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1: 4.14 Definitions and Periodicity
4.14.2 cos z = e i z + e - i z 2 ,
4.14.4 tan z = sin z cos z ,
4.14.5 csc z = 1 sin z ,
4.14.6 sec z = 1 cos z ,
4.14.7 cot z = cos z sin z = 1 tan z .
2: 4.28 Definitions and Periodicity
4.28.2 cosh z = e z + e - z 2 ,
4.28.4 tanh z = sinh z cosh z ,
4.28.5 csch z = 1 sinh z ,
4.28.6 sech z = 1 cosh z ,
4.28.7 coth z = 1 tanh z .
3: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
4.23.7 arccsc z = arcsin ( 1 / z ) ,
4.23.8 arcsec z = arccos ( 1 / z ) .
4.23.9 arccot z = arctan ( 1 / z ) , z ± i .
4: 4.37 Inverse Hyperbolic Functions
4.37.4 Arccsch z = Arcsinh ( 1 / z ) ,
4.37.7 arccsch z = arcsinh ( 1 / z ) ,
4.37.8 arcsech z = arccosh ( 1 / z ) .
4.37.9 arccoth z = arctanh ( 1 / z ) , z ± 1 .
5: 17.3 q -Elementary and q -Special Functions
17.3.3 sin q ( x ) = 1 2 i ( e q ( i x ) - e q ( - i x ) ) = n = 0 ( 1 - q ) 2 n + 1 ( - 1 ) n x 2 n + 1 ( q ; q ) 2 n + 1 ,
17.3.4 Sin q ( x ) = 1 2 i ( E q ( i x ) - E q ( - i x ) ) = n = 0 ( 1 - q ) 2 n + 1 q n ( 2 n + 1 ) ( - 1 ) n x 2 n + 1 ( q ; q ) 2 n + 1 .
17.3.5 cos q ( x ) = 1 2 ( e q ( i x ) + e q ( - i x ) ) = n = 0 ( 1 - q ) 2 n ( - 1 ) n x 2 n ( q ; q ) 2 n ,
17.3.6 Cos q ( x ) = 1 2 ( E q ( i x ) + E q ( - i x ) ) = n = 0 ( 1 - q ) 2 n q n ( 2 n - 1 ) ( - 1 ) n x 2 n ( q ; q ) 2 n .
6: 6.2 Definitions and Interrelations
This is also true of the functions Ci ( z ) and Chi ( z ) defined in §6.2(ii). … Si ( z ) is an odd entire function. … Cin ( z ) is an even entire function. …
6.2.17 f ( z ) = Ci ( z ) sin z - si ( z ) cos z ,
6.2.18 g ( z ) = - Ci ( z ) cos z - si ( z ) sin z .
7: 7.2 Definitions
7.2.7 C ( z ) = 0 z cos ( 1 2 π t 2 ) d t ,
7.2.8 S ( z ) = 0 z sin ( 1 2 π t 2 ) d t ,
( z ) , C ( z ) , and S ( z ) are entire functions of z , as are f ( z ) and g ( z ) in the next subsection. …
7.2.10 f ( z ) = ( 1 2 - S ( z ) ) cos ( 1 2 π z 2 ) - ( 1 2 - C ( z ) ) sin ( 1 2 π z 2 ) ,
7.2.11 g ( z ) = ( 1 2 - C ( z ) ) cos ( 1 2 π z 2 ) + ( 1 2 - S ( z ) ) sin ( 1 2 π z 2 ) .
8: 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).
9: 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. …
10: 4.47 Approximations
§4.47 Approximations
§4.47(i) Chebyshev-Series Expansions