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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: 18.2 General Orthogonal Polynomials
    A system (or set) of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , is said to be orthogonal on ( a , b ) with respect to the weight function w ( x ) ( 0 ) ifIt is assumed throughout this chapter that for each polynomial p n ( x ) that is orthogonal on an open interval ( a , b ) the variable x is confined to the closure of ( a , b ) unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … Then a system of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , is said to be orthogonal on X with respect to the weights w x if … The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials p n ( x ) uniquely up to constant factors, which may be fixed by suitable normalization. … Conversely, if a system of polynomials { p n ( x ) } satisfies (18.2.10) with a n - 1 c n > 0 ( n 1 ), then { p n ( x ) } is orthogonal with respect to some positive measure on (Favard’s theorem). …
    5: 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 …
    6: 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 ) . …
    Szegő–Askey
    See Askey (1982) and Pastro (1985) for special cases extending (18.33.13)–(18.33.14) and (18.33.15)–(18.33.16), respectively. …
    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 polynomials of degree 2 n - 1 . …
    §3.5(vii) Oscillatory Integrals
    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 .