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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: 8.5 Confluent Hypergeometric Representations
8.5.1 γ ( a , z ) = a 1 z a e z M ( 1 , 1 + a , z ) = a 1 z a M ( a , 1 + a , z ) , a 0 , 1 , 2 , .
8.5.2 γ ( a , z ) = e z 𝐌 ( 1 , 1 + a , z ) = 𝐌 ( a , 1 + a , z ) .
8.5.3 Γ ( a , z ) = e z U ( 1 a , 1 a , z ) = z a e z U ( 1 , 1 + a , z ) .
8.5.4 γ ( a , z ) = a 1 z 1 2 a 1 2 e 1 2 z M 1 2 a 1 2 , 1 2 a ( z ) .
8.5.5 Γ ( a , z ) = e 1 2 z z 1 2 a 1 2 W 1 2 a 1 2 , 1 2 a ( z ) .
10: 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 .