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1: 8.22 Mathematical Applications
§8.22(i) Terminant Function
The so-called terminant function F p ( z ) , defined by
8.22.1 F p ( z ) = Γ ( p ) 2 π z 1 - p E p ( z ) = Γ ( p ) 2 π Γ ( 1 - p , z ) ,
2: 2.11 Remainder Terms; Stokes Phenomenon
2.11.10 E p ( z ) = e - z z s = 0 n - 1 ( - 1 ) s ( p ) s z s + ( - 1 ) n 2 π Γ ( p ) z p - 1 F n + p ( z ) ,
2.11.11 F n + p ( z ) = e - z 2 π 0 e - z t t n + p - 1 1 + t d t = Γ ( n + p ) 2 π E n + p ( z ) z n + p - 1 .
Owing to the factor e - ρ , that is, e - | z | in (2.11.13), F n + p ( z ) is uniformly exponentially small compared with E p ( z ) . …
3: 9.7 Asymptotic Expansions
9.7.20 R n ( z ) = ( - 1 ) n k = 0 m - 1 ( - 1 ) k u k G n - k ( 2 ζ ) ζ k + R m , n ( z ) ,
9.7.21 S n ( z ) = ( - 1 ) n - 1 k = 0 m - 1 ( - 1 ) k v k G n - k ( 2 ζ ) ζ k + S m , n ( z ) ,
9.7.22 G p ( z ) = e z 2 π Γ ( p ) Γ ( 1 - p , z ) .
4: 10.17 Asymptotic Expansions for Large Argument
10.17.17 R ± ( ν , z ) = ( - 1 ) 2 cos ( ν π ) ( k = 0 m - 1 ( ± i ) k a k ( ν ) z k G - k ( 2 i z ) + R m , ± ( ν , z ) ) ,
5: 10.40 Asymptotic Expansions for Large Argument
10.40.13 R ( ν , z ) = ( - 1 ) 2 cos ( ν π ) ( k = 0 m - 1 a k ( ν ) z k G - k ( 2 z ) + R m , ( ν , z ) ) ,
where G p ( z ) is given by (10.17.16). …
6: 16.2 Definition and Analytic Properties
Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in z . … However, when one or more of the top parameters a j is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in z . Note that if - m is the value of the numerically largest a j that is a nonpositive integer, then the identity …
7: 34.6 Definition: 9 j Symbol
The 9 j symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …
8: 10.74 Methods of Computation
In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. …
9: 29.15 Fourier Series and Chebyshev Series
When ν = 2 n , m = 0 , 1 , , n , the Fourier series (29.6.1) terminates: … When ν = 2 n + 1 , m = 0 , 1 , , n , the Fourier series (29.6.16) terminates: … When ν = 2 n + 1 , m = 0 , 1 , , n , the Fourier series (29.6.31) terminates: … When ν = 2 n + 1 , m = 0 , 1 , , n , the Fourier series (29.6.8) terminates: … When ν = 2 n + 2 , m = 0 , 1 , , n , the Fourier series (29.6.46) terminates: …
10: 17.4 Basic Hypergeometric Functions
The series (17.4.1) is said to be balanced or Saalschützian when it terminates, r = s , z = q , and …