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1: 3.5 Quadrature
Gauss Formula for a Logarithmic Weight Function
Table 3.5.14: Nodes and weights for the 5-point Gauss formula for the logarithmic weight function.
x k w k
Table 3.5.15: Nodes and weights for the 10-point Gauss formula for the logarithmic weight function.
x k w k
Table 3.5.16: Nodes and weights for the 15-point Gauss formula for the logarithmic weight function.
x k w k
Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.
x k w k
2: 18.19 Hahn Class: Definitions
18.19.7 w ( λ ) ( z ; ϕ ) = Γ ( λ + i z ) Γ ( λ - i z ) e ( 2 ϕ - π ) z ,
18.19.8 w ( x ) = w ( λ ) ( x ; ϕ ) = | Γ ( λ + i x ) | 2 e ( 2 ϕ - π ) x , λ > 0 , 0 < ϕ < π ,
3: 18.35 Pollaczek Polynomials
18.35.6 w ( λ ) ( cos θ ; a , b ) = π - 1 2 2 λ - 1 e ( 2 θ - π ) τ a , b ( θ ) ( sin θ ) 2 λ - 1 | Γ ( λ + i τ a , b ( θ ) ) | 2 , a b - a , λ > - 1 2 , 0 < θ < π .
4: 18.34 Bessel Polynomials
§18.34(i) Definitions and Recurrence Relation
For the confluent hypergeometric function F 1 1 and the generalized hypergeometric function F 0 2 see §16.2(ii) and §16.2(iv). … Because the coefficients C n in (18.34.4) are not all positive, the polynomials y n ( x ; a ) cannot be orthogonal on the line with respect to a positive weight function. … Orthogonality can also be expressed in terms of moment functionals; see Durán (1993), Evans et al. (1993), and Maroni (1995). … For uniform asymptotic expansions of y n ( x ; a ) as n in terms of Airy functions9.2) see Wong and Zhang (1997) and Dunster (2001c). …
5: 18.28 Askey–Wilson Class
18.28.3 2 π sin θ w ( cos θ ) = | ( e 2 i θ ; q ) ( a e i θ , b e i θ , c e i θ , d e i θ ; q ) | 2 ,
6: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
18.3.2 x N + 1 , n = cos ( ( n - 1 2 ) π / ( N + 1 ) ) .
Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions14.7(i)). …
7: 1.7 Inequalities
1.7.8 min ( a 1 , a 2 , , a n ) M ( r ) max ( a 1 , a 2 , , a n ) ,
1.7.9 M ( r ) M ( s ) , r < s ,
For f integrable on [ 0 , 1 ] , a < f ( x ) < b , and ϕ convex on ( a , b ) 1.4(viii)), …
1.7.11 exp ( 0 1 ln ( f ( x ) ) d x ) < 0 1 f ( x ) d x .
For exp and ln see §4.2.
8: 18.38 Mathematical Applications
In consequence, expansions of functions that are infinitely differentiable on [ - 1 , 1 ] in series of Chebyshev polynomials usually converge extremely rapidly. … If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the n th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding 2 n - 1 . … While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. …
Complex Function Theory
Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. …
9: 3.11 Approximation Techniques
§3.11(iii) Minimax Rational Approximations
Then the minimax (or best uniform) rational approximation … w ( x ) being a given positive weight function, and again J n + 1 . Then (3.11.29) is replaced by …
10: 31.9 Orthogonality
§31.9(i) Single Orthogonality
The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. … For corresponding orthogonality relations for Heun functions31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64).
§31.9(ii) Double Orthogonality
31.9.6 ρ ( s , t ) = ( s - t ) ( s t ) γ - 1 ( ( s - 1 ) ( t - 1 ) ) δ - 1 ( ( s - a ) ( t - a ) ) ϵ - 1 ,