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1: 27.9 Quadratic Characters
27.9.2 ( 2 | p ) = ( - 1 ) ( p 2 - 1 ) / 8 .
If p , q are distinct odd primes, then the quadratic reciprocity law states that … Both (27.9.1) and (27.9.2) are valid with p replaced by P ; the reciprocity law (27.9.3) holds if p , q are replaced by any two relatively prime odd integers P , Q .
2: 5.3 Graphics
See accompanying text
Figure 5.3.1: Γ ( x ) and 1 / Γ ( x ) . … Magnify
See accompanying text
Figure 5.3.5: 1 / | Γ ( x + i y ) | . Magnify 3D Help
3: 8.23 Statistical Applications
In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of Q ( a , x ) ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …
4: 5.2 Definitions
5.2.1 Γ ( z ) = 0 e - t t z - 1 d t , z > 0 .
5: 27.14 Unrestricted Partitions
Euler introduced the reciprocal of the infinite product
27.14.2 f ( x ) = m = 1 ( 1 - x m ) = ( x ; x ) , | x | < 1 ,
27.14.3 1 f ( x ) = n = 0 p ( n ) x n ,
27.14.4 f ( x ) = 1 - x - x 2 + x 5 + x 7 - x 12 - x 15 + = 1 + k = 1 ( - 1 ) k ( x ω ( k ) + x ω ( - k ) ) ,
27.14.15 5 ( f ( x 5 ) ) 5 ( f ( x ) ) 6 = n = 0 p ( 5 n + 4 ) x n
6: 5.7 Series Expansions
§5.7(i) Maclaurin and Taylor Series
7: 5.8 Infinite Products
5.8.3 | Γ ( x ) Γ ( x + i y ) | 2 = k = 0 ( 1 + y 2 ( x + k ) 2 ) , x 0 , - 1 , .
8: 5.23 Approximations
Clenshaw (1962) also gives 20D Chebyshev-series coefficients for Γ ( 1 + x ) and its reciprocal for 0 x 1 . …
9: 19.31 Probability Distributions
R G ( x , y , z ) and R F ( x , y , z ) occur as the expectation values, relative to a normal probability distribution in 2 or 3 , of the square root or reciprocal square root of a quadratic form. …
10: 27.16 Cryptography
To do this, let s denote the reciprocal of r modulo ϕ ( n ) , so that r s = 1 + t ϕ ( n ) for some integer t . …