About the Project
NIST

terminating

AdvancedHelp

(0.002 seconds)

1—10 of 30 matching pages

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 ) . … In this context the F -functions are called terminants, a name introduced by Dingle (1973). …
3: 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 …
4: 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 ) .
5: 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. …
6: 10.17 Asymptotic Expansions for Large Argument
If these expansions are terminated when k = - 1 , then the remainder term is bounded in absolute value by the first neglected term, provided that max ( ν - 1 2 , 1 ) . …
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 ) ) ,
7: 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: …
8: 4.6 Power Series
If a = 0 , 1 , 2 , , then the series terminates and z is unrestricted. …
9: 16.4 Argument Unity
See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … when ( 2 c - a - b ) > - 1 , or when the series terminates with a = - n . … when ( b + c + d - a ) < 1 , or when the series terminates with d = - n . … when the series on the right terminates and the series on the left converges. When the series on the right does not terminate, a second term appears. …
10: 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). …