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integrals with respect to degree

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1: 14.20 Conical (or Mehler) Functions
§14.20(x) Zeros and Integrals
2: 7.23 Tables
  • Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of erf z , x [ 0 , 5 ] , y = 0.5 ( .5 ) 3 , 7D and 8D, respectively; the real and imaginary parts of x e ± i t 2 d t , ( 1 / π ) e i ( x 2 + ( π / 4 ) ) x e ± i t 2 d t , x = 0 ( .5 ) 20 ( 1 ) 25 , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

  • 3: Bibliography C
  • H. S. Cohl (2010) Derivatives with respect to the degree and order of associated Legendre functions for | z | > 1 using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.
  • 4: 18.33 Polynomials Orthogonal on the Unit Circle
    A system of polynomials { ϕ n ( z ) } , n = 0 , 1 , , where ϕ n ( z ) is of proper degree n , is orthonormal on the unit circle with respect to the weight function w ( z ) ( 0 ) if … Let { p n ( x ) } and { q n ( x ) } , n = 0 , 1 , , be OP’s with weight functions w 1 ( x ) and w 2 ( x ) , respectively, on ( 1 , 1 ) . … See Askey (1982a) and Pastro (1985) for special cases extending (18.33.13)–(18.33.14) and (18.33.15)–(18.33.16), respectively. … A system of monic polynomials { Φ n ( z ) } , n = 0 , 1 , , where Φ n ( x ) is of proper degree n , is orthogonal on the unit circle with respect to the measure μ if … with complex coefficients c k and of a certain degree n define the reversed polynomial p ( z ) by …
    5: 18.2 General Orthogonal Polynomials
    A system (or set) of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , where p n ( x ) has degree n as in §18.1(i), is said to be orthogonal on ( a , b ) with respect to the weight function w ( x ) ( 0 ) ifFor OP’s { p n ( x ) } on with respect to an even weight function w ( x ) we have … As a slight variant let { p n ( x ) } be OP’s with respect to an even weight function w ( x ) on ( 1 , 1 ) . … The monic OP’s p n ( x ) with respect to the measure d μ ( x ) can be expressed in terms of the moments by …
    Degree lowering and raising differentiation formulas and structure relations
    6: 12.10 Uniform Asymptotic Expansions for Large Parameter
    In this section we give asymptotic expansions of PCFs for large values of the parameter a that are uniform with respect to the variable z , when both a and z ( = x ) are real. … where u s ( t ) and v s ( t ) are polynomials in t of degree 3 s , ( s odd), 3 s 2 ( s even, s 2 ). … and the coefficients 𝖠 s ( τ ) are the product of τ s and a polynomial in τ of degree 2 s . … The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when μ uniformly with respect to t [ 1 + δ , ) . … where ξ , η are given by (12.10.7), (12.10.23), respectively, and …
    7: 8.20 Asymptotic Expansions of E p ( z )
    §8.20 Asymptotic Expansions of E p ( z )
    §8.20(i) Large z
    Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). …
    §8.20(ii) Large p
    so that A k ( λ ) is a polynomial in λ of degree k 1 when k 1 . …
    8: 3.5 Quadrature
    Let { p n } denote the set of monic polynomials p n of degree n (coefficient of x n equal to 1 ) that are orthogonal with respect to a positive weight function w on a finite or infinite interval ( a , b ) ; compare §18.2(i). …As a consequence, the rule is exact for any polynomial f ( x ) of degree 2 n 1 , that is, …In particular, with h m = a b p m ( x ) 2 w ( x ) d x , we have a finite system of orthogonal polynomials p m ( x ) ( m = 0 , 1 , , n 1 ) on { x 1 , x 2 , , x n } with respect to the weights w k : … The standard Monte Carlo method samples points uniformly from the integration region to estimate the integral and its error. In more advanced methods points are sampled from a probability distribution, so that they are concentrated in regions that make the largest contribution to the integral. …
    9: 14.6 Integer Order
    10: 30.11 Radial Spheroidal Wave Functions
    For fixed γ , as z in the sector | ph z | π δ ( < π ), …
    30.11.10 K n m ( γ ) = π 2 ( γ 2 ) m ( 1 ) m a n , 1 2 ( m n ) m ( γ 2 ) Γ ( 3 2 + m ) A n m ( γ 2 ) 𝖯𝗌 n m ( 0 , γ 2 ) , n m even,
    30.11.11 K n m ( γ ) = π 2 ( γ 2 ) m + 1 ( 1 ) m a n , 1 2 ( m n + 1 ) m ( γ 2 ) Γ ( 5 2 + m ) A n m ( γ 2 ) ( d 𝖯𝗌 n m ( z , γ 2 ) / d z | z = 0 ) , n m odd.
    §30.11(vi) Integral Representations
    30.11.12 A n m ( γ 2 ) S n m ( 1 ) ( z , γ ) = 1 2 i m + n γ m ( n m ) ! ( n + m ) ! z m ( 1 z 2 ) 1 2 m 1 1 e i γ z t ( 1 t 2 ) 1 2 m 𝖯𝗌 n m ( t , γ 2 ) d t .