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1: 4.2 Definitions
§4.2(iii) The Exponential Function
4.2.19 exp z = 1 + z 1 ! + z 2 2 ! + z 3 3 ! + .
The function exp is an entire function of z , with no real or complex zeros. …
4.2.32 e z = exp z ,
2: 4.11 Sums
§4.11 Sums
3: 6.11 Relations to Other Functions
Incomplete Gamma Function
6.11.1 E 1 ( z ) = Γ ( 0 , z ) .
Confluent Hypergeometric Function
6.11.2 E 1 ( z ) = e - z U ( 1 , 1 , z ) ,
4: 4.47 Approximations
§4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
5: 17.17 Physical Applications
See Kassel (1995). …
6: 4.46 Tables
§4.46 Tables
7: 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. …
8: 4.3 Graphics
See accompanying text
Figure 4.3.1: ln x and e x . … Magnify
§4.3(ii) Complex Arguments: Conformal Maps
See accompanying text
Figure 4.3.4: e x + i y . Magnify 3D Help
9: 4.12 Generalized Logarithms and Exponentials
A generalized exponential function ϕ ( x ) satisfies the equations …
4.12.6 ϕ ( x ) = ln ( x + 1 ) , - 1 < x < 0 ,
4.12.7 ϕ ( x ) = exp exp x  times ( x - x ) , x > 1 .
10: 17.3 q -Elementary and q -Special Functions
q -Exponential Functions
17.3.1 e q ( x ) = n = 0 ( 1 - q ) n x n ( q ; q ) n = 1 ( ( 1 - q ) x ; q ) ,
17.3.2 E q ( x ) = n = 0 ( 1 - q ) n q ( n 2 ) x n ( q ; q ) n = ( - ( 1 - q ) x ; q ) .
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 .
17.3.6 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 .