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1: 4.1 Special Notation
k , m , n integers.
The main functions treated in this chapter are the logarithm ln z , Ln z ; the exponential exp z , e z ; the circular trigonometric (or just trigonometric) functions sin z , cos z , tan z , csc z , sec z , cot z ; the inverse trigonometric functions arcsin z , Arcsin z , etc. ; the hyperbolic trigonometric (or just hyperbolic) functions sinh z , cosh z , tanh z , csch z , sech z , coth z ; the inverse hyperbolic functions arcsinh z , Arcsinh z , etc. …
2: 4.29 Graphics
§4.29(i) Real Arguments
See accompanying text
Figure 4.29.5: csch x and sech x . Magnify
See accompanying text
Figure 4.29.6: Principal values of arccsch x and arcsech x . … Magnify
§4.29(ii) Complex Arguments
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. …
3: 4.16 Elementary Properties
§4.16 Elementary Properties
Table 4.16.1: Signs of the trigonometric functions in the four quadrants.
Quadrant sin θ , csc θ cos θ , sec θ tan θ , cot θ
Table 4.16.2: Trigonometric functions: quarter periods and change of sign.
x θ 1 2 π ± θ π ± θ 3 2 π ± θ 2 π ± θ
Table 4.16.3: Trigonometric functions: interrelations. …
sin θ = a cos θ = a tan θ = a csc θ = a sec θ = a cot θ = a
4: 22.10 Maclaurin Series
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 ) .
5: 4.28 Definitions and Periodicity
4.28.6 sech z = 1 cosh z ,
4.28.12 sec ( i z ) = sech z ,
6: 4.31 Special Values and Limits
§4.31 Special Values and Limits
Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 π i .
z 0 1 2 π i π i 3 2 π i
4.31.1 lim z 0 sinh z z = 1 ,
4.31.2 lim z 0 tanh z z = 1 ,
7: 4.14 Definitions and Periodicity
4.14.6 sec z = 1 cos z ,
The functions tan z , csc z , sec z , and cot z are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …
8: 4.15 Graphics
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).
9: 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 .
10: 4.30 Elementary Properties
§4.30 Elementary Properties
Table 4.30.1: Hyperbolic functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.
sinh θ = a cosh θ = a tanh θ = a csch θ = a sech θ = a coth θ = a
sech θ ( 1 + a 2 ) 1 / 2 a 1 ( 1 a 2 ) 1 / 2 a ( 1 + a 2 ) 1 / 2 a a 1 ( a 2 1 ) 1 / 2