) ,
22.10.8 cn ( z , k ) = sech z + k 2 4 ( z - sinh z cosh z ) tanh z sech z + O ( k 4 ) ,
22.10.9 dn ( z , k ) = sech z + k 2 4 ( z + sinh z cosh z ) tanh z sech z + O ( k 4 ) .
13: 4.35 Identities
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 ,
4.35.36 tanh z = sinh ( 2 x ) + i sin ( 2 y ) cosh ( 2 x ) + cos ( 2 y ) ,
4.35.37 coth z = sinh ( 2 x ) - i sin ( 2 y ) cosh ( 2 x ) - cos ( 2 y ) .
4.35.38 | sinh z | = ( sinh 2 x + sin 2 y ) 1 / 2 = ( 1 2 ( cosh ( 2 x ) - cos ( 2 y ) ) ) 1 / 2 ,
14: 4.42 Solution of Triangles
§4.42 Solution of Triangles
4.42.1 sin A = a c = 1 csc A ,
4.42.2 cos A = b c = 1 sec A ,
4.42.3 tan A = a b = 1 cot A .
4.42.11 cos a cos C = sin a cot b - sin C cot B ,
15: 4.20 Derivatives and Differential Equations
§4.20 Derivatives and Differential Equations
4.20.3 d d z tan z = sec 2 z ,
4.20.5 d d z sec z = sec z tan z ,
With a 0 , the general solutions of the differential equations …
4.20.12 w = A cos ( a z ) + B sin ( a z ) ,
16: 4.17 Special Values and Limits
Table 4.17.1: Trigonometric functions: values at multiples of 1 12 π .
θ sin θ cos θ tan θ csc θ sec θ cot θ
4.17.1 lim z 0 sin z z = 1 ,
4.17.2 lim z 0 tan z z = 1 .
17: 4.31 Special Values and Limits
4.31.1 lim z 0 sinh z z = 1 ,
4.31.2 lim z 0 tanh z z = 1 ,
4.31.3 lim z 0 cosh z - 1 z 2 = 1 2 .
18: 4.22 Infinite Products and Partial Fractions
§4.22 Infinite Products and Partial Fractions
4.22.1 sin z = z n = 1 ( 1 - z 2 n 2 π 2 ) ,
4.22.2 cos z = n = 1 ( 1 - 4 z 2 ( 2 n - 1 ) 2 π 2 ) .
4.22.3 cot z = 1 z + 2 z n = 1 1 z 2 - n 2 π 2 ,
4.22.4 csc 2 z = n = - 1 ( z - n π ) 2 ,
19: 4.29 Graphics
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. …
20: 4.18 Inequalities
§4.18 Inequalities
4.18.3 cos x sin x x 1 , 0 x π ,
4.18.5 | sinh y | | sin z | cosh y ,
4.18.6 | sinh y | | cos z | cosh y ,
4.18.8 | cos z | cosh | z | ,