About the Project
NIST

parameter constraint

AdvancedHelp

(0.002 seconds)

1—10 of 19 matching pages

1: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …For further implications of the parameter constraints see the Note in §18.5(iii).
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
2: 18.5 Explicit Representations
For corresponding formulas for Chebyshev, Legendre, and the Hermite He n polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …
3: 18.25 Wilson Class: Definitions
Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials W n ( x ; a , b , c , d ) , continuous dual Hahn polynomials S n ( x ; a , b , c ) , Racah polynomials R n ( x ; α , β , γ , δ ) , and dual Hahn polynomials R n ( x ; γ , δ , N ) .
Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
p n ( x ) x = λ ( y ) Orthogonality range for y Constraints
4: 18.19 Hahn Class: Definitions
Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, normalizations, and parameter constraints.
p n ( x ) X w x h n
5: 20.10 Integrals
§20.10(i) Mellin Transforms with respect to the Lattice Parameter
20.10.1 0 x s - 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 - 2 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
20.10.2 0 x s - 1 ( θ 3 ( 0 | i x 2 ) - 1 ) d x = π - s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
Then …
6: 11.11 Asymptotic Expansions of Anger–Weber Functions
11.11.8 A ν ( λ ν ) 1 π k = 0 ( 2 k ) ! a k ( λ ) ν 2 k + 1 , ν , | ph ν | π - δ ,
11.11.10 A - ν ( λ ν ) - 1 π k = 0 ( 2 k ) ! a k ( - λ ) ν 2 k + 1 , ν , | ph ν | π - δ .
11.11.11 A - ν ( λ ν ) ( 2 π ν ) 1 / 2 e - ν μ k = 0 ( 1 2 ) k b k ( λ ) ν k , ν , | ph ν | π 2 - δ ,
11.11.14 A - ν ( λ ν ) 1 π ν ( λ - 1 ) , λ > 1 , | ph ν | π - δ ,
11.11.15 A - ν ( λ ν ) ( 2 π ν ) 1 / 2 ( 1 + 1 - λ 2 λ ) ν e - ν 1 - λ 2 ( 1 - λ 2 ) 1 / 4 , 0 < λ < 1 , | ph ν | π 2 - δ .
7: 25.14 Lerch’s Transcendent
25.14.1 Φ ( z , s , a ) n = 0 z n ( a + n ) s , | z | < 1 ; s > 1 , | z | = 1 .
25.14.6 Φ ( z , s , a ) = 1 2 a - s + 0 z x ( a + x ) s d x - 2 0 sin ( x ln z - s arctan ( x / a ) ) ( a 2 + x 2 ) s / 2 ( e 2 π x - 1 ) d x , a > 0 if | z | < 1 ; s > 1 , a > 0 if | z | = 1 .
8: 15.6 Integral Representations
15.6.1 F ( a , b ; c ; z ) = 1 Γ ( b ) Γ ( c - b ) 0 1 t b - 1 ( 1 - t ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; c > b > 0 .
15.6.2 F ( a , b ; c ; z ) = Γ ( 1 + b - c ) 2 π i Γ ( b ) 0 ( 1 + ) t b - 1 ( t - 1 ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; c - b 1 , 2 , 3 , , b > 0 .
15.6.6 F ( a , b ; c ; z ) = 1 2 π i Γ ( a ) Γ ( b ) - i i Γ ( a + t ) Γ ( b + t ) Γ ( - t ) Γ ( c + t ) ( - z ) t d t , | ph ( - z ) | < π ; a , b 0 , - 1 , - 2 , .
15.6.7 F ( a , b ; c ; z ) = 1 2 π i Γ ( a ) Γ ( b ) Γ ( c - a ) Γ ( c - b ) - i i Γ ( a + t ) Γ ( b + t ) Γ ( c - a - b - t ) Γ ( - t ) ( 1 - z ) t d t , | ph ( 1 - z ) | < π ; a , b , c - a , c - b 0 , - 1 , - 2 , .
15.6.8 F ( a , b ; c ; z ) = 1 Γ ( c - d ) 0 1 F ( a , b ; d ; z t ) t d - 1 ( 1 - t ) c - d - 1 d t , | ph ( 1 - z ) | < π ; c > d > 0 .
9: 33.14 Definitions and Basic Properties
33.14.13 0 s ( ϵ 1 , ; r ) s ( ϵ 2 , ; r ) d r = δ ( ϵ 1 - ϵ 2 ) , ϵ 1 , ϵ 2 > 0 ,
10: 10.32 Integral Representations
10.32.13 K ν ( z ) = ( 1 2 z ) ν 4 π i c - i c + i Γ ( t ) Γ ( t - ν ) ( 1 2 z ) - 2 t d t , c > max ( ν , 0 ) , | ph z | < 1 2 π .