integrals with respect to degree
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1: 14.20 Conical (or Mehler) Functions
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§14.20(x) Zeros and Integrals
…2: 7.23 Tables
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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
3: Bibliography C
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Derivatives with respect to the degree and order of associated Legendre functions for using modified Bessel functions.
Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.
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4: 18.33 Polynomials Orthogonal on the Unit Circle
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►A system of polynomials , , where is of proper degree
, is orthonormal on the unit circle with respect
to the weight function
() if
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►Let and , , be OP’s with weight functions and , respectively, on .
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►See Askey (1982a) and Pastro (1985) for special cases extending (18.33.13)–(18.33.14) and (18.33.15)–(18.33.16), respectively.
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►A system of monic polynomials , , where is of proper degree
, is orthogonal on the unit circle with respect
to the measure
if
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►with complex coefficients and of a certain degree
define the reversed polynomial
by
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5: 18.2 General Orthogonal Polynomials
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►A system (or set) of polynomials , , where has degree
as in §18.1(i), is said to be orthogonal on
with respect to the weight function
() if
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►For OP’s on with respect to an even weight function we have
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►As a slight variant let be OP’s with respect to an even weight function on .
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►The monic OP’s with respect to the measure can be expressed in terms of the moments by
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Degree lowering and raising differentiation formulas and structure relations
…6: 12.10 Uniform Asymptotic Expansions for Large Parameter
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►In this section we give asymptotic expansions of PCFs for large values of the parameter that are uniform with respect to the variable , when both and
are real.
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►where and are polynomials in of degree
, ( odd), ( even, ).
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►and the coefficients are the product of and a polynomial in of degree
.
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►The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when uniformly with respect to
.
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►where are given by (12.10.7), (12.10.23), respectively, and
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7: 8.20 Asymptotic Expansions of
§8.20 Asymptotic Expansions of
►§8.20(i) Large
… ►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
… ►so that is a polynomial in of degree when . …8: 3.5 Quadrature
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►Let denote the set of monic polynomials of degree
(coefficient of equal to
) that are orthogonal with respect to a positive weight function on a finite or infinite interval ; compare §18.2(i).
…As a consequence, the rule is exact for any polynomial of degree
, that is,
…In particular, with , we have a finite system of orthogonal polynomials () on with respect to the weights :
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►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.
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