relation to inverse phase functions
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1: 10.21 Zeros
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►The functions
and are related to the inverses of the phase functions
and defined in §10.18(i): if , then
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2: 9.8 Modulus and Phase
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§9.8(i) Definitions
… ►(These definitions of and differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).) … ►§9.8(ii) Identities
… ►§9.8(iii) Monotonicity
… ►3: Errata
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Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function
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Paragraph Inversion Formula (in §35.2)
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Equation (9.5.6)
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Chapter 25 Zeta and Related Functions
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Figure 4.3.1
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The wording was changed to make the integration variable more apparent.
The validity constraint was added. Additionally, specific source citations are now given in the metadata for all equations in Chapter 9 Airy and Related Functions.
A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.
This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.
Reported 2015-11-12 by James W. Pitman.
4: 32.11 Asymptotic Approximations for Real Variables
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►Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to
, for some nonzero real , where denotes the Airy function (§9.2).
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►where is the gamma function (§5.2(i)), and the branch of the
function is immaterial.
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►The connection formulas relating (32.11.25) and (32.11.26) are
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►Now suppose .
…and the branch of the
function is immaterial.
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5: 10.68 Modulus and Phase Functions
§10.68 Modulus and Phase Functions
►§10.68(i) Definitions
… ►where , , , and are continuous real functions of and , with the branches of and chosen to satisfy (10.68.18) and (10.68.21) as . … ►§10.68(ii) Basic Properties
… ►However, care needs to be exercised with the branches of the phases. …6: 15.9 Relations to Other Functions
§15.9 Relations to Other Functions
►§15.9(i) Orthogonal Polynomials
… ►Jacobi
… ►Legendre
… ►Meixner
…7: 36.7 Zeros
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►The zeros in Table 36.7.1 are points in the plane, where is undetermined.
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►The zeros are lines in space where is undetermined.
…Near , and for small and , the modulus has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose and repeat distances are given by
…Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral.
There are also three sets of zero lines in the plane
related by rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates is given by
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8: 25.14 Lerch’s Transcendent
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►If is not an integer then ; if is a positive integer then ; if is a non-positive integer then can be any complex number.
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►The Hurwitz zeta function
(§25.11) and the polylogarithm (§25.12(ii)) are special cases:
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25.14.2
, ,
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25.14.5
, if ;
, if .
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25.14.6
if ;
, if .
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9: 13.8 Asymptotic Approximations for Large Parameters
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►as in , where and
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►For the parabolic cylinder function
see §12.2, and for an extension to an asymptotic expansion see Temme (1978).
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►where , and .
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►For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i).
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►When in and and fixed,
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10: 3.5 Quadrature
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►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3.
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►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi).
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►are related to Bessel polynomials (§§10.49(ii) and 18.34).
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