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1: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
§4.23(i) General Definitions
Figure 4.23.1: z -plane. …
§4.23(iv) Logarithmic Forms
§4.23(vii) Special Values and Interrelations
2: 4.27 Sums
§4.27 Sums
For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).
3: 4.1 Special Notation
k , m , n integers.
The main purpose of the present chapter is to extend these definitions and properties to complex arguments z . 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. …
4: 4.14 Definitions and Periodicity
§4.14 Definitions and Periodicity
4.14.4 tan z = sin z cos z ,
4.14.7 cot z = cos z sin z = 1 tan z .
The functions sin z and cos z are entire. …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). …
5: 4.47 Approximations
§4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
6: 4.46 Tables
§4.46 Tables
7: 4.28 Definitions and Periodicity
4.28.4 tanh z = sinh z cosh z ,
Relations to Trigonometric Functions
As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions.
Periodicity and Zeros
The functions sinh z and cosh z have period 2 π i , and tanh z has period π i . …
8: 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
9: 4.21 Identities
§4.21 Identities
§4.21(i) Addition Formulas
§4.21(ii) Squares and Products
§4.21(iii) Multiples of the Argument
§4.21(iv) Real and Imaginary Parts; Moduli
10: 4.32 Inequalities
4.32.1 cosh x ( sinh x x ) 3 ,
4.32.2 sin x cos x < tanh x < x , x > 0 ,
4.32.3 | cosh x - cosh y | | x - y | sinh x sinh y , x > 0 , y > 0 ,
4.32.4 arctan x 1 2 π tanh x , x 0 .