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1: 5.21 Methods of Computation
An effective way of computing Γ ( z ) in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). …
2: 8.24 Physical Applications
With more general values of p , E p ( x ) supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).
3: 8.21 Generalized Sine and Cosine Integrals
§8.21(i) Definitions: General Values
With γ and Γ denoting here the general values of the incomplete gamma functions (§8.2(i)), we define …
§8.21(ii) Definitions: Principal Values
When ph z = 0 (and when a - 1 , - 3 , - 5 , , in the case of Si ( a , z ) , or a 0 , - 2 , - 4 , , in the case of Ci ( a , z ) ) the principal values of si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values8.2(i)). …
§8.21(v) Special Values
4: 25.18 Methods of Computation
The principal tools for computing ζ ( s ) are the expansion (25.2.9) for general values of s , and the Riemann–Siegel formula (25.10.3) (extended to higher terms) for ζ ( 1 2 + i t ) . …
5: 6.4 Analytic Continuation
The general value of E 1 ( z ) is given by … The general values of the other functions are defined in a similar manner, and …
6: 4.2 Definitions
§4.2(i) The Logarithm
The general value of the phase is given by …
Powers with General Bases
but the general value of e z is … If z a has its general value, with a 0 , and if w 0 , then …
7: 8.2 Definitions and Basic Properties
The general values of the incomplete gamma functions γ ( a , z ) and Γ ( a , z ) are defined by … In this subsection the functions γ and Γ have their general values. …
8: 4.37 Inverse Hyperbolic Functions
§4.37(i) General Definitions
The general values of the inverse hyperbolic functions are defined by …
9: 4.23 Inverse Trigonometric Functions
§4.23(i) General Definitions
The general values of the inverse trigonometric functions are defined by …
4.23.34 arcsin z = arcsin β + i sign ( y ) ln ( α + ( α 2 - 1 ) 1 / 2 ) ,
4.23.35 arccos z = arccos β - i sign ( y ) ln ( α + ( α 2 - 1 ) 1 / 2 ) ,
4.23.36 arctan z = 1 2 arctan ( 2 x 1 - x 2 - y 2 ) + 1 4 i ln ( x 2 + ( y + 1 ) 2 x 2 + ( y - 1 ) 2 ) ,
10: 8.19 Generalized Exponential Integral
8.19.1 E p ( z ) = z p - 1 Γ ( 1 - p , z ) .
8.19.2 E p ( z ) = z p - 1 z e - t t p d t .
§8.19(iii) Special Values
8.19.6 E p ( 0 ) = 1 p - 1 , p > 1 ,
8.19.13 d d z E p ( z ) = - E p - 1 ( z ) ,